3.17.33 \(\int (A+B x) (d+e x)^{3/2} (a^2+2 a b x+b^2 x^2)^{5/2} \, dx\)
Optimal. Leaf size=452 \[ -\frac {20 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{11 e^7 (a+b x)}+\frac {10 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{9 e^7 (a+b x)}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{7 e^7 (a+b x)}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^5 (B d-A e)}{5 e^7 (a+b x)}-\frac {2 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{15/2} (-5 a B e-A b e+6 b B d)}{15 e^7 (a+b x)}+\frac {10 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{13/2} (b d-a e) (-2 a B e-A b e+3 b B d)}{13 e^7 (a+b x)}+\frac {2 b^5 B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{17/2}}{17 e^7 (a+b x)} \]
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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)^{15/2} (-5 a B e-A b e+6 b B d)}{15 e^7 (a+b x)}+\frac {10 b^3 \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+3 b B d)}{13 e^7 (a+b x)}-\frac {20 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{11 e^7 (a+b x)}+\frac {10 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^3 (-a B e-2 A b e+3 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)^4 (-a B e-5 A b e+6 b B d)}{7 e^7 (a+b x)}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^5 (B d-A e)}{5 e^7 (a+b x)}+\frac {2 b^5 B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{17/2}}{17 e^7 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(A + B*x)*(d + e*x)^(3/2)*(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)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*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)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*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)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*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)^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^7*(a + b*x)) +
(10*b^3*(b*d - a*e)*(3*b*B*d - A*b*e - 2*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^4*(6*b*B*d - A*b*e - 5*a*B*e)*(d + e*x)^(15/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(15*e^7*(a + b*x)) +
(2*b^5*B*(d + e*x)^(17/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(17*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)^{3/2} \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)^{3/2} \, 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)^{3/2}}{e^6}+\frac {b^5 (b d-a e)^4 (-6 b B d+5 A b e+a B e) (d+e x)^{5/2}}{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)^{7/2}}{e^6}+\frac {10 b^7 (b d-a e)^2 (-2 b B d+A b e+a B e) (d+e x)^{9/2}}{e^6}-\frac {5 b^8 (b d-a e) (-3 b B d+A b e+2 a B e) (d+e x)^{11/2}}{e^6}+\frac {b^9 (-6 b B d+A b e+5 a B e) (d+e x)^{13/2}}{e^6}+\frac {b^{10} B (d+e x)^{15/2}}{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)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 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)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 