3.802 \(\int x^{3/2} (A+B x) (a^2+2 a b x+b^2 x^2)^{5/2} \, dx\)

Optimal. Leaf size=320 \[ \frac{2 b^4 x^{15/2} \sqrt{a^2+2 a b x+b^2 x^2} (5 a B+A b)}{15 (a+b x)}+\frac{10 a b^3 x^{13/2} \sqrt{a^2+2 a b x+b^2 x^2} (2 a B+A b)}{13 (a+b x)}+\frac{20 a^2 b^2 x^{11/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{11 (a+b x)}+\frac{10 a^3 b x^{9/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+2 A b)}{9 (a+b x)}+\frac{2 a^4 x^{7/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+5 A b)}{7 (a+b x)}+\frac{2 a^5 A x^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac{2 b^5 B x^{17/2} \sqrt{a^2+2 a b x+b^2 x^2}}{17 (a+b x)} \]

[Out]

(2*a^5*A*x^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*(a + b*x)) + (2*a^4*(5*A*b + a*B)*x^(7/2)*Sqrt[a^2 + 2*a*b*
x + b^2*x^2])/(7*(a + b*x)) + (10*a^3*b*(2*A*b + a*B)*x^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*(a + b*x)) + (
20*a^2*b^2*(A*b + a*B)*x^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*(a + b*x)) + (10*a*b^3*(A*b + 2*a*B)*x^(13/
2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*(a + b*x)) + (2*b^4*(A*b + 5*a*B)*x^(15/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]
)/(15*(a + b*x)) + (2*b^5*B*x^(17/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(17*(a + b*x))

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Rubi [A]  time = 0.12043, antiderivative size = 320, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {770, 76} \[ \frac{2 b^4 x^{15/2} \sqrt{a^2+2 a b x+b^2 x^2} (5 a B+A b)}{15 (a+b x)}+\frac{10 a b^3 x^{13/2} \sqrt{a^2+2 a b x+b^2 x^2} (2 a B+A b)}{13 (a+b x)}+\frac{20 a^2 b^2 x^{11/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{11 (a+b x)}+\frac{10 a^3 b x^{9/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+2 A b)}{9 (a+b x)}+\frac{2 a^4 x^{7/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+5 A b)}{7 (a+b x)}+\frac{2 a^5 A x^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac{2 b^5 B x^{17/2} \sqrt{a^2+2 a b x+b^2 x^2}}{17 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[x^(3/2)*(A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(2*a^5*A*x^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*(a + b*x)) + (2*a^4*(5*A*b + a*B)*x^(7/2)*Sqrt[a^2 + 2*a*b*
x + b^2*x^2])/(7*(a + b*x)) + (10*a^3*b*(2*A*b + a*B)*x^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*(a + b*x)) + (
20*a^2*b^2*(A*b + a*B)*x^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*(a + b*x)) + (10*a*b^3*(A*b + 2*a*B)*x^(13/
2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*(a + b*x)) + (2*b^4*(A*b + 5*a*B)*x^(15/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]
)/(15*(a + b*x)) + (2*b^5*B*x^(17/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(17*(a + b*x))

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]

Rule 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

Rubi steps

\begin{align*} \int x^{3/2} (A+B x) \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 x^{3/2} \left (a b+b^2 x\right )^5 (A+B 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 (a^5 A b^5 x^{3/2}+a^4 b^5 (5 A b+a B) x^{5/2}+5 a^3 b^6 (2 A b+a B) x^{7/2}+10 a^2 b^7 (A b+a B) x^{9/2}+5 a b^8 (A b+2 a B) x^{11/2}+b^9 (A b+5 a B) x^{13/2}+b^{10} B x^{15/2}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac{2 a^5 A x^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac{2 a^4 (5 A b+a B) x^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}{7 (a+b x)}+\frac{10 a^3 b (2 A b+a B) x^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}}{9 (a+b x)}+\frac{20 a^2 b^2 (A b+a B) x^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}}{11 (a+b x)}+\frac{10 a b^3 (A b+2 a B) x^{13/2} \sqrt{a^2+2 a b x+b^2 x^2}}{13 (a+b x)}+\frac{2 b^4 (A b+5 a B) x^{15/2} \sqrt{a^2+2 a b x+b^2 x^2}}{15 (a+b x)}+\frac{2 b^5 B x^{17/2} \sqrt{a^2+2 a b x+b^2 x^2}}{17 (a+b x)}\\ \end{align*}

