3.17.32 \(\int (A+B x) (d+e x)^{5/2} (a^2+2 a b x+b^2 x^2)^{5/2} \, dx\)
Optimal. Leaf size=452 \[ -\frac {20 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{13/2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{13 e^7 (a+b x)}+\frac {10 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{11 e^7 (a+b x)}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{9 e^7 (a+b x)}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^5 (B d-A e)}{7 e^7 (a+b x)}-\frac {2 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{17/2} (-5 a B e-A b e+6 b B d)}{17 e^7 (a+b x)}+\frac {2 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{15/2} (b d-a e) (-2 a B e-A b e+3 b B d)}{3 e^7 (a+b x)}+\frac {2 b^5 B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{19/2}}{19 e^7 (a+b x)} \]
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Rubi [A] time = 0.22, antiderivative size = 452, normalized size of antiderivative = 1.00,
number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used =
{770, 77} \begin {gather*} -\frac {2 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{17/2} (-5 a B e-A b e+6 b B d)}{17 e^7 (a+b x)}+\frac {2 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{15/2} (b d-a e) (-2 a B e-A b e+3 b B d)}{3 e^7 (a+b x)}-\frac {20 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{13/2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{13 e^7 (a+b x)}+\frac {10 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{11 e^7 (a+b x)}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{9 e^7 (a+b x)}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^5 (B d-A e)}{7 e^7 (a+b x)}+\frac {2 b^5 B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{19/2}}{19 e^7 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(A + B*x)*(d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]
[Out]
(2*(b*d - a*e)^5*(B*d - A*e)*(d + e*x)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^7*(a + b*x)) - (2*(b*d - a*e)
^4*(6*b*B*d - 5*A*b*e - a*B*e)*(d + e*x)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^7*(a + b*x)) + (10*b*(b*d -
a*e)^3*(3*b*B*d - 2*A*b*e - a*B*e)*(d + e*x)^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^7*(a + b*x)) - (20*b
^2*(b*d - a*e)^2*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^(13/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*e^7*(a + b*x))
+ (2*b^3*(b*d - a*e)*(3*b*B*d - A*b*e - 2*a*B*e)*(d + e*x)^(15/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^7*(a + b
*x)) - (2*b^4*(6*b*B*d - A*b*e - 5*a*B*e)*(d + e*x)^(17/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(17*e^7*(a + b*x)) +
(2*b^5*B*(d + e*x)^(19/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(19*e^7*(a + b*x))
Rule 77
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Rule 770
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dis
t[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(f + g*x)*(b/2 + c
*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]
Rubi steps
\begin {align*} \int (A+B x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (a b+b^2 x\right )^5 (A+B x) (d+e x)^{5/2} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (-\frac {b^5 (b d-a e)^5 (-B d+A e) (d+e x)^{5/2}}{e^6}+\frac {b^5 (b d-a e)^4 (-6 b B d+5 A b e+a B e) (d+e x)^{7/2}}{e^6}-\frac {5 b^6 (b d-a e)^3 (-3 b B d+2 A b e+a B e) (d+e x)^{9/2}}{e^6}+\frac {10 b^7 (b d-a e)^2 (-2 b B d+A b e+a B e) (d+e x)^{11/2}}{e^6}-\frac {5 b^8 (b d-a e) (-3 b B d+A b e+2 a B e) (d+e x)^{13/2}}{e^6}+\frac {b^9 (-6 b B d+A b e+5 a B e) (d+e x)^{15/2}}{e^6}+\frac {b^{10} B (d+e x)^{17/2}}{e^6}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {2 (b d-a e)^5 (B d-A e) (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x)}-\frac {2 (b d-a e)^4 (6 b B d-5 A b e-a B e) (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^7 (a+b x)}+\frac {10 b (b d-a e)^3 (3 b B d-2 