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)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 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)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x)}+\frac {10 b^3 (b d-a e) (3 b B d-A b e-2 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^4 (6 b B d-A b e-5 a B e) (d+e x)^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}}{15 e^7 (a+b x)}+\frac {2 b^5 B (d+e x)^{17/2} \sqrt {a^2+2 a b x+b^2 x^2}}{17 e^7 (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 239, normalized size = 0.53 \begin {gather*} \frac {2 \sqrt {(a+b x)^2} (d+e x)^{5/2} \left (-51051 b^4 (d+e x)^5 (-5 a B e-A b e+6 b B d)+294525 b^3 (d+e x)^4 (b d-a e) (-2 a B e-A b e+3 b B d)-696150 b^2 (d+e x)^3 (b d-a e)^2 (-a B e-A b e+2 b B d)+425425 b (d+e x)^2 (b d-a e)^3 (-a B e-2 A b e+3 b B d)-109395 (d+e x) (b d-a e)^4 (-a B e-5 A b e+6 b B d)+153153 (b d-a e)^5 (B d-A e)+45045 b^5 B (d+e x)^6\right )}{765765 e^7 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(A + B*x)*(d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]
[Out]
(2*Sqrt[(a + b*x)^2]*(d + e*x)^(5/2)*(153153*(b*d - a*e)^5*(B*d - A*e) - 109395*(b*d - a*e)^4*(6*b*B*d - 5*A*b
*e - a*B*e)*(d + e*x) + 425425*b*(b*d - a*e)^3*(3*b*B*d - 2*A*b*e - a*B*e)*(d + e*x)^2 - 696150*b^2*(b*d - a*e
)^2*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^3 + 294525*b^3*(b*d - a*e)*(3*b*B*d - A*b*e - 2*a*B*e)*(d + e*x)^4 - 5
1051*b^4*(6*b*B*d - A*b*e - 5*a*B*e)*(d + e*x)^5 + 45045*b^5*B*(d + e*x)^6))/(765765*e^7*(a + b*x))
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IntegrateAlgebraic [A] time = 54.55, size = 812, normalized size = 1.80 \begin {gather*} \frac {2 (d+e x)^{5/2} \sqrt {\frac {(a e+b x e)^2}{e^2}} \left (153153 b^5 B d^6-153153 A b^5 e d^5-765765 a b^4 B e d^5-656370 b^5 B (d+e x) d^5+765765 a A b^4 e^2 d^4+1531530 a^2 b^3 B e^2 d^4+1276275 b^5 B (d+e x)^2 d^4+546975 A b^5 e (d+e x) d^4+2734875 a b^4 B e (d+e x) d^4-1531530 a^2 A b^3 e^3 d^3-1531530 a^3 b^2 B e^3 d^3-1392300 b^5 B (d+e x)^3 d^3-850850 A b^5 e (d+e x)^2 d^3-4254250 a b^4 B e (d+e x)^2 d^3-2187900 a A b^4 e^2 (d+e x) d^3-4375800 a^2 b^3 B e^2 (d+e x) d^3+1531530 a^3 A b^2 e^4 d^2+765765 a^4 b B e^4 d^2+883575 b^5 B (d+e x)^4 d^2+696150 A b^5 e (d+e x)^3 d^2+3480750 a b^4 B e (d+e x)^3 d^2+2552550 a A b^4 e^2 (d+e x)^2 d^2+5105100 a^2 b^3 B e^2 (d+e x)^2 d^2+3281850 a^2 A b^3 e^3 (d+e x) d^2+3281850 a^3 b^2 B e^3 (d+e x) d^2-765765 a^4 A b e^5 d-153153 a^5 B e^5 d-306306 b^5 B (d+e x)^5 d-294525 A b^5 e (d+e x)^4 d-1472625 a b^4 B e (d+e x)^4 d-1392300 a A b^4 e^2 (d+e x)^3 d-2784600 a^2 b^3 B e^2 (d+e x)^3 d-2552550 a^2 A b^3 e^3 (d+e x)^2 d-2552550 a^3 b^2 B e^3 (d+e x)^2 d-2187900 a^3 A b^2 e^4 (d+e x) d-1093950 a^4 b B e^4 (d+e x) d+153153 a^5 A e^6+45045 b^5 B (d+e x)^6+51051 A b^5 e (d+e x)^5+255255 a b^4 B e (d+e x)^5+294525 a A b^4 e^2 (d+e x)^4+589050 a^2 b^3 B e^2 (d+e x)^4+696150 a^2 A b^3 e^3 (d+e x)^3+696150 a^3 b^2 B e^3 (d+e x)^3+850850 a^3 A b^2 e^4 (d+e x)^2+425425 a^4 b B e^4 (d+e x)^2+546975 a^4 A b e^5 (d+e x)+109395 a^5 B e^5 (d+e x)\right )}{765765 e^6 (a e+b x e)} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(A + B*x)*(d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]
[Out]
(2*(d + e*x)^(5/2)*Sqrt[(a*e + b*e*x)^2/e^2]*(153153*b^5*B*d^6 - 153153*A*b^5*d^5*e - 765765*a*b^4*B*d^5*e + 7
65765*a*A*b^4*d^4*e^2 + 1531530*a^2*b^3*B*d^4*e^2 - 1531530*a^2*A*b^3*d^3*e^3 - 1531530*a^3*b^2*B*d^3*e^3 + 15
31530*a^3*A*b^2*d^2*e^4 + 765765*a^4*b*B*d^2*e^4 - 765765*a^4*A*b*d*e^5 - 153153*a^5*B*d*e^5 + 153153*a^5*A*e^
6 - 656370*b^5*B*d^5*(d + e*x) + 546975*A*b^5*d^4*e*(d + e*x) + 2734875*a*b^4*B*d^4*e*(d + e*x) - 2187900*a*A*
b^4*d^3*e^2*(d + e*x) - 4375800*a^2*b^3*B*d^3*e^2*(d + e*x) + 3281850*a^2*A*b^3*d^2*e^3*(d + e*x) + 3281850*a^
3*b^2*B*d^2*e^3*(d + e*x) - 2187900*a^3*A*b^2*d*e^4*(d + e*x) - 1093950*a^4*b*B*d*e^4*(d + e*x) + 546975*a^4*A
*b*e^5*(d + e*x) + 109395*a^5*B*e^5*(d + e*x) + 1276275*b^5*B*d^4*(d + e*x)^2 - 850850*A*b^5*d^3*e*(d + e*x)^2
- 4254250*a*b^4*B*d^3*e*(d + e*x)^2 + 2552550*a*A*b^4*d^2*e^2*(d + e*x)^2 + 5105100*a^2*b^3*B*d^2*e^2*(d + e*
x)^2 - 2552550*a^2*A*b^3*d*e^3*(d + e*x)^2 - 2552550*a^3*b^2*B*d*e^3*(d + e*x)^2 + 850850*a^3*A*b^2*e^4*(d + e
*x)^2 + 425425*a^4*b*B*e^4*(d + e*x)^2 - 1392300*b^5*B*d^3*(d + e*x)^3 + 696150*A*b^5*d^2*e*(d + e*x)^3 + 3480
750*a*b^4*B*d^2*e*(d + e*x)^3 - 1392300*a*A*b^4*d*e^2*(d + e*x)^3 - 2784600*a^2*b^3*B*d*e^2*(d + e*x)^3 + 6961
50*a^2*A*b^3*e^3*(d + e*x)^3 + 696150*a^3*b^2*B*e^3*(d + e*x)^3 + 883575*b^5*B*d^2*(d + e*x)^4 - 294525*A*b^5*
d*e*(d + e*x)^4 - 1472625*a*b^4*B*d*e*(d + e*x)^4 + 294525*a*A*b^4*e^2*(d + e*x)^4 + 589050*a^2*b^3*B*e^2*(d +
e*x)^4 - 306306*b^5*B*d*(d + e*x)^5 + 51051*A*b^5*e*(d + e*x)^5 + 255255*a*b^4*B*e*(d + e*x)^5 + 45045*b^5*B*
(d + e*x)^6))/(765765*e^6*(a*e + b*e*x))
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fricas [B] time = 0.44, size = 848, normalized size = 1.88 \begin {gather*} \frac {2 \, {\left (45045 \, B b^{5} e^{8} x^{8} + 3072 \, B b^{5} d^{8} + 153153 \, A a^{5} d^{2} e^{6} - 4352 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{7} e + 32640 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{6} e^{2} - 106080 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{5} e^{3} + 97240 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{4} e^{4} - 43758 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d^{3} e^{5} + 3003 \, {\left (18 \, B b^{5} d e^{7} + 17 \, {\left (5 \, B a b^{4} + A b^{5}\right )} e^{8}\right )} x^{7} + 231 \, {\left (3 \, B b^{5} d^{2} e^{6} + 272 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d e^{7} + 1275 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e^{8}\right )} x^{6} - 63 \, {\left (12 \, B b^{5} d^{3} e^{5} - 17 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e^{6} - 5950 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{7} - 11050 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{8}\right )} x^{5} + 35 \, {\left (24 \, B b^{5} d^{4} e^{4} - 34 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} e^{5} + 255 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{6} + 26520 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{7} + 12155 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e^{8}\right )} x^{4} - 5 \, {\left (192 \, B b^{5} d^{5} e^{3} - 272 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} e^{4} + 2040 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e^{5} - 6630 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e^{6} - 121550 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d e^{7} - 21879 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} e^{8}\right )} x^{3} + 3 \, {\left (384 \, B b^{5} d^{6} e^{2} + 51051 \, A a^{5} e^{8} - 544 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} e^{3} + 4080 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} e^{4} - 13260 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} e^{5} + 12155 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} e^{6} + 58344 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d e^{7}\right )} x^{2} - {\left (1536 \, B b^{5} d^{7} e - 306306 \, A a^{5} d e^{7} - 2176 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{6} e^{2} + 16320 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{5} e^{3} - 53040 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{4} e^{4} + 48620 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{3} e^{5} - 21879 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d^{2} e^{6}\right )} x\right )} \sqrt {e x + d}}{765765 \, e^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(e*x+d)^(3/2)*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")
[Out]
2/765765*(45045*B*b^5*e^8*x^8 + 3072*B*b^5*d^8 + 153153*A*a^5*d^2*e^6 - 4352*(5*B*a*b^4 + A*b^5)*d^7*e + 32640
*(2*B*a^2*b^3 + A*a*b^4)*d^6*e^2 - 106080*(B*a^3*b^2 + A*a^2*b^3)*d^5*e^3 + 97240*(B*a^4*b + 2*A*a^3*b^2)*d^4*
e^4 - 43758*(B*a^5 + 5*A*a^4*b)*d^3*e^5 + 3003*(18*B*b^5*d*e^7 + 17*(5*B*a*b^4 + A*b^5)*e^8)*x^7 + 231*(3*B*b^
5*d^2*e^6 + 272*(5*B*a*b^4 + A*b^5)*d*e^7 + 1275*(2*B*a^2*b^3 + A*a*b^4)*e^8)*x^6 - 63*(12*B*b^5*d^3*e^5 - 17*
(5*B*a*b^4 + A*b^5)*d^2*e^6 - 5950*(2*B*a^2*b^3 + A*a*b^4)*d*e^7 - 11050*(B*a^3*b^2 + A*a^2*b^3)*e^8)*x^5 + 35
*(24*B*b^5*d^4*e^4 - 34*(5*B*a*b^4 + A*b^5)*d^3*e^5 + 255*(2*B*a^2*b^3 + A*a*b^4)*d^2*e^6 + 26520*(B*a^3*b^2 +
A*a^2*b^3)*d*e^7 + 12155*(B*a^4*b + 2*A*a^3*b^2)*e^8)*x^4 - 5*(192*B*b^5*d^5*e^3 - 272*(5*B*a*b^4 + A*b^5)*d^
4*e^4 + 2040*(2*B*a^2*b^3 + A*a*b^4)*d^3*e^5 - 6630*(B*a^3*b^2 + A*a^2*b^3)*d^2*e^6 - 121550*(B*a^4*b + 2*A*a^
3*b^2)*d*e^7 - 21879*(B*a^5 + 5*A*a^4*b)*e^8)*x^3 + 3*(384*B*b^5*d^6*e^2 + 51051*A*a^5*e^8 - 544*(5*B*a*b^4 +
A*b^5)*d^5*e^3 + 4080*(2*B*a^2*b^3 + A*a*b^4)*d^4*e^4 - 13260*(B*a^3*b^2 + A*a^2*b^3)*d^3*e^5 + 12155*(B*a^4*b
+ 2*A*a^3*b^2)*d^2*e^6 + 58344*(B*a^5 + 5*A*a^4*b)*d*e^7)*x^2 - (1536*B*b^5*d^7*e - 306306*A*a^5*d*e^7 - 2176
*(5*B*a*b^4 + A*b^5)*d^6*e^2 + 16320*(2*B*a^2*b^3 + A*a*b^4)*d^5*e^3 - 53040*(B*a^3*b^2 + A*a^2*b^3)*d^4*e^4 +
48620*(B*a^4*b + 2*A*a^3*b^2)*d^3*e^5 - 21879*(B*a^5 + 5*A*a^4*b)*d^2*e^6)*x)*sqrt(e*x + d)/e^7
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giac [B] time = 0.48, size = 2788, normalized size = 6.17
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(e*x+d)^(3/2)*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")
[Out]
2/765765*(255255*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*B*a^5*d^2*e^(-1)*sgn(b*x + a) + 1276275*((x*e + d)^(3/2
) - 3*sqrt(x*e + d)*d)*A*a^4*b*d^2*e^(-1)*sgn(b*x + a) + 255255*(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^2*e^(-2)*sgn(b*x + a) + 510510*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sq
rt(x*e + d)*d^2)*A*a^3*b^2*d^2*e^(-2)*sgn(b*x + a) + 218790*(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^2*e^(-3)*sgn(b*x + a) + 218790*(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^2*e^(-3)*sgn(b*x + a) + 24310*(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^2*b^3*d^2*e^(-4)*sgn(b*x + a) + 12155*(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^2*e^(-4)*sgn(b*x + a) + 5525*(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^2*e^(-5)*sgn(b*x + a) + 1105*(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^2*e^(-5)*sgn(b*x + a) + 255*(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^2*e^(-6)*sgn(b*x + a) + 102102*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*B*a^5*d*
e^(-1)*sgn(b*x + a) + 510510*(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*e^(-1
)*sgn(b*x + a) + 218790*(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*e^(-2)*sgn(b*x + a) + 437580*(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*e^(-2)*sgn(b*x + a) + 48620*(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*e^(-3)*sgn(b*x + a) +
48620*(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)*A*a^2*b^3*d*e^(-3)*sgn(b*x + a) + 22100*(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^2*b^3*d*e
^(-4)*sgn(b*x + a) + 11050*(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*e^(-4)*sgn(b*x + a) + 2550*(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*e^(-5)*sgn(b*x + a) + 510*(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*e^(-5)*sgn(b*x + a) + 238*(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*e^(-6)*sg
n(b*x + a) + 765765*sqrt(x*e + d)*A*a^5*d^2*sgn(b*x + a) + 510510*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*A*a^5*
d*sgn(b*x + a) + 21879*(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*e^(-1)*sgn(b*x + a) + 109395*(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*e^(-1)*sgn(b*x + a) + 12155*(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*e^(-2)*sgn(b*x + a) + 24310*(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^3*b^2*e^(-2)*sgn(b*x + a) + 11050*(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*e^(-3)*sgn(b*x + a
) + 11050*(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^2*b^3*e^(-3)*sgn(b*x + a) + 2550*(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 - 60
06*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*B*a^2*b^3*e^(-4)*sgn(b*x + a) + 1275*(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 - 600
6*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*A*a*b^4*e^(-4)*sgn(b*x + a) + 595*(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 - 270
27*(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*e^(-5)*sgn(b*x + a) + 119
*(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 + 32
175*(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*e^(-5)*sgn(b*x + a) + 7*(6435*(x*e + d)^(17/2) - 58344*(x*e + d)^(15/2)*d + 235620*(x*e + d)^(13/2)*d^2 - 55
6920*(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*e^(-6)*sgn(b*x + a) + 51051*(3*(x*e + d)^(5/
2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*A*a^5*sgn(b*x + a))*e^(-1)
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maple [A] time = 0.06, size = 689, normalized size = 1.52 \begin {gather*} \frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (45045 B \,b^{5} e^{6} x^{6}+51051 A \,b^{5} e^{6} x^{5}+255255 B a \,b^{4} e^{6} x^{5}-36036 B \,b^{5} d \,e^{5} x^{5}+294525 A a \,b^{4} e^{6} x^{4}-39270 A \,b^{5} d \,e^{5} x^{4}+589050 B \,a^{2} b^{3} e^{6} x^{4}-196350 B a \,b^{4} d \,e^{5} x^{4}+27720 B \,b^{5} d^{2} e^{4} x^{4}+696150 A \,a^{2} b^{3} e^{6} x^{3}-214200 A a \,b^{4} d \,e^{5} x^{3}+28560 A \,b^{5} d^{2} e^{4} x^{3}+696150 B \,a^{3} b^{2} e^{6} x^{3}-428400 B \,a^{2} b^{3} d \,e^{5} x^{3}+142800 B a \,b^{4} d^{2} e^{4} x^{3}-20160 B \,b^{5} d^{3} e^{3} x^{3}+850850 A \,a^{3} b^{2} e^{6} x^{2}-464100 A \,a^{2} b^{3} d \,e^{5} x^{2}+142800 A a \,b^{4} d^{2} e^{4} x^{2}-19040 A \,b^{5} d^{3} e^{3} x^{2}+425425 B \,a^{4} b \,e^{6} x^{2}-464100 B \,a^{3} b^{2} d \,e^{5} x^{2}+285600 B \,a^{2} b^{3} d^{2} e^{4} x^{2}-95200 B a \,b^{4} d^{3} e^{3} x^{2}+13440 B \,b^{5} d^{4} e^{2} x^{2}+546975 A \,a^{4} b \,e^{6} x -486200 A \,a^{3} b^{2} d \,e^{5} x +265200 A \,a^{2} b^{3} d^{2} e^{4} x -81600 A a \,b^{4} d^{3} e^{3} x +10880 A \,b^{5} d^{4} e^{2} x +109395 B \,a^{5} e^{6} x -243100 B \,a^{4} b d \,e^{5} x +265200 B \,a^{3} b^{2} d^{2} e^{4} x -163200 B \,a^{2} b^{3} d^{3} e^{3} x +54400 B a \,b^{4} d^{4} e^{2} x -7680 B \,b^{5} d^{5} e x +153153 A \,a^{5} e^{6}-218790 A \,a^{4} b d \,e^{5}+194480 A \,a^{3} b^{2} d^{2} e^{4}-106080 A \,a^{2} b^{3} d^{3} e^{3}+32640 A a \,b^{4} d^{4} e^{2}-4352 A \,b^{5} d^{5} e -43758 B \,a^{5} d \,e^{5}+97240 B \,a^{4} b \,d^{2} e^{4}-106080 B \,a^{3} b^{2} d^{3} e^{3}+65280 B \,a^{2} b^{3} d^{4} e^{2}-21760 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}}}{765765 \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)^(3/2)*(b^2*x^2+2*a*b*x+a^2)^(5/2),x)
[Out]
2/765765*(e*x+d)^(5/2)*(45045*B*b^5*e^6*x^6+51051*A*b^5*e^6*x^5+255255*B*a*b^4*e^6*x^5-36036*B*b^5*d*e^5*x^5+2
94525*A*a*b^4*e^6*x^4-39270*A*b^5*d*e^5*x^4+589050*B*a^2*b^3*e^6*x^4-196350*B*a*b^4*d*e^5*x^4+27720*B*b^5*d^2*
e^4*x^4+696150*A*a^2*b^3*e^6*x^3-214200*A*a*b^4*d*e^5*x^3+28560*A*b^5*d^2*e^4*x^3+696150*B*a^3*b^2*e^6*x^3-428
400*B*a^2*b^3*d*e^5*x^3+142800*B*a*b^4*d^2*e^4*x^3-20160*B*b^5*d^3*e^3*x^3+850850*A*a^3*b^2*e^6*x^2-464100*A*a
^2*b^3*d*e^5*x^2+142800*A*a*b^4*d^2*e^4*x^2-19040*A*b^5*d^3*e^3*x^2+425425*B*a^4*b*e^6*x^2-464100*B*a^3*b^2*d*
e^5*x^2+285600*B*a^2*b^3*d^2*e^4*x^2-95200*B*a*b^4*d^3*e^3*x^2+13440*B*b^5*d^4*e^2*x^2+546975*A*a^4*b*e^6*x-48
6200*A*a^3*b^2*d*e^5*x+265200*A*a^2*b^3*d^2*e^4*x-81600*A*a*b^4*d^3*e^3*x+10880*A*b^5*d^4*e^2*x+109395*B*a^5*e
^6*x-243100*B*a^4*b*d*e^5*x+265200*B*a^3*b^2*d^2*e^4*x-163200*B*a^2*b^3*d^3*e^3*x+54400*B*a*b^4*d^4*e^2*x-7680
*B*b^5*d^5*e*x+153153*A*a^5*e^6-218790*A*a^4*b*d*e^5+194480*A*a^3*b^2*d^2*e^4-106080*A*a^2*b^3*d^3*e^3+32640*A
*a*b^4*d^4*e^2-4352*A*b^5*d^5*e-43758*B*a^5*d*e^5+97240*B*a^4*b*d^2*e^4-106080*B*a^3*b^2*d^3*e^3+65280*B*a^2*b
^3*d^4*e^2-21760*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 = 0.77, size = 921, normalized size = 2.04 \begin {gather*} \frac {2 \, {\left (3003 \, b^{5} e^{7} x^{7} - 256 \, b^{5} d^{7} + 1920 \, a b^{4} d^{6} e - 6240 \, a^{2} b^{3} d^{5} e^{2} + 11440 \, a^{3} b^{2} d^{4} e^{3} - 12870 \, a^{4} b d^{3} e^{4} + 9009 \, a^{5} d^{2} e^{5} + 231 \, {\left (16 \, b^{5} d e^{6} + 75 \, a b^{4} e^{7}\right )} x^{6} + 63 \, {\left (b^{5} d^{2} e^{5} + 350 \, a b^{4} d e^{6} + 650 \, a^{2} b^{3} e^{7}\right )} x^{5} - 35 \, {\left (2 \, b^{5} d^{3} e^{4} - 15 \, a b^{4} d^{2} e^{5} - 1560 \, a^{2} b^{3} d e^{6} - 1430 \, a^{3} b^{2} e^{7}\right )} x^{4} + 5 \, {\left (16 \, b^{5} d^{4} e^{3} - 120 \, a b^{4} d^{3} e^{4} + 390 \, a^{2} b^{3} d^{2} e^{5} + 14300 \, a^{3} b^{2} d e^{6} + 6435 \, a^{4} b e^{7}\right )} x^{3} - 3 \, {\left (32 \, b^{5} d^{5} e^{2} - 240 \, a b^{4} d^{4} e^{3} + 780 \, a^{2} b^{3} d^{3} e^{4} - 1430 \, a^{3} b^{2} d^{2} e^{5} - 17160 \, a^{4} b d e^{6} - 3003 \, a^{5} e^{7}\right )} x^{2} + {\left (128 \, b^{5} d^{6} e - 960 \, a b^{4} d^{5} e^{2} + 3120 \, a^{2} b^{3} d^{4} e^{3} - 5720 \, a^{3} b^{2} d^{3} e^{4} + 6435 \, a^{4} b d^{2} e^{5} + 18018 \, a^{5} d e^{6}\right )} x\right )} \sqrt {e x + d} A}{45045 \, e^{6}} + \frac {2 \, {\left (45045 \, b^{5} e^{8} x^{8} + 3072 \, b^{5} d^{8} - 21760 \, a b^{4} d^{7} e + 65280 \, a^{2} b^{3} d^{6} e^{2} - 106080 \, a^{3} b^{2} d^{5} e^{3} + 97240 \, a^{4} b d^{4} e^{4} - 43758 \, a^{5} d^{3} e^{5} + 3003 \, {\left (18 \, b^{5} d e^{7} + 85 \, a b^{4} e^{8}\right )} x^{7} + 231 \, {\left (3 \, b^{5} d^{2} e^{6} + 1360 \, a b^{4} d e^{7} + 2550 \, a^{2} b^{3} e^{8}\right )} x^{6} - 63 \, {\left (12 \, b^{5} d^{3} e^{5} - 85 \, a b^{4} d^{2} e^{6} - 11900 \, a^{2} b^{3} d e^{7} - 11050 \, a^{3} b^{2} e^{8}\right )} x^{5} + 35 \, {\left (24 \, b^{5} d^{4} e^{4} - 170 \, a b^{4} d^{3} e^{5} + 510 \, a^{2} b^{3} d^{2} e^{6} + 26520 \, a^{3} b^{2} d e^{7} + 12155 \, a^{4} b e^{8}\right )} x^{4} - 5 \, {\left (192 \, b^{5} d^{5} e^{3} - 1360 \, a b^{4} d^{4} e^{4} + 4080 \, a^{2} b^{3} d^{3} e^{5} - 6630 \, a^{3} b^{2} d^{2} e^{6} - 121550 \, a^{4} b d e^{7} - 21879 \, a^{5} e^{8}\right )} x^{3} + 3 \, {\left (384 \, b^{5} d^{6} e^{2} - 2720 \, a b^{4} d^{5} e^{3} + 8160 \, a^{2} b^{3} d^{4} e^{4} - 13260 \, a^{3} b^{2} d^{3} e^{5} + 12155 \, a^{4} b d^{2} e^{6} + 58344 \, a^{5} d e^{7}\right )} x^{2} - {\left (1536 \, b^{5} d^{7} e - 10880 \, a b^{4} d^{6} e^{2} + 32640 \, a^{2} b^{3} d^{5} e^{3} - 53040 \, a^{3} b^{2} d^{4} e^{4} + 48620 \, a^{4} b d^{3} e^{5} - 21879 \, a^{5} d^{2} e^{6}\right )} x\right )} \sqrt {e x + d} B}{765765 \, e^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(e*x+d)^(3/2)*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="maxima")
[Out]
2/45045*(3003*b^5*e^7*x^7 - 256*b^5*d^7 + 1920*a*b^4*d^6*e - 6240*a^2*b^3*d^5*e^2 + 11440*a^3*b^2*d^4*e^3 - 12
870*a^4*b*d^3*e^4 + 9009*a^5*d^2*e^5 + 231*(16*b^5*d*e^6 + 75*a*b^4*e^7)*x^6 + 63*(b^5*d^2*e^5 + 350*a*b^4*d*e
^6 + 650*a^2*b^3*e^7)*x^5 - 35*(2*b^5*d^3*e^4 - 15*a*b^4*d^2*e^5 - 1560*a^2*b^3*d*e^6 - 1430*a^3*b^2*e^7)*x^4
+ 5*(16*b^5*d^4*e^3 - 120*a*b^4*d^3*e^4 + 390*a^2*b^3*d^2*e^5 + 14300*a^3*b^2*d*e^6 + 6435*a^4*b*e^7)*x^3 - 3*
(32*b^5*d^5*e^2 - 240*a*b^4*d^4*e^3 + 780*a^2*b^3*d^3*e^4 - 1430*a^3*b^2*d^2*e^5 - 17160*a^4*b*d*e^6 - 3003*a^
5*e^7)*x^2 + (128*b^5*d^6*e - 960*a*b^4*d^5*e^2 + 3120*a^2*b^3*d^4*e^3 - 5720*a^3*b^2*d^3*e^4 + 6435*a^4*b*d^2
*e^5 + 18018*a^5*d*e^6)*x)*sqrt(e*x + d)*A/e^6 + 2/765765*(45045*b^5*e^8*x^8 + 3072*b^5*d^8 - 21760*a*b^4*d^7*
e + 65280*a^2*b^3*d^6*e^2 - 106080*a^3*b^2*d^5*e^3 + 97240*a^4*b*d^4*e^4 - 43758*a^5*d^3*e^5 + 3003*(18*b^5*d*
e^7 + 85*a*b^4*e^8)*x^7 + 231*(3*b^5*d^2*e^6 + 1360*a*b^4*d*e^7 + 2550*a^2*b^3*e^8)*x^6 - 63*(12*b^5*d^3*e^5 -
85*a*b^4*d^2*e^6 - 11900*a^2*b^3*d*e^7 - 11050*a^3*b^2*e^8)*x^5 + 35*(24*b^5*d^4*e^4 - 170*a*b^4*d^3*e^5 + 51
0*a^2*b^3*d^2*e^6 + 26520*a^3*b^2*d*e^7 + 12155*a^4*b*e^8)*x^4 - 5*(192*b^5*d^5*e^3 - 1360*a*b^4*d^4*e^4 + 408
0*a^2*b^3*d^3*e^5 - 6630*a^3*b^2*d^2*e^6 - 121550*a^4*b*d*e^7 - 21879*a^5*e^8)*x^3 + 3*(384*b^5*d^6*e^2 - 2720
*a*b^4*d^5*e^3 + 8160*a^2*b^3*d^4*e^4 - 13260*a^3*b^2*d^3*e^5 + 12155*a^4*b*d^2*e^6 + 58344*a^5*d*e^7)*x^2 - (
1536*b^5*d^7*e - 10880*a*b^4*d^6*e^2 + 32640*a^2*b^3*d^5*e^3 - 53040*a^3*b^2*d^4*e^4 + 48620*a^4*b*d^3*e^5 - 2
1879*a^5*d^2*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 )}^{3/2}\,{\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)^(3/2)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2),x)
[Out]
int((A + B*x)*(d + e*x)^(3/2)*(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)**(3/2)*(b**2*x**2+2*a*b*x+a**2)**(5/2),x)
[Out]
Timed out
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