Mathematica [A]  time = 0.0520219, size = 127, normalized size = 0.4 \[ \frac{2 x^{5/2} \sqrt{(a+b x)^2} \left (77350 a^3 b^2 x^2 (11 A+9 B x)+53550 a^2 b^3 x^3 (13 A+11 B x)+60775 a^4 b x (9 A+7 B x)+21879 a^5 (7 A+5 B x)+19635 a b^4 x^4 (15 A+13 B x)+3003 b^5 x^5 (17 A+15 B x)\right )}{765765 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(3/2)*(A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(2*x^(5/2)*Sqrt[(a + b*x)^2]*(21879*a^5*(7*A + 5*B*x) + 60775*a^4*b*x*(9*A + 7*B*x) + 77350*a^3*b^2*x^2*(11*A
+ 9*B*x) + 53550*a^2*b^3*x^3*(13*A + 11*B*x) + 19635*a*b^4*x^4*(15*A + 13*B*x) + 3003*b^5*x^5*(17*A + 15*B*x))
)/(765765*(a + b*x))

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Maple [A]  time = 0.007, size = 140, normalized size = 0.4 \begin{align*}{\frac{90090\,B{b}^{5}{x}^{6}+102102\,A{x}^{5}{b}^{5}+510510\,B{x}^{5}a{b}^{4}+589050\,A{x}^{4}a{b}^{4}+1178100\,B{x}^{4}{a}^{2}{b}^{3}+1392300\,A{x}^{3}{a}^{2}{b}^{3}+1392300\,B{x}^{3}{a}^{3}{b}^{2}+1701700\,A{x}^{2}{a}^{3}{b}^{2}+850850\,B{x}^{2}{a}^{4}b+1093950\,A{a}^{4}bx+218790\,B{a}^{5}x+306306\,A{a}^{5}}{765765\, \left ( bx+a \right ) ^{5}}{x}^{{\frac{5}{2}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(3/2)*(B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

2/765765*x^(5/2)*(45045*B*b^5*x^6+51051*A*b^5*x^5+255255*B*a*b^4*x^5+294525*A*a*b^4*x^4+589050*B*a^2*b^3*x^4+6
96150*A*a^2*b^3*x^3+696150*B*a^3*b^2*x^3+850850*A*a^3*b^2*x^2+425425*B*a^4*b*x^2+546975*A*a^4*b*x+109395*B*a^5
*x+153153*A*a^5)*((b*x+a)^2)^(5/2)/(b*x+a)^5

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Maxima [A]  time = 1.08868, size = 325, normalized size = 1.02 \begin{align*} \frac{2}{45045} \,{\left (231 \,{\left (13 \, b^{5} x^{2} + 15 \, a b^{4} x\right )} x^{\frac{11}{2}} + 1260 \,{\left (11 \, a b^{4} x^{2} + 13 \, a^{2} b^{3} x\right )} x^{\frac{9}{2}} + 2730 \,{\left (9 \, a^{2} b^{3} x^{2} + 11 \, a^{3} b^{2} x\right )} x^{\frac{7}{2}} + 2860 \,{\left (7 \, a^{3} b^{2} x^{2} + 9 \, a^{4} b x\right )} x^{\frac{5}{2}} + 1287 \,{\left (5 \, a^{4} b x^{2} + 7 \, a^{5} x\right )} x^{\frac{3}{2}}\right )} A + \frac{2}{765765} \,{\left (3003 \,{\left (15 \, b^{5} x^{2} + 17 \, a b^{4} x\right )} x^{\frac{13}{2}} + 15708 \,{\left (13 \, a b^{4} x^{2} + 15 \, a^{2} b^{3} x\right )} x^{\frac{11}{2}} + 32130 \,{\left (11 \, a^{2} b^{3} x^{2} + 13 \, a^{3} b^{2} x\right )} x^{\frac{9}{2}} + 30940 \,{\left (9 \, a^{3} b^{2} x^{2} + 11 \, a^{4} b x\right )} x^{\frac{7}{2}} + 12155 \,{\left (7 \, a^{4} b x^{2} + 9 \, a^{5} x\right )} x^{\frac{5}{2}}\right )} B \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(3/2)*(B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="maxima")

[Out]

2/45045*(231*(13*b^5*x^2 + 15*a*b^4*x)*x^(11/2) + 1260*(11*a*b^4*x^2 + 13*a^2*b^3*x)*x^(9/2) + 2730*(9*a^2*b^3
*x^2 + 11*a^3*b^2*x)*x^(7/2) + 2860*(7*a^3*b^2*x^2 + 9*a^4*b*x)*x^(5/2) + 1287*(5*a^4*b*x^2 + 7*a^5*x)*x^(3/2)
)*A + 2/765765*(3003*(15*b^5*x^2 + 17*a*b^4*x)*x^(13/2) + 15708*(13*a*b^4*x^2 + 15*a^2*b^3*x)*x^(11/2) + 32130
*(11*a^2*b^3*x^2 + 13*a^3*b^2*x)*x^(9/2) + 30940*(9*a^3*b^2*x^2 + 11*a^4*b*x)*x^(7/2) + 12155*(7*a^4*b*x^2 + 9
*a^5*x)*x^(5/2))*B

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Fricas [A]  time = 1.27652, size = 312, normalized size = 0.98 \begin{align*} \frac{2}{765765} \,{\left (45045 \, B b^{5} x^{8} + 153153 \, A a^{5} x^{2} + 51051 