3.17.35 \(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)^{5/2}}{\sqrt {d+e x}} \, dx\)

Optimal. Leaf size=448 \[ -\frac {20 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{7 e^7 (a+b x)}+\frac {2 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{e^7 (a+b x)}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{3 e^7 (a+b x)}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^5 (B d-A e)}{e^7 (a+b x)}-\frac {2 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (-5 a B e-A b e+6 b B d)}{11 e^7 (a+b x)}+\frac {10 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e) (-2 a B e-A b e+3 b B d)}{9 e^7 (a+b x)}+\frac {2 b^5 B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{13/2}}{13 e^7 (a+b x)} \]

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Rubi [A]  time = 0.22, antiderivative size = 448, 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)^{11/2} (-5 a B e-A b e+6 b B d)}{11 e^7 (a+b x)}+\frac {10 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e) (-2 a B e-A b e+3 b B d)}{9 e^7 (a+b x)}-\frac {20 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{7 e^7 (a+b x)}+\frac {2 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{e^7 (a+b x)}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{3 e^7 (a+b x)}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^5 (B d-A e)}{e^7 (a+b x)}+\frac {2 b^5 B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{13/2}}{13 e^7 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/Sqrt[d + e*x],x]

[Out]

(2*(b*d - a*e)^5*(B*d - A*e)*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(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)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^7*(a + b*x)) + (2*b*(b*d - a*e)
^3*(3*b*B*d - 2*A*b*e - a*B*e)*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(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)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*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)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^7*(a + b*x)) - (2*b
^4*(6*b*B*d - A*b*e - 5*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)) + (2*b^5*B*(
d + e*x)^(13/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*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 \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^5 (A+B x)}{\sqrt {d+e x}} \, 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)}{e^6 \sqrt {d+e x}}+\frac {b^5 (b d-a e)^4 (-6 b B d+5 A b e+a B e) \sqrt {d+e x}}{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)^{3/2}}{e^6}+\frac {10 b^7 (b d-a e)^2 (-2 b B d+A b e+a B e) (d+e x)^{5/2}}{e^6}-\frac {5 b^8 (b d-a e) (-3 b B d+A b e+2 a B e) (d+e x)^{7/2}}{e^6}+\frac {b^9 (-6 b B d+A b e+5 a B e) (d+e x)^{9/2}}{e^6}+\frac {b^{10} B (d+e x)^{11/2}}{e^6}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {2 (b d-a e)^5 (B d-A e) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}{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)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x)}+\frac {2 b (b d-a e)^3 (3 b B d-2 A b e-a B e) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{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)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 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)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^7 (a+b x)}-\frac {2 b^4 (6 b B d-A b e-5 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 {2 b^5 B (d+e x)^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^7 (a+b x)}\\ \end {align*}

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Mathematica [A]  time = 0.19, size = 239, normalized size = 0.53 \begin {gather*} \frac {2 \sqrt {(a+b x)^2} \sqrt {d+e x} \left (-819 b^4 (d+e x)^5 (-5 a B e-A b e+6 b B d)+5005 b^3 (d+e x)^4 (b d-a e) (-2 a B e-A b e+3 b B d)-12870 b^2 (d+e x)^3 (b d-a e)^2 (-a B e-A b e+2 b B d)+9009 b (d+e x)^2 (b d-a e)^3 (-a B e-2 A b e+3 b B d)-3003 (d+e x) (b d-a e)^4 (-a B e-5 A b e+6 b B d)+9009 (b d-a e)^5 (B d-A e)+693 b^5 B (d+e x)^6\right )}{9009 e^7 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/Sqrt[d + e*x],x]

[Out]

(2*Sqrt[(a + b*x)^2]*Sqrt[d + e*x]*(9009*(b*d - a*e)^5*(B*d - A*e) - 3003*(b*d - a*e)^4*(6*b*B*d - 5*A*b*e - a
*B*e)*(d + e*x) + 9009*b*(b*d - a*e)^3*(3*b*B*d - 2*A*b*e - a*B*e)*(d + e*x)^2 - 12870*b^2*(b*d - a*e)^2*(2*b*
B*d - A*b*e - a*B*e)*(d + e*x)^3 + 5005*b^3*(b*d - a*e)*(3*b*B*d - A*b*e - 2*a*B*e)*(d + e*x)^4 - 819*b^4*(6*b
*B*d - A*b*e - 5*a*B*e)*(d + e*x)^5 + 693*b^5*B*(d + e*x)^6))/(9009*e^7*(a + b*x))