A b e-a B e) (d+e x)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x)}-\frac {20 b^2 (b d-a e)^2 (2 b B d-A b e-a B e) (d+e x)^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^7 (a+b x)}+\frac {2 b^3 (b d-a e) (3 b B d-A b e-2 a B e) (d+e x)^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x)}-\frac {2 b^4 (6 b B d-A b e-5 a B e) (d+e x)^{17/2} \sqrt {a^2+2 a b x+b^2 x^2}}{17 e^7 (a+b x)}+\frac {2 b^5 B (d+e x)^{19/2} \sqrt {a^2+2 a b x+b^2 x^2}}{19 e^7 (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 239, normalized size = 0.53 \begin {gather*} \frac {2 \sqrt {(a+b x)^2} (d+e x)^{7/2} \left (-171171 b^4 (d+e x)^5 (-5 a B e-A b e+6 b B d)+969969 b^3 (d+e x)^4 (b d-a e) (-2 a B e-A b e+3 b B d)-2238390 b^2 (d+e x)^3 (b d-a e)^2 (-a B e-A b e+2 b B d)+1322685 b (d+e x)^2 (b d-a e)^3 (-a B e-2 A b e+3 b B d)-323323 (d+e x) (b d-a e)^4 (-a B e-5 A b e+6 b B d)+415701 (b d-a e)^5 (B d-A e)+153153 b^5 B (d+e x)^6\right )}{2909907 e^7 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(A + B*x)*(d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]
[Out]
(2*Sqrt[(a + b*x)^2]*(d + e*x)^(7/2)*(415701*(b*d - a*e)^5*(B*d - A*e) - 323323*(b*d - a*e)^4*(6*b*B*d - 5*A*b
*e - a*B*e)*(d + e*x) + 1322685*b*(b*d - a*e)^3*(3*b*B*d - 2*A*b*e - a*B*e)*(d + e*x)^2 - 2238390*b^2*(b*d - a
*e)^2*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^3 + 969969*b^3*(b*d - a*e)*(3*b*B*d - A*b*e - 2*a*B*e)*(d + e*x)^4 -
171171*b^4*(6*b*B*d - A*b*e - 5*a*B*e)*(d + e*x)^5 + 153153*b^5*B*(d + e*x)^6))/(2909907*e^7*(a + b*x))
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IntegrateAlgebraic [A] time = 55.03, size = 812, normalized size = 1.80 \begin {gather*} \frac {2 (d+e x)^{7/2} \sqrt {\frac {(a e+b x e)^2}{e^2}} \left (415701 b^5 B d^6-415701 A b^5 e d^5-2078505 a b^4 B e d^5-1939938 b^5 B (d+e x) d^5+2078505 a A b^4 e^2 d^4+4157010 a^2 b^3 B e^2 d^4+3968055 b^5 B (d+e x)^2 d^4+1616615 A b^5 e (d+e x) d^4+8083075 a b^4 B e (d+e x) d^4-4157010 a^2 A b^3 e^3 d^3-4157010 a^3 b^2 B e^3 d^3-4476780 b^5 B (d+e x)^3 d^3-2645370 A b^5 e (d+e x)^2 d^3-13226850 a b^4 B e (d+e x)^2 d^3-6466460 a A b^4 e^2 (d+e x) d^3-12932920 a^2 b^3 B e^2 (d+e x) d^3+4157010 a^3 A b^2 e^4 d^2+2078505 a^4 b B e^4 d^2+2909907 b^5 B (d+e x)^4 d^2+2238390 A b^5 e (d+e x)^3 d^2+11191950 a b^4 B e (d+e x)^3 d^2+7936110 a A b^4 e^2 (d+e x)^2 d^2+15872220 a^2 b^3 B e^2 (d+e x)^2 d^2+9699690 a^2 A b^3 e^3 (d+e x) d^2+9699690 a^3 b^2 B e^3 (d+e x) d^2-2078505 a^4 A b e^5 d-415701 a^5 B e^5 d-1027026 b^5 B (d+e x)^5 d-969969 A b^5 e (d+e x)^4 d-4849845 a b^4 B e (d+e x)^4 d-4476780 a A b^4 e^2 (d+e x)^3 d-8953560 a^2 b^3 B e^2 (d+e x)^3 d-7936110 a^2 A b^3 e^3 (d+e x)^2 d-7936110 a^3 b^2 B e^3 (d+e x)^2 d-6466460 a^3 A b^2 e^4 (d+e x) d-3233230 a^4 b B e^4 (d+e x) d+415701 a^5 A e^6+153153 b^5 B (d+e x)^6+171171 A b^5 e (d+e x)^5+855855 a b^4 B e (d+e x)^5+969969 a A b^4 e^2 (d+e x)^4+1939938 a^2 b^3 B e^2 (d+e x)^4+2238390 a^2 A b^3 e^3 (d+e x)^3+2238390 a^3 b^2 B e^3 (d+e x)^3+2645370 a^3 A b^2 e^4 (d+e x)^2+1322685 a^4 b B e^4 (d+e x)^2+1616615 a^4 A b e^5 (d+e x)+323323 a^5 B e^5 (d+e x)\right )}{2909907 e^6 (a e+b x e)} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(A + B*x)*(d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]
[Out]
(2*(d + e*x)^(7/2)*Sqrt[(a*e + b*e*x)^2/e^2]*(415701*b^5*B*d^6 - 415701*A*b^5*d^5*e - 2078505*a*b^4*B*d^5*e +
2078505*a*A*b^4*d^4*e^2 + 4157010*a^2*b^3*B*d^4*e^2 - 4157010*a^2*A*b^3*d^3*e^3 - 4157010*a^3*b^2*B*d^3*e^3 +
4157010*a^3*A*b^2*d^2*e^4 + 2078505*a^4*b*B*d^2*e^4 - 2078505*a^4*A*b*d*e^5 - 415701*a^5*B*d*e^5 + 415701*a^5*
A*e^6 - 1939938*b^5*B*d^5*(d + e*x) + 1616615*A*b^5*d^4*e*(d + e*x) + 8083075*a*b^4*B*d^4*e*(d + e*x) - 646646
0*a*A*b^4*d^3*e^2*(d + e*x) - 12932920*a^2*b^3*B*d^3*e^2*(d + e*x) + 9699690*a^2*A*b^3*d^2*e^3*(d + e*x) + 969
9690*a^3*b^2*B*d^2*e^3*(d + e*x) - 6466460*a^3*A*b^2*d*e^4*(d + e*x) - 3233230*a^4*b*B*d*e^4*(d + e*x) + 16166
15*a^4*A*b*e^5*(d + e*x) + 323323*a^5*B*e^5*(d + e*x) + 3968055*b^5*B*d^4*(d + e*x)^2 - 2645370*A*b^5*d^3*e*(d
+ e*x)^2 - 13226850*a*b^4*B*d^3*e*(d + e*x)^2 + 7936110*a*A*b^4*d^2*e^2*(d + e*x)^2 + 15872220*a^2*b^3*B*d^2*
e^2*(d + e*x)^2 - 7936110*a^2*A*b^3*d*e^3*(d + e*x)^2 - 7936110*a^3*b^2*B*d*e^3*(d + e*x)^2 + 2645370*a^3*A*b^
2*e^4*(d + e*x)^2 + 1322685*a^4*b*B*e^4*(d + e*x)^2 - 4476780*b^5*B*d^3*(d + e*x)^3 + 2238390*A*b^5*d^2*e*(d +
e*x)^3 + 11191950*a*b^4*B*d^2*e*(d + e*x)^3 - 4476780*a*A*b^4*d*e^2*(d + e*x)^3 - 8953560*a^2*b^3*B*d*e^2*(d
+ e*x)^3 + 2238390*a^2*A*b^3*e^3*(d + e*x)^3 + 2238390*a^3*b^2*B*e^3*(d + e*x)^3 + 2909907*b^5*B*d^2*(d + e*x)
^4 - 969969*A*b^5*d*e*(d + e*x)^4 - 4849845*a*b^4*B*d*e*(d + e*x)^4 + 969969*a*A*b^4*e^2*(d + e*x)^4 + 1939938
*a^2*b^3*B*e^2*(d + e*x)^4 - 1027026*b^5*B*d*(d + e*x)^5 + 171171*A*b^5*e*(d + e*x)^5 + 855855*a*b^4*B*e*(d +
e*x)^5 + 153153*b^5*B*(d + e*x)^6))/(2909907*e^6*(a*e + b*e*x))
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fricas [B] time = 0.45, size = 993, normalized size = 2.20 \begin {gather*} \frac {2 \, {\left (153153 \, B b^{5} e^{9} x^{9} + 3072 \, B b^{5} d^{9} + 415701 \, A a^{5} d^{3} e^{6} - 4864 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{8} e + 41344 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{7} e^{2} - 155040 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{6} e^{3} + 167960 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{5} e^{4} - 92378 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d^{4} e^{5} + 9009 \, {\left (39 \, B b^{5} d e^{8} + 19 \, {\left (5 \, B a b^{4} + A b^{5}\right )} e^{9}\right )} x^{8} + 3003 \, {\left (69 \, B b^{5} d^{2} e^{7} + 133 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d e^{8} + 323 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e^{9}\right )} x^{7} + 231 \, {\left (3 \, B b^{5} d^{3} e^{6} + 1045 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e^{7} + 10013 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{8} + 9690 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{9}\right )} x^{6} - 63 \, {\left (12 \, B b^{5} d^{4} e^{5} - 19 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} e^{6} - 22933 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{7} - 87210 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{8} - 20995 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e^{9}\right )} x^{5} + 7 \, {\left (120 \, B b^{5} d^{5} e^{4} - 190 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} e^{5} + 1615 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e^{6} + 513570 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e^{7} + 482885 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d e^{8} + 46189 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} e^{9}\right )} x^{4} - {\left (960 \, B b^{5} d^{6} e^{3} - 415701 \, A a^{5} e^{9} - 1520 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} e^{4} + 12920 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} e^{5} - 48450 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} e^{6} - 2372435 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} e^{7} - 877591 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d e^{8}\right )} x^{3} + 3 \, {\left (384 \, B b^{5} d^{7} e^{2} + 415701 \, A a^{5} d e^{8} - 608 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{6} e^{3} + 5168 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{5} e^{4} - 19380 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{4} e^{5} + 20995 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{3} e^{6} + 230945 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d^{2} e^{7}\right )} x^{2} - {\left (1536 \, B b^{5} d^{8} e - 1247103 \, A a^{5} d^{2} e^{7} - 2432 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{7} e^{2} + 20672 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{6} e^{3} - 77520 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{5} e^{4} + 83980 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{4} e^{5} - 46189 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d^{3} e^{6}\right )} x\right )} \sqrt {e x + d}}{2909907 \, e^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(e*x+d)^(5/2)*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")
[Out]
2/2909907*(153153*B*b^5*e^9*x^9 + 3072*B*b^5*d^9 + 415701*A*a^5*d^3*e^6 - 4864*(5*B*a*b^4 + A*b^5)*d^8*e + 413
44*(2*B*a^2*b^3 + A*a*b^4)*d^7*e^2 - 155040*(B*a^3*b^2 + A*a^2*b^3)*d^6*e^3 + 167960*(B*a^4*b + 2*A*a^3*b^2)*d
^5*e^4 - 92378*(B*a^5 + 5*A*a^4*b)*d^4*e^5 + 9009*(39*B*b^5*d*e^8 + 19*(5*B*a*b^4 + A*b^5)*e^9)*x^8 + 3003*(69
*B*b^5*d^2*e^7 + 133*(5*B*a*b^4 + A*b^5)*d*e^8 + 323*(2*B*a^2*b^3 + A*a*b^4)*e^9)*x^7 + 231*(3*B*b^5*d^3*e^6 +
1045*(5*B*a*b^4 + A*b^5)*d^2*e^7 + 10013*(2*B*a^2*b^3 + A*a*b^4)*d*e^8 + 9690*(B*a^3*b^2 + A*a^2*b^3)*e^9)*x^
6 - 63*(12*B*b^5*d^4*e^5 - 19*(5*B*a*b^4 + A*b^5)*d^3*e^6 - 22933*(2*B*a^2*b^3 + A*a*b^4)*d^2*e^7 - 87210*(B*a
^3*b^2 + A*a^2*b^3)*d*e^8 - 20995*(B*a^4*b + 2*A*a^3*b^2)*e^9)*x^5 + 7*(120*B*b^5*d^5*e^4 - 190*(5*B*a*b^4 + A
*b^5)*d^4*e^5 + 1615*(2*B*a^2*b^3 + A*a*b^4)*d^3*e^6 + 513570*(B*a^3*b^2 + A*a^2*b^3)*d^2*e^7 + 482885*(B*a^4*
b + 2*A*a^3*b^2)*d*e^8 + 46189*(B*a^5 + 5*A*a^4*b)*e^9)*x^4 - (960*B*b^5*d^6*e^3 - 415701*A*a^5*e^9 - 1520*(5*
B*a*b^4 + A*b^5)*d^5*e^4 + 12920*(2*B*a^2*b^3 + A*a*b^4)*d^4*e^5 - 48450*(B*a^3*b^2 + A*a^2*b^3)*d^3*e^6 - 237
2435*(B*a^4*b + 2*A*a^3*b^2)*d^2*e^7 - 877591*(B*a^5 + 5*A*a^4*b)*d*e^8)*x^3 + 3*(384*B*b^5*d^7*e^2 + 415701*A
*a^5*d*e^8 - 608*(5*B*a*b^4 + A*b^5)*d^6*e^3 + 5168*(2*B*a^2*b^3 + A*a*b^4)*d^5*e^4 - 19380*(B*a^3*b^2 + A*a^2
*b^3)*d^4*e^5 + 20995*(B*a^4*b + 2*A*a^3*b^2)*d^3*e^6 + 230945*(B*a^5 + 5*A*a^4*b)*d^2*e^7)*x^2 - (1536*B*b^5*
d^8*e - 1247103*A*a^5*d^2*e^7 - 2432*(5*B*a*b^4 + A*b^5)*d^7*e^2 + 20672*(2*B*a^2*b^3 + A*a*b^4)*d^6*e^3 - 775
20*(B*a^3*b^2 + A*a^2*b^3)*d^5*e^4 + 83980*(B*a^4*b + 2*A*a^3*b^2)*d^4*e^5 - 46189*(B*a^5 + 5*A*a^4*b)*d^3*e^6
)*x)*sqrt(e*x + d)/e^7
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giac [B] time = 0.62, size = 4050, normalized size = 8.96
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(e*x+d)^(5/2)*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")
[Out]
2/14549535*(4849845*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*B*a^5*d^3*e^(-1)*sgn(b*x + a) + 24249225*((x*e + d)^
(3/2) - 3*sqrt(x*e + d)*d)*A*a^4*b*d^3*e^(-1)*sgn(b*x + a) + 4849845*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d
+ 15*sqrt(x*e + d)*d^2)*B*a^4*b*d^3*e^(-2)*sgn(b*x + a) + 9699690*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d +
15*sqrt(x*e + d)*d^2)*A*a^3*b^2*d^3*e^(-2)*sgn(b*x + a) + 4157010*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d +
35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*B*a^3*b^2*d^3*e^(-3)*sgn(b*x + a) + 4157010*(5*(x*e + d)^(7/2)
- 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*A*a^2*b^3*d^3*e^(-3)*sgn(b*x + a) + 4
61890*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sq
rt(x*e + d)*d^4)*B*a^2*b^3*d^3*e^(-4)*sgn(b*x + a) + 230945*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*
(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*A*a*b^4*d^3*e^(-4)*sgn(b*x + a) + 10497
5*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*
e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*B*a*b^4*d^3*e^(-5)*sgn(b*x + a) + 20995*(63*(x*e + d)^(11/2) - 385*(
x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*
e + d)*d^5)*A*b^5*d^3*e^(-5)*sgn(b*x + a) + 4845*(231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e +
d)^(9/2)*d^2 - 8580*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e
+ d)*d^6)*B*b^5*d^3*e^(-6)*sgn(b*x + a) + 2909907*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d
)*d^2)*B*a^5*d^2*e^(-1)*sgn(b*x + a) + 14549535*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d
^2)*A*a^4*b*d^2*e^(-1)*sgn(b*x + a) + 6235515*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d
^2 - 35*sqrt(x*e + d)*d^3)*B*a^4*b*d^2*e^(-2)*sgn(b*x + a) + 12471030*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*
d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*A*a^3*b^2*d^2*e^(-2)*sgn(b*x + a) + 1385670*(35*(x*e + d)^(
9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*B*a^
3*b^2*d^2*e^(-3)*sgn(b*x + a) + 1385670*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2
- 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*A*a^2*b^3*d^2*e^(-3)*sgn(b*x + a) + 629850*(63*(x*e + d)^(1
1/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 -
693*sqrt(x*e + d)*d^5)*B*a^2*b^3*d^2*e^(-4)*sgn(b*x + a) + 314925*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*
d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*A*a
*b^4*d^2*e^(-4)*sgn(b*x + a) + 72675*(231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^
2 - 8580*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*B
*a*b^4*d^2*e^(-5)*sgn(b*x + a) + 14535*(231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*
d^2 - 8580*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)
*A*b^5*d^2*e^(-5)*sgn(b*x + a) + 6783*(429*(x*e + d)^(15/2) - 3465*(x*e + d)^(13/2)*d + 12285*(x*e + d)^(11/2)
*d^2 - 25025*(x*e + d)^(9/2)*d^3 + 32175*(x*e + d)^(7/2)*d^4 - 27027*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/
2)*d^6 - 6435*sqrt(x*e + d)*d^7)*B*b^5*d^2*e^(-6)*sgn(b*x + a) + 14549535*sqrt(x*e + d)*A*a^5*d^3*sgn(b*x + a)
+ 14549535*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*A*a^5*d^2*sgn(b*x + a) + 1247103*(5*(x*e + d)^(7/2) - 21*(x*
e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*B*a^5*d*e^(-1)*sgn(b*x + a) + 6235515*(5*(x*e
+ d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*A*a^4*b*d*e^(-1)*sgn(b*x +
a) + 692835*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 +
315*sqrt(x*e + d)*d^4)*B*a^4*b*d*e^(-2)*sgn(b*x + a) + 1385670*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 3
78*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*A*a^3*b^2*d*e^(-2)*sgn(b*x + a) + 62
9850*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*
(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*B*a^3*b^2*d*e^(-3)*sgn(b*x + a) + 629850*(63*(x*e + d)^(11/2) - 3
85*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqr
t(x*e + d)*d^5)*A*a^2*b^3*d*e^(-3)*sgn(b*x + a) + 145350*(231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 500
5*(x*e + d)^(9/2)*d^2 - 8580*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 3003*
sqrt(x*e + d)*d^6)*B*a^2*b^3*d*e^(-4)*sgn(b*x + a) + 72675*(231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 5
005*(x*e + d)^(9/2)*d^2 - 8580*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 300
3*sqrt(x*e + d)*d^6)*A*a*b^4*d*e^(-4)*sgn(b*x + a) + 33915*(429*(x*e + d)^(15/2) - 3465*(x*e + d)^(13/2)*d + 1
2285*(x*e + d)^(11/2)*d^2 - 25025*(x*e + d)^(9/2)*d^3 + 32175*(x*e + d)^(7/2)*d^4 - 27027*(x*e + d)^(5/2)*d^5
+ 15015*(x*e + d)^(3/2)*d^6 - 6435*sqrt(x*e + d)*d^7)*B*a*b^4*d*e^(-5)*sgn(b*x + a) + 6783*(429*(x*e + d)^(15/
2) - 3465*(x*e + d)^(13/2)*d + 12285*(x*e + d)^(11/2)*d^2 - 25025*(x*e + d)^(9/2)*d^3 + 32175*(x*e + d)^(7/2)*
d^4 - 27027*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6 - 6435*sqrt(x*e + d)*d^7)*A*b^5*d*e^(-5)*sgn(b*x +
a) + 399*(6435*(x*e + d)^(17/2) - 58344*(x*e + d)^(15/2)*d + 235620*(x*e + d)^(13/2)*d^2 - 556920*(x*e + d)^(
11/2)*d^3 + 850850*(x*e + d)^(9/2)*d^4 - 875160*(x*e + d)^(7/2)*d^5 + 612612*(x*e + d)^(5/2)*d^6 - 291720*(x*e
+ d)^(3/2)*d^7 + 109395*sqrt(x*e + d)*d^8)*B*b^5*d*e^(-6)*sgn(b*x + a) + 2909907*(3*(x*e + d)^(5/2) - 10*(x*e
+ d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*A*a^5*d*sgn(b*x + a) + 46189*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d
+ 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*B*a^5*e^(-1)*sgn(b*x + a) + 2309
45*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(
x*e + d)*d^4)*A*a^4*b*e^(-1)*sgn(b*x + a) + 104975*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d
)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*B*a^4*b*e^(-2)*sgn(
b*x + a) + 209950*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2
)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*A*a^3*b^2*e^(-2)*sgn(b*x + a) + 48450*(231*(x*e + d)
^(13/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 8580*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)
*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*B*a^3*b^2*e^(-3)*sgn(b*x + a) + 48450*(231*(x*e + d)
^(13/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 8580*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)
*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*A*a^2*b^3*e^(-3)*sgn(b*x + a) + 22610*(429*(x*e + d)
^(15/2) - 3465*(x*e + d)^(13/2)*d + 12285*(x*e + d)^(11/2)*d^2 - 25025*(x*e + d)^(9/2)*d^3 + 32175*(x*e + d)^(
7/2)*d^4 - 27027*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6 - 6435*sqrt(x*e + d)*d^7)*B*a^2*b^3*e^(-4)*sg
n(b*x + a) + 11305*(429*(x*e + d)^(15/2) - 3465*(x*e + d)^(13/2)*d + 12285*(x*e + d)^(11/2)*d^2 - 25025*(x*e +
d)^(9/2)*d^3 + 32175*(x*e + d)^(7/2)*d^4 - 27027*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6 - 6435*sqrt(
x*e + d)*d^7)*A*a*b^4*e^(-4)*sgn(b*x + a) + 665*(6435*(x*e + d)^(17/2) - 58344*(x*e + d)^(15/2)*d + 235620*(x*
e + d)^(13/2)*d^2 - 556920*(x*e + d)^(11/2)*d^3 + 850850*(x*e + d)^(9/2)*d^4 - 875160*(x*e + d)^(7/2)*d^5 + 61
2612*(x*e + d)^(5/2)*d^6 - 291720*(x*e + d)^(3/2)*d^7 + 109395*sqrt(x*e + d)*d^8)*B*a*b^4*e^(-5)*sgn(b*x + a)
+ 133*(6435*(x*e + d)^(17/2) - 58344*(x*e + d)^(15/2)*d + 235620*(x*e + d)^(13/2)*d^2 - 556920*(x*e + d)^(11/2
)*d^3 + 850850*(x*e + d)^(9/2)*d^4 - 875160*(x*e + d)^(7/2)*d^5 + 612612*(x*e + d)^(5/2)*d^6 - 291720*(x*e + d
)^(3/2)*d^7 + 109395*sqrt(x*e + d)*d^8)*A*b^5*e^(-5)*sgn(b*x + a) + 63*(12155*(x*e + d)^(19/2) - 122265*(x*e +
d)^(17/2)*d + 554268*(x*e + d)^(15/2)*d^2 - 1492260*(x*e + d)^(13/2)*d^3 + 2645370*(x*e + d)^(11/2)*d^4 - 323
3230*(x*e + d)^(9/2)*d^5 + 2771340*(x*e + d)^(7/2)*d^6 - 1662804*(x*e + d)^(5/2)*d^7 + 692835*(x*e + d)^(3/2)*
d^8 - 230945*sqrt(x*e + d)*d^9)*B*b^5*e^(-6)*sgn(b*x + a) + 415701*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d +
35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*A*a^5*sgn(b*x + a))*e^(-1)
________________________________________________________________________________________
maple [A] time = 0.05, size = 689, normalized size = 1.52 \begin {gather*} \frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (153153 B \,b^{5} e^{6} x^{6}+171171 A \,b^{5} e^{6} x^{5}+855855 B a \,b^{4} e^{6} x^{5}-108108 B \,b^{5} d \,e^{5} x^{5}+969969 A a \,b^{4} e^{6} x^{4}-114114 A \,b^{5} d \,e^{5} x^{4}+1939938 B \,a^{2} b^{3} e^{6} x^{4}-570570 B a \,b^{4} d \,e^{5} x^{4}+72072 B \,b^{5} d^{2} e^{4} x^{4}+2238390 A \,a^{2} b^{3} e^{6} x^{3}-596904 A a \,b^{4} d \,e^{5} x^{3}+70224 A \,b^{5} d^{2} e^{4} x^{3}+2238390 B \,a^{3} b^{2} e^{6} x^{3}-1193808 B \,a^{2} b^{3} d \,e^{5} x^{3}+351120 B a \,b^{4} d^{2} e^{4} x^{3}-44352 B \,b^{5} d^{3} e^{3} x^{3}+2645370 A \,a^{3} b^{2} e^{6} x^{2}-1220940 A \,a^{2} b^{3} d \,e^{5} x^{2}+325584 A a \,b^{4} d^{2} e^{4} x^{2}-38304 A \,b^{5} d^{3} e^{3} x^{2}+1322685 B \,a^{4} b \,e^{6} x^{2}-1220940 B \,a^{3} b^{2} d \,e^{5} x^{2}+651168 B \,a^{2} b^{3} d^{2} e^{4} x^{2}-191520 B a \,b^{4} d^{3} e^{3} x^{2}+24192 B \,b^{5} d^{4} e^{2} x^{2}+1616615 A \,a^{4} b \,e^{6} x -1175720 A \,a^{3} b^{2} d \,e^{5} x +542640 A \,a^{2} b^{3} d^{2} e^{4} x -144704 A a \,b^{4} d^{3} e^{3} x +17024 A \,b^{5} d^{4} e^{2} x +323323 B \,a^{5} e^{6} x -587860 B \,a^{4} b d \,e^{5} x +542640 B \,a^{3} b^{2} d^{2} e^{4} x -289408 B \,a^{2} b^{3} d^{3} e^{3} x +85120 B a \,b^{4} d^{4} e^{2} x -10752 B \,b^{5} d^{5} e x +415701 A \,a^{5} e^{6}-461890 A \,a^{4} b d \,e^{5}+335920 A \,a^{3} b^{2} d^{2} e^{4}-155040 A \,a^{2} b^{3} d^{3} e^{3}+41344 A a \,b^{4} d^{4} e^{2}-4864 A \,b^{5} d^{5} e -92378 B \,a^{5} d \,e^{5}+167960 B \,a^{4} b \,d^{2} e^{4}-155040 B \,a^{3} b^{2} d^{3} e^{3}+82688 B \,a^{2} b^{3} d^{4} e^{2}-24320 B a \,b^{4} d^{5} e +3072 B \,b^{5} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{2909907 \left (b x +a \right )^{5} e^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((B*x+A)*(e*x+d)^(5/2)*(b^2*x^2+2*a*b*x+a^2)^(5/2),x)
[Out]
2/2909907*(e*x+d)^(7/2)*(153153*B*b^5*e^6*x^6+171171*A*b^5*e^6*x^5+855855*B*a*b^4*e^6*x^5-108108*B*b^5*d*e^5*x
^5+969969*A*a*b^4*e^6*x^4-114114*A*b^5*d*e^5*x^4+1939938*B*a^2*b^3*e^6*x^4-570570*B*a*b^4*d*e^5*x^4+72072*B*b^
5*d^2*e^4*x^4+2238390*A*a^2*b^3*e^6*x^3-596904*A*a*b^4*d*e^5*x^3+70224*A*b^5*d^2*e^4*x^3+2238390*B*a^3*b^2*e^6
*x^3-1193808*B*a^2*b^3*d*e^5*x^3+351120*B*a*b^4*d^2*e^4*x^3-44352*B*b^5*d^3*e^3*x^3+2645370*A*a^3*b^2*e^6*x^2-
1220940*A*a^2*b^3*d*e^5*x^2+325584*A*a*b^4*d^2*e^4*x^2-38304*A*b^5*d^3*e^3*x^2+1322685*B*a^4*b*e^6*x^2-1220940
*B*a^3*b^2*d*e^5*x^2+651168*B*a^2*b^3*d^2*e^4*x^2-191520*B*a*b^4*d^3*e^3*x^2+24192*B*b^5*d^4*e^2*x^2+1616615*A
*a^4*b*e^6*x-1175720*A*a^3*b^2*d*e^5*x+542640*A*a^2*b^3*d^2*e^4*x-144704*A*a*b^4*d^3*e^3*x+17024*A*b^5*d^4*e^2
*x+323323*B*a^5*e^6*x-587860*B*a^4*b*d*e^5*x+542640*B*a^3*b^2*d^2*e^4*x-289408*B*a^2*b^3*d^3*e^3*x+85120*B*a*b
^4*d^4*e^2*x-10752*B*b^5*d^5*e*x+415701*A*a^5*e^6-461890*A*a^4*b*d*e^5+335920*A*a^3*b^2*d^2*e^4-155040*A*a^2*b
^3*d^3*e^3+41344*A*a*b^4*d^4*e^2-4864*A*b^5*d^5*e-92378*B*a^5*d*e^5+167960*B*a^4*b*d^2*e^4-155040*B*a^3*b^2*d^
3*e^3+82688*B*a^2*b^3*d^4*e^2-24320*B*a*b^4*d^5*e+3072*B*b^5*d^6)*((b*x+a)^2)^(5/2)/e^7/(b*x+a)^5
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maxima [B] time = 0.85, size = 1080, normalized size = 2.39 \begin {gather*} \frac {2 \, {\left (9009 \, b^{5} e^{8} x^{8} - 256 \, b^{5} d^{8} + 2176 \, a b^{4} d^{7} e - 8160 \, a^{2} b^{3} d^{6} e^{2} + 17680 \, a^{3} b^{2} d^{5} e^{3} - 24310 \, a^{4} b d^{4} e^{4} + 21879 \, a^{5} d^{3} e^{5} + 3003 \, {\left (7 \, b^{5} d e^{7} + 17 \, a b^{4} e^{8}\right )} x^{7} + 231 \, {\left (55 \, b^{5} d^{2} e^{6} + 527 \, a b^{4} d e^{7} + 510 \, a^{2} b^{3} e^{8}\right )} x^{6} + 63 \, {\left (b^{5} d^{3} e^{5} + 1207 \, a b^{4} d^{2} e^{6} + 4590 \, a^{2} b^{3} d e^{7} + 2210 \, a^{3} b^{2} e^{8}\right )} x^{5} - 35 \, {\left (2 \, b^{5} d^{4} e^{4} - 17 \, a b^{4} d^{3} e^{5} - 5406 \, a^{2} b^{3} d^{2} e^{6} - 10166 \, a^{3} b^{2} d e^{7} - 2431 \, a^{4} b e^{8}\right )} x^{4} + {\left (80 \, b^{5} d^{5} e^{3} - 680 \, a b^{4} d^{4} e^{4} + 2550 \, a^{2} b^{3} d^{3} e^{5} + 249730 \, a^{3} b^{2} d^{2} e^{6} + 230945 \, a^{4} b d e^{7} + 21879 \, a^{5} e^{8}\right )} x^{3} - 3 \, {\left (32 \, b^{5} d^{6} e^{2} - 272 \, a b^{4} d^{5} e^{3} + 1020 \, a^{2} b^{3} d^{4} e^{4} - 2210 \, a^{3} b^{2} d^{3} e^{5} - 60775 \, a^{4} b d^{2} e^{6} - 21879 \, a^{5} d e^{7}\right )} x^{2} + {\left (128 \, b^{5} d^{7} e - 1088 \, a b^{4} d^{6} e^{2} + 4080 \, a^{2} b^{3} d^{5} e^{3} - 8840 \, a^{3} b^{2} d^{4} e^{4} + 12155 \, a^{4} b d^{3} e^{5} + 65637 \, a^{5} d^{2} e^{6}\right )} x\right )} \sqrt {e x + d} A}{153153 \, e^{6}} + \frac {2 \, {\left (153153 \, b^{5} e^{9} x^{9} + 3072 \, b^{5} d^{9} - 24320 \, a b^{4} d^{8} e + 82688 \, a^{2} b^{3} d^{7} e^{2} - 155040 \, a^{3} b^{2} d^{6} e^{3} + 167960 \, a^{4} b d^{5} e^{4} - 92378 \, a^{5} d^{4} e^{5} + 9009 \, {\left (39 \, b^{5} d e^{8} + 95 \, a b^{4} e^{9}\right )} x^{8} + 3003 \, {\left (69 \, b^{5} d^{2} e^{7} + 665 \, a b^{4} d e^{8} + 646 \, a^{2} b^{3} e^{9}\right )} x^{7} + 231 \, {\left (3 \, b^{5} d^{3} e^{6} + 5225 \, a b^{4} d^{2} e^{7} + 20026 \, a^{2} b^{3} d e^{8} + 9690 \, a^{3} b^{2} e^{9}\right )} x^{6} - 63 \, {\left (12 \, b^{5} d^{4} e^{5} - 95 \, a b^{4} d^{3} e^{6} - 45866 \, a^{2} b^{3} d^{2} e^{7} - 87210 \, a^{3} b^{2} d e^{8} - 20995 \, a^{4} b e^{9}\right )} x^{5} + 7 \, {\left (120 \, b^{5} d^{5} e^{4} - 950 \, a b^{4} d^{4} e^{5} + 3230 \, a^{2} b^{3} d^{3} e^{6} + 513570 \, a^{3} b^{2} d^{2} e^{7} + 482885 \, a^{4} b d e^{8} + 46189 \, a^{5} e^{9}\right )} x^{4} - {\left (960 \, b^{5} d^{6} e^{3} - 7600 \, a b^{4} d^{5} e^{4} + 25840 \, a^{2} b^{3} d^{4} e^{5} - 48450 \, a^{3} b^{2} d^{3} e^{6} - 2372435 \, a^{4} b d^{2} e^{7} - 877591 \, a^{5} d e^{8}\right )} x^{3} + 3 \, {\left (384 \, b^{5} d^{7} e^{2} - 3040 \, a b^{4} d^{6} e^{3} + 10336 \, a^{2} b^{3} d^{5} e^{4} - 19380 \, a^{3} b^{2} d^{4} e^{5} + 20995 \, a^{4} b d^{3} e^{6} + 230945 \, a^{5} d^{2} e^{7}\right )} x^{2} - {\left (1536 \, b^{5} d^{8} e - 12160 \, a b^{4} d^{7} e^{2} + 41344 \, a^{2} b^{3} d^{6} e^{3} - 77520 \, a^{3} b^{2} d^{5} e^{4} + 83980 \, a^{4} b d^{4} e^{5} - 46189 \, a^{5} d^{3} e^{6}\right )} x\right )} \sqrt {e x + d} B}{2909907 \, e^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(e*x+d)^(5/2)*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="maxima")
[Out]
2/153153*(9009*b^5*e^8*x^8 - 256*b^5*d^8 + 2176*a*b^4*d^7*e - 8160*a^2*b^3*d^6*e^2 + 17680*a^3*b^2*d^5*e^3 - 2
4310*a^4*b*d^4*e^4 + 21879*a^5*d^3*e^5 + 3003*(7*b^5*d*e^7 + 17*a*b^4*e^8)*x^7 + 231*(55*b^5*d^2*e^6 + 527*a*b
^4*d*e^7 + 510*a^2*b^3*e^8)*x^6 + 63*(b^5*d^3*e^5 + 1207*a*b^4*d^2*e^6 + 4590*a^2*b^3*d*e^7 + 2210*a^3*b^2*e^8
)*x^5 - 35*(2*b^5*d^4*e^4 - 17*a*b^4*d^3*e^5 - 5406*a^2*b^3*d^2*e^6 - 10166*a^3*b^2*d*e^7 - 2431*a^4*b*e^8)*x^
4 + (80*b^5*d^5*e^3 - 680*a*b^4*d^4*e^4 + 2550*a^2*b^3*d^3*e^5 + 249730*a^3*b^2*d^2*e^6 + 230945*a^4*b*d*e^7 +
21879*a^5*e^8)*x^3 - 3*(32*b^5*d^6*e^2 - 272*a*b^4*d^5*e^3 + 1020*a^2*b^3*d^4*e^4 - 2210*a^3*b^2*d^3*e^5 - 60
775*a^4*b*d^2*e^6 - 21879*a^5*d*e^7)*x^2 + (128*b^5*d^7*e - 1088*a*b^4*d^6*e^2 + 4080*a^2*b^3*d^5*e^3 - 8840*a
^3*b^2*d^4*e^4 + 12155*a^4*b*d^3*e^5 + 65637*a^5*d^2*e^6)*x)*sqrt(e*x + d)*A/e^6 + 2/2909907*(153153*b^5*e^9*x
^9 + 3072*b^5*d^9 - 24320*a*b^4*d^8*e + 82688*a^2*b^3*d^7*e^2 - 155040*a^3*b^2*d^6*e^3 + 167960*a^4*b*d^5*e^4
- 92378*a^5*d^4*e^5 + 9009*(39*b^5*d*e^8 + 95*a*b^4*e^9)*x^8 + 3003*(69*b^5*d^2*e^7 + 665*a*b^4*d*e^8 + 646*a^
2*b^3*e^9)*x^7 + 231*(3*b^5*d^3*e^6 + 5225*a*b^4*d^2*e^7 + 20026*a^2*b^3*d*e^8 + 9690*a^3*b^2*e^9)*x^6 - 63*(1
2*b^5*d^4*e^5 - 95*a*b^4*d^3*e^6 - 45866*a^2*b^3*d^2*e^7 - 87210*a^3*b^2*d*e^8 - 20995*a^4*b*e^9)*x^5 + 7*(120
*b^5*d^5*e^4 - 950*a*b^4*d^4*e^5 + 3230*a^2*b^3*d^3*e^6 + 513570*a^3*b^2*d^2*e^7 + 482885*a^4*b*d*e^8 + 46189*
a^5*e^9)*x^4 - (960*b^5*d^6*e^3 - 7600*a*b^4*d^5*e^4 + 25840*a^2*b^3*d^4*e^5 - 48450*a^3*b^2*d^3*e^6 - 2372435
*a^4*b*d^2*e^7 - 877591*a^5*d*e^8)*x^3 + 3*(384*b^5*d^7*e^2 - 3040*a*b^4*d^6*e^3 + 10336*a^2*b^3*d^5*e^4 - 193
80*a^3*b^2*d^4*e^5 + 20995*a^4*b*d^3*e^6 + 230945*a^5*d^2*e^7)*x^2 - (1536*b^5*d^8*e - 12160*a*b^4*d^7*e^2 + 4
1344*a^2*b^3*d^6*e^3 - 77520*a^3*b^2*d^5*e^4 + 83980*a^4*b*d^4*e^5 - 46189*a^5*d^3*e^6)*x)*sqrt(e*x + d)*B/e^7
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \left (A+B\,x\right )\,{\left (d+e\,x\right )}^{5/2}\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((A + B*x)*(d + e*x)^(5/2)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2),x)
[Out]
int((A + B*x)*(d + e*x)^(5/2)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2), x)
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(e*x+d)**(5/2)*(b**2*x**2+2*a*b*x+a**2)**(5/2),x)
[Out]
Timed out
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