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IntegrateAlgebraic [A]  time = 53.12, size = 812, normalized size = 1.81 \begin {gather*} \frac {2 \sqrt {d+e x} \sqrt {\frac {(a e+b x e)^2}{e^2}} \left (9009 b^5 B d^6-9009 A b^5 e d^5-45045 a b^4 B e d^5-18018 b^5 B (d+e x) d^5+45045 a A b^4 e^2 d^4+90090 a^2 b^3 B e^2 d^4+27027 b^5 B (d+e x)^2 d^4+15015 A b^5 e (d+e x) d^4+75075 a b^4 B e (d+e x) d^4-90090 a^2 A b^3 e^3 d^3-90090 a^3 b^2 B e^3 d^3-25740 b^5 B (d+e x)^3 d^3-18018 A b^5 e (d+e x)^2 d^3-90090 a b^4 B e (d+e x)^2 d^3-60060 a A b^4 e^2 (d+e x) d^3-120120 a^2 b^3 B e^2 (d+e x) d^3+90090 a^3 A b^2 e^4 d^2+45045 a^4 b B e^4 d^2+15015 b^5 B (d+e x)^4 d^2+12870 A b^5 e (d+e x)^3 d^2+64350 a b^4 B e (d+e x)^3 d^2+54054 a A b^4 e^2 (d+e x)^2 d^2+108108 a^2 b^3 B e^2 (d+e x)^2 d^2+90090 a^2 A b^3 e^3 (d+e x) d^2+90090 a^3 b^2 B e^3 (d+e x) d^2-45045 a^4 A b e^5 d-9009 a^5 B e^5 d-4914 b^5 B (d+e x)^5 d-5005 A b^5 e (d+e x)^4 d-25025 a b^4 B e (d+e x)^4 d-25740 a A b^4 e^2 (d+e x)^3 d-51480 a^2 b^3 B e^2 (d+e x)^3 d-54054 a^2 A b^3 e^3 (d+e x)^2 d-54054 a^3 b^2 B e^3 (d+e x)^2 d-60060 a^3 A b^2 e^4 (d+e x) d-30030 a^4 b B e^4 (d+e x) d+9009 a^5 A e^6+693 b^5 B (d+e x)^6+819 A b^5 e (d+e x)^5+4095 a b^4 B e (d+e x)^5+5005 a A b^4 e^2 (d+e x)^4+10010 a^2 b^3 B e^2 (d+e x)^4+12870 a^2 A b^3 e^3 (d+e x)^3+12870 a^3 b^2 B e^3 (d+e x)^3+18018 a^3 A b^2 e^4 (d+e x)^2+9009 a^4 b B e^4 (d+e x)^2+15015 a^4 A b e^5 (d+e x)+3003 a^5 B e^5 (d+e x)\right )}{9009 e^6 (a e+b x e)} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/Sqrt[d + e*x],x]

[Out]

(2*Sqrt[d + e*x]*Sqrt[(a*e + b*e*x)^2/e^2]*(9009*b^5*B*d^6 - 9009*A*b^5*d^5*e - 45045*a*b^4*B*d^5*e + 45045*a*
A*b^4*d^4*e^2 + 90090*a^2*b^3*B*d^4*e^2 - 90090*a^2*A*b^3*d^3*e^3 - 90090*a^3*b^2*B*d^3*e^3 + 90090*a^3*A*b^2*
d^2*e^4 + 45045*a^4*b*B*d^2*e^4 - 45045*a^4*A*b*d*e^5 - 9009*a^5*B*d*e^5 + 9009*a^5*A*e^6 - 18018*b^5*B*d^5*(d
+ e*x) + 15015*A*b^5*d^4*e*(d + e*x) + 75075*a*b^4*B*d^4*e*(d + e*x) - 60060*a*A*b^4*d^3*e^2*(d + e*x) - 1201
20*a^2*b^3*B*d^3*e^2*(d + e*x) + 90090*a^2*A*b^3*d^2*e^3*(d + e*x) + 90090*a^3*b^2*B*d^2*e^3*(d + e*x) - 60060
*a^3*A*b^2*d*e^4*(d + e*x) - 30030*a^4*b*B*d*e^4*(d + e*x) + 15015*a^4*A*b*e^5*(d + e*x) + 3003*a^5*B*e^5*(d +
e*x) + 27027*b^5*B*d^4*(d + e*x)^2 - 18018*A*b^5*d^3*e*(d + e*x)^2 - 90090*a*b^4*B*d^3*e*(d + e*x)^2 + 54054*
a*A*b^4*d^2*e^2*(d + e*x)^2 + 108108*a^2*b^3*B*d^2*e^2*(d + e*x)^2 - 54054*a^2*A*b^3*d*e^3*(d + e*x)^2 - 54054
*a^3*b^2*B*d*e^3*(d + e*x)^2 + 18018*a^3*A*b^2*e^4*(d + e*x)^2 + 9009*a^4*b*B*e^4*(d + e*x)^2 - 25740*b^5*B*d^
3*(d + e*x)^3 + 12870*A*b^5*d^2*e*(d + e*x)^3 + 64350*a*b^4*B*d^2*e*(d + e*x)^3 - 25740*a*A*b^4*d*e^2*(d + e*x
)^3 - 51480*a^2*b^3*B*d*e^2*(d + e*x)^3 + 12870*a^2*A*b^3*e^3*(d + e*x)^3 + 12870*a^3*b^2*B*e^3*(d + e*x)^3 +
15015*b^5*B*d^2*(d + e*x)^4 - 5005*A*b^5*d*e*(d + e*x)^4 - 25025*a*b^4*B*d*e*(d + e*x)^4 + 5005*a*A*b^4*e^2*(d
+ e*x)^4 + 10010*a^2*b^3*B*e^2*(d + e*x)^4 - 4914*b^5*B*d*(d + e*x)^5 + 819*A*b^5*e*(d + e*x)^5 + 4095*a*b^4*
B*e*(d + e*x)^5 + 693*b^5*B*(d + e*x)^6))/(9009*e^6*(a*e + b*e*x))

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fricas [A]  time = 0.44, size = 560, normalized size = 1.25 \begin {gather*} \frac {2 \, {\left (693 \, B b^{5} e^{6} x^{6} + 3072 \, B b^{5} d^{6} + 9009 \, A a^{5} e^{6} - 3328 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} e + 18304 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} e^{2} - 41184 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} e^{3} + 24024 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} e^{4} - 6006 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d e^{5} - 63 \, {\left (12 \, B b^{5} d e^{5} - 13 \, {\left (5 \, B a b^{4} + A b^{5}\right )} e^{6}\right )} x^{5} + 35 \, {\left (24 \, B b^{5} d^{2} e^{4} - 26 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d e^{5} + 143 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e^{6}\right )} x^{4} - 10 \, {\left (96 \, B b^{5} d^{3} e^{3} - 104 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e^{4} + 572 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{5} - 1287 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{6}\right )} x^{3} + 3 \, {\left (384 \, B b^{5} d^{4} e^{2} - 416 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} e^{3} + 2288 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{4} - 5148 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{5} + 3003 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e^{6}\right )} x^{2} - {\left (1536 \, B b^{5} d^{5} e - 1664 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} e^{2} + 9152 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e^{3} - 20592 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e^{4} + 12012 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d e^{5} - 3003 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} e^{6}\right )} x\right )} \sqrt {e x + d}}{9009 \, e^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

2/9009*(693*B*b^5*e^6*x^6 + 3072*B*b^5*d^6 + 9009*A*a^5*e^6 - 3328*(5*B*a*b^4 + A*b^5)*d^5*e + 18304*(2*B*a^2*
b^3 + A*a*b^4)*d^4*e^2 - 41184*(B*a^3*b^2 + A*a^2*b^3)*d^3*e^3 + 24024*(B*a^4*b + 2*A*a^3*b^2)*d^2*e^4 - 6006*
(B*a^5 + 5*A*a^4*b)*d*e^5 - 63*(12*B*b^5*d*e^5 - 13*(5*B*a*b^4 + A*b^5)*e^6)*x^5 + 35*(24*B*b^5*d^2*e^4 - 26*(
5*B*a*b^4 + A*b^5)*d*e^5 + 143*(2*B*a^2*b^3 + A*a*b^4)*e^6)*x^4 - 10*(96*B*b^5*d^3*e^3 - 104*(5*B*a*b^4 + A*b^
5)*d^2*e^4 + 572*(2*B*a^2*b^3 + A*a*b^4)*d*e^5 - 1287*(B*a^3*b^2 + A*a^2*b^3)*e^6)*x^3 + 3*(384*B*b^5*d^4*e^2
- 416*(5*B*a*b^4 + A*b^5)*d^3*e^3 + 2288*(2*B*a^2*b^3 + A*a*b^4)*d^2*e^4 - 5148*(B*a^3*b^2 + A*a^2*b^3)*d*e^5
+ 3003*(B*a^4*b + 2*A*a^3*b^2)*e^6)*x^2 - (1536*B*b^5*d^5*e - 1664*(5*B*a*b^4 + A*b^5)*d^4*e^2 + 9152*(2*B*a^2
*b^3 + A*a*b^4)*d^3*e^3 - 20592*(B*a^3*b^2 + A*a^2*b^3)*d^2*e^4 + 12012*(B*a^4*b + 2*A*a^3*b^2)*d*e^5 - 3003*(
B*a^5 + 5*A*a^4*b)*e^6)*x)*sqrt(e*x + d)/e^7

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giac [B]  time = 0.25, size = 758, normalized size = 1.69

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

2/9009*(3003*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*B*a^5*e^(-1)*sgn(b*x + a) + 15015*((x*e + d)^(3/2) - 3*sqrt
(x*e + d)*d)*A*a^4*b*e^(-1)*sgn(b*x + a) + 3003*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d
^2)*B*a^4*b*e^(-2)*sgn(b*x + a) + 6006*(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*e^(-2)*sgn(b*x + a) + 2574*(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*e^(-3)*sgn(b*x + a) + 2574*(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*e^(-3)*sgn(b*x + a) + 286*(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*e^(-4)*sgn(b*x + a) +
143*(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*e^(-4)*sgn(b*x + a) + 65*(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*e^(-5)*sgn(b*x
+ a) + 13*(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*e^(-5)*sgn(b*x + a) + 3*(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*e^(-6)*sgn(b*x + a) + 9009*sqrt(x*e + d)*A*a^5*sgn(b*x + a))*
e^(-1)

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maple [A]  time = 0.06, size = 689, normalized size = 1.54 \begin {gather*} \frac {2 \sqrt {e x +d}\, \left (693 B \,b^{5} e^{6} x^{6}+819 A \,b^{5} e^{6} x^{5}+4095 B a \,b^{4} e^{6} x^{5}-756 B \,b^{5} d \,e^{5} x^{5}+5005 A a \,b^{4} e^{6} x^{4}-910 A \,b^{5} d \,e^{5} x^{4}+10010 B \,a^{2} b^{3} e^{6} x^{4}-4550 B a \,b^{4} d \,e^{5} x^{4}+840 B \,b^{5} d^{2} e^{4} x^{4}+12870 A \,a^{2} b^{3} e^{6} x^{3}-5720 A a \,b^{4} d \,e^{5} x^{3}+1040 A \,b^{5} d^{2} e^{4} x^{3}+12870 B \,a^{3} b^{2} e^{6} x^{3}-11440 B \,a^{2} b^{3} d \,e^{5} x^{3}+5200 B a \,b^{4} d^{2} e^{4} x^{3}-960 B \,b^{5} d^{3} e^{3} x^{3}+18018 A \,a^{3} b^{2} e^{6} x^{2}-15444 A \,a^{2} b^{3} d \,e^{5} x^{2}+6864 A a \,b^{4} d^{2} e^{4} x^{2}-1248 A \,b^{5} d^{3} e^{3} x^{2}+9009 B \,a^{4} b \,e^{6} x^{2}-15444 B \,a^{3} b^{2} d \,e^{5} x^{2}+13728 B \,a^{2} b^{3} d^{2} e^{4} x^{2}-6240 B a \,b^{4} d^{3} e^{3} x^{2}+1152 B \,b^{5} d^{4} e^{2} x^{2}+15015 A \,a^{4} b \,e^{6} x -24024 A \,a^{3} b^{2} d \,e^{5} x +20592 A \,a^{2} b^{3} d^{2} e^{4} x -9152 A a \,b^{4} d^{3} e^{3} x +1664 A \,b^{5} d^{4} e^{2} x +3003 B \,a^{5} e^{6} x -12012 B \,a^{4} b d \,e^{5} x +20592 B \,a^{3} b^{2} d^{2} e^{4} x -18304 B \,a^{2} b^{3} d^{3} e^{3} x +8320 B a \,b^{4} d^{4} e^{2} x -1536 B \,b^{5} d^{5} e x +9009 A \,a^{5} e^{6}-30030 A \,a^{4} b d \,e^{5}+48048 A \,a^{3} b^{2} d^{2} e^{4}-41184 A \,a^{2} b^{3} d^{3} e^{3}+18304 A a \,b^{4} d^{4} e^{2}-3328 A \,b^{5} d^{5} e -6006 B \,a^{5} d \,e^{5}+24024 B \,a^{4} b \,d^{2} e^{4}-41184 B \,a^{3} b^{2} d^{3} e^{3}+36608 B \,a^{2} b^{3} d^{4} e^{2}-16640 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}}}{9009 \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)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^(1/2),x)

[Out]

2/9009*(e*x+d)^(1/2)*(693*B*b^5*e^6*x^6+819*A*b^5*e^6*x^5+4095*B*a*b^4*e^6*x^5-756*B*b^5*d*e^5*x^5+5005*A*a*b^
4*e^6*x^4-910*A*b^5*d*e^5*x^4+10010*B*a^2*b^3*e^6*x^4-4550*B*a*b^4*d*e^5*x^4+840*B*b^5*d^2*e^4*x^4+12870*A*a^2
*b^3*e^6*x^3-5720*A*a*b^4*d*e^5*x^3+1040*A*b^5*d^2*e^4*x^3+12870*B*a^3*b^2*e^6*x^3-11440*B*a^2*b^3*d*e^5*x^3+5
200*B*a*b^4*d^2*e^4*x^3-960*B*b^5*d^3*e^3*x^3+18018*A*a^3*b^2*e^6*x^2-15444*A*a^2*b^3*d*e^5*x^2+6864*A*a*b^4*d
^2*e^4*x^2-1248*A*b^5*d^3*e^3*x^2+9009*B*a^4*b*e^6*x^2-15444*B*a^3*b^2*d*e^5*x^2+13728*B*a^2*b^3*d^2*e^4*x^2-6
240*B*a*b^4*d^3*e^3*x^2+1152*B*b^5*d^4*e^2*x^2+15015*A*a^4*b*e^6*x-24024*A*a^3*b^2*d*e^5*x+20592*A*a^2*b^3*d^2
*e^4*x-9152*A*a*b^4*d^3*e^3*x+1664*A*b^5*d^4*e^2*x+3003*B*a^5*e^6*x-12012*B*a^4*b*d*e^5*x+20592*B*a^3*b^2*d^2*
e^4*x-18304*B*a^2*b^3*d^3*e^3*x+8320*B*a*b^4*d^4*e^2*x-1536*B*b^5*d^5*e*x+9009*A*a^5*e^6-30030*A*a^4*b*d*e^5+4
8048*A*a^3*b^2*d^2*e^4-41184*A*a^2*b^3*d^3*e^3+18304*A*a*b^4*d^4*e^2-3328*A*b^5*d^5*e-6006*B*a^5*d*e^5+24024*B
*a^4*b*d^2*e^4-41184*B*a^3*b^2*d^3*e^3+36608*B*a^2*b^3*d^4*e^2-16640*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 = 0.74, size = 758, normalized size = 1.69 \begin {gather*} \frac {2 \, {\left (63 \, b^{5} e^{6} x^{6} - 256 \, b^{5} d^{6} + 1408 \, a b^{4} d^{5} e - 3168 \, a^{2} b^{3} d^{4} e^{2} + 3696 \, a^{3} b^{2} d^{3} e^{3} - 2310 \, a^{4} b d^{2} e^{4} + 693 \, a^{5} d e^{5} - 7 \, {\left (b^{5} d e^{5} - 55 \, a b^{4} e^{6}\right )} x^{5} + 5 \, {\left (2 \, b^{5} d^{2} e^{4} - 11 \, a b^{4} d e^{5} + 198 \, a^{2} b^{3} e^{6}\right )} x^{4} - 2 \, {\left (8 \, b^{5} d^{3} e^{3} - 44 \, a b^{4} d^{2} e^{4} + 99 \, a^{2} b^{3} d e^{5} - 693 \, a^{3} b^{2} e^{6}\right )} x^{3} + {\left (32 \, b^{5} d^{4} e^{2} - 176 \, a b^{4} d^{3} e^{3} + 396 \, a^{2} b^{3} d^{2} e^{4} - 462 \, a^{3} b^{2} d e^{5} + 1155 \, a^{4} b e^{6}\right )} x^{2} - {\left (128 \, b^{5} d^{5} e - 704 \, a b^{4} d^{4} e^{2} + 1584 \, a^{2} b^{3} d^{3} e^{3} - 1848 \, a^{3} b^{2} d^{2} e^{4} + 1155 \, a^{4} b d e^{5} - 693 \, a^{5} e^{6}\right )} x\right )} A}{693 \, \sqrt {e x + d} e^{6}} + \frac {2 \, {\left (693 \, b^{5} e^{7} x^{7} + 3072 \, b^{5} d^{7} - 16640 \, a b^{4} d^{6} e + 36608 \, a^{2} b^{3} d^{5} e^{2} - 41184 \, a^{3} b^{2} d^{4} e^{3} + 24024 \, a^{4} b d^{3} e^{4} - 6006 \, a^{5} d^{2} e^{5} - 63 \, {\left (b^{5} d e^{6} - 65 \, a b^{4} e^{7}\right )} x^{6} + 7 \, {\left (12 \, b^{5} d^{2} e^{5} - 65 \, a b^{4} d e^{6} + 1430 \, a^{2} b^{3} e^{7}\right )} x^{5} - 10 \, {\left (12 \, b^{5} d^{3} e^{4} - 65 \, a b^{4} d^{2} e^{5} + 143 \, a^{2} b^{3} d e^{6} - 1287 \, a^{3} b^{2} e^{7}\right )} x^{4} + {\left (192 \, b^{5} d^{4} e^{3} - 1040 \, a b^{4} d^{3} e^{4} + 2288 \, a^{2} b^{3} d^{2} e^{5} - 2574 \, a^{3} b^{2} d e^{6} + 9009 \, a^{4} b e^{7}\right )} x^{3} - {\left (384 \, b^{5} d^{5} e^{2} - 2080 \, a b^{4} d^{4} e^{3} + 4576 \, a^{2} b^{3} d^{3} e^{4} - 5148 \, a^{3} b^{2} d^{2} e^{5} + 3003 \, a^{4} b d e^{6} - 3003 \, a^{5} e^{7}\right )} x^{2} + {\left (1536 \, b^{5} d^{6} e - 8320 \, a b^{4} d^{5} e^{2} + 18304 \, a^{2} b^{3} d^{4} e^{3} - 20592 \, a^{3} b^{2} d^{3} e^{4} + 12012 \, a^{4} b d^{2} e^{5} - 3003 \, a^{5} d e^{6}\right )} x\right )} B}{9009 \, \sqrt {e x + d} e^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

2/693*(63*b^5*e^6*x^6 - 256*b^5*d^6 + 1408*a*b^4*d^5*e - 3168*a^2*b^3*d^4*e^2 + 3696*a^3*b^2*d^3*e^3 - 2310*a^
4*b*d^2*e^4 + 693*a^5*d*e^5 - 7*(b^5*d*e^5 - 55*a*b^4*e^6)*x^5 + 5*(2*b^5*d^2*e^4 - 11*a*b^4*d*e^5 + 198*a^2*b
^3*e^6)*x^4 - 2*(8*b^5*d^3*e^3 - 44*a*b^4*d^2*e^4 + 99*a^2*b^3*d*e^5 - 693*a^3*b^2*e^6)*x^3 + (32*b^5*d^4*e^2
- 176*a*b^4*d^3*e^3 + 396*a^2*b^3*d^2*e^4 - 462*a^3*b^2*d*e^5 + 1155*a^4*b*e^6)*x^2 - (128*b^5*d^5*e - 704*a*b
^4*d^4*e^2 + 1584*a^2*b^3*d^3*e^3 - 1848*a^3*b^2*d^2*e^4 + 1155*a^4*b*d*e^5 - 693*a^5*e^6)*x)*A/(sqrt(e*x + d)
*e^6) + 2/9009*(693*b^5*e^7*x^7 + 3072*b^5*d^7 - 16640*a*b^4*d^6*e + 36608*a^2*b^3*d^5*e^2 - 41184*a^3*b^2*d^4
*e^3 + 24024*a^4*b*d^3*e^4 - 6006*a^5*d^2*e^5 - 63*(b^5*d*e^6 - 65*a*b^4*e^7)*x^6 + 7*(12*b^5*d^2*e^5 - 65*a*b
^4*d*e^6 + 1430*a^2*b^3*e^7)*x^5 - 10*(12*b^5*d^3*e^4 - 65*a*b^4*d^2*e^5 + 143*a^2*b^3*d*e^6 - 1287*a^3*b^2*e^
7)*x^4 + (192*b^5*d^4*e^3 - 1040*a*b^4*d^3*e^4 + 2288*a^2*b^3*d^2*e^5 - 2574*a^3*b^2*d*e^6 + 9009*a^4*b*e^7)*x
^3 - (384*b^5*d^5*e^2 - 2080*a*b^4*d^4*e^3 + 4576*a^2*b^3*d^3*e^4 - 5148*a^3*b^2*d^2*e^5 + 3003*a^4*b*d*e^6 -
3003*a^5*e^7)*x^2 + (1536*b^5*d^6*e - 8320*a*b^4*d^5*e^2 + 18304*a^2*b^3*d^4*e^3 - 20592*a^3*b^2*d^3*e^4 + 120
12*a^4*b*d^2*e^5 - 3003*a^5*d*e^6)*x)*B/(sqrt(e*x + d)*e^7)

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mupad [B]  time = 3.26, size = 826, normalized size = 1.84 \begin {gather*} \frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (\frac {2\,B\,b^4\,x^7}{13}+\frac {-12012\,B\,a^5\,d^2\,e^5+18018\,A\,a^5\,d\,e^6+48048\,B\,a^4\,b\,d^3\,e^4-60060\,A\,a^4\,b\,d^2\,e^5-82368\,B\,a^3\,b^2\,d^4\,e^3+96096\,A\,a^3\,b^2\,d^3\,e^4+73216\,B\,a^2\,b^3\,d^5\,e^2-82368\,A\,a^2\,b^3\,d^4\,e^3-33280\,B\,a\,b^4\,d^6\,e+36608\,A\,a\,b^4\,d^5\,e^2+6144\,B\,b^5\,d^7-6656\,A\,b^5\,d^6\,e}{9009\,b\,e^7}+\frac {x^3\,\left (18018\,B\,a^4\,b\,e^7-5148\,B\,a^3\,b^2\,d\,e^6+36036\,A\,a^3\,b^2\,e^7+4576\,B\,a^2\,b^3\,d^2\,e^5-5148\,A\,a^2\,b^3\,d\,e^6-2080\,B\,a\,b^4\,d^3\,e^4+2288\,A\,a\,b^4\,d^2\,e^5+384\,B\,b^5\,d^4\,e^3-416\,A\,b^5\,d^3\,e^4\right )}{9009\,b\,e^7}+\frac {x^4\,\left (25740\,B\,a^3\,b^2\,e^7-2860\,B\,a^2\,b^3\,d\,e^6+25740\,A\,a^2\,b^3\,e^7+1300\,B\,a\,b^4\,d^2\,e^5-1430\,A\,a\,b^4\,d\,e^6-240\,B\,b^5\,d^3\,e^4+260\,A\,b^5\,d^2\,e^5\right )}{9009\,b\,e^7}+\frac {2\,b^3\,x^6\,\left (13\,A\,b\,e+65\,B\,a\,e-B\,b\,d\right )}{143\,e}+\frac {x\,\left (-6006\,B\,a^5\,d\,e^6+18018\,A\,a^5\,e^7+24024\,B\,a^4\,b\,d^2\,e^5-30030\,A\,a^4\,b\,d\,e^6-41184\,B\,a^3\,b^2\,d^3\,e^4+48048\,A\,a^3\,b^2\,d^2\,e^5+36608\,B\,a^2\,b^3\,d^4\,e^3-41184\,A\,a^2\,b^3\,d^3\,e^4-16640\,B\,a\,b^4\,d^5\,e^2+18304\,A\,a\,b^4\,d^4\,e^3+3072\,B\,b^5\,d^6\,e-3328\,A\,b^5\,d^5\,e^2\right )}{9009\,b\,e^7}+\frac {x^2\,\left (6006\,B\,a^5\,e^7-6006\,B\,a^4\,b\,d\,e^6+30030\,A\,a^4\,b\,e^7+10296\,B\,a^3\,b^2\,d^2\,e^5-12012\,A\,a^3\,b^2\,d\,e^6-9152\,B\,a^2\,b^3\,d^3\,e^4+10296\,A\,a^2\,b^3\,d^2\,e^5+4160\,B\,a\,b^4\,d^4\,e^3-4576\,A\,a\,b^4\,d^3\,e^4-768\,B\,b^5\,d^5\,e^2+832\,A\,b^5\,d^4\,e^3\right )}{9009\,b\,e^7}+\frac {2\,b^2\,x^5\,\left (1430\,B\,a^2\,e^2-65\,B\,a\,b\,d\,e+715\,A\,a\,b\,e^2+12\,B\,b^2\,d^2-13\,A\,b^2\,d\,e\right )}{1287\,e^2}\right )}{x\,\sqrt {d+e\,x}+\frac {a\,\sqrt {d+e\,x}}{b}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2))/(d + e*x)^(1/2),x)

[Out]

((a^2 + b^2*x^2 + 2*a*b*x)^(1/2)*((2*B*b^4*x^7)/13 + (6144*B*b^5*d^7 + 18018*A*a^5*d*e^6 - 6656*A*b^5*d^6*e -
12012*B*a^5*d^2*e^5 + 36608*A*a*b^4*d^5*e^2 - 60060*A*a^4*b*d^2*e^5 + 48048*B*a^4*b*d^3*e^4 - 82368*A*a^2*b^3*
d^4*e^3 + 96096*A*a^3*b^2*d^3*e^4 + 73216*B*a^2*b^3*d^5*e^2 - 82368*B*a^3*b^2*d^4*e^3 - 33280*B*a*b^4*d^6*e)/(
9009*b*e^7) + (x^3*(18018*B*a^4*b*e^7 + 36036*A*a^3*b^2*e^7 - 416*A*b^5*d^3*e^4 + 384*B*b^5*d^4*e^3 + 2288*A*a
*b^4*d^2*e^5 - 5148*A*a^2*b^3*d*e^6 - 2080*B*a*b^4*d^3*e^4 - 5148*B*a^3*b^2*d*e^6 + 4576*B*a^2*b^3*d^2*e^5))/(
9009*b*e^7) + (x^4*(25740*A*a^2*b^3*e^7 + 25740*B*a^3*b^2*e^7 + 260*A*b^5*d^2*e^5 - 240*B*b^5*d^3*e^4 + 1300*B
*a*b^4*d^2*e^5 - 2860*B*a^2*b^3*d*e^6 - 1430*A*a*b^4*d*e^6))/(9009*b*e^7) + (2*b^3*x^6*(13*A*b*e + 65*B*a*e -
B*b*d))/(143*e) + (x*(18018*A*a^5*e^7 - 6006*B*a^5*d*e^6 + 3072*B*b^5*d^6*e - 3328*A*b^5*d^5*e^2 + 18304*A*a*b
^4*d^4*e^3 - 16640*B*a*b^4*d^5*e^2 + 24024*B*a^4*b*d^2*e^5 - 41184*A*a^2*b^3*d^3*e^4 + 48048*A*a^3*b^2*d^2*e^5
+ 36608*B*a^2*b^3*d^4*e^3 - 41184*B*a^3*b^2*d^3*e^4 - 30030*A*a^4*b*d*e^6))/(9009*b*e^7) + (x^2*(6006*B*a^5*e
^7 + 30030*A*a^4*b*e^7 + 832*A*b^5*d^4*e^3 - 768*B*b^5*d^5*e^2 - 4576*A*a*b^4*d^3*e^4 - 12012*A*a^3*b^2*d*e^6
+ 4160*B*a*b^4*d^4*e^3 + 10296*A*a^2*b^3*d^2*e^5 - 9152*B*a^2*b^3*d^3*e^4 + 10296*B*a^3*b^2*d^2*e^5 - 6006*B*a
^4*b*d*e^6))/(9009*b*e^7) + (2*b^2*x^5*(1430*B*a^2*e^2 + 12*B*b^2*d^2 + 715*A*a*b*e^2 - 13*A*b^2*d*e - 65*B*a*
b*d*e))/(1287*e^2)))/(x*(d + e*x)^(1/2) + (a*(d + e*x)^(1/2))/b)

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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)*(b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**(1/2),x)

[Out]

Timed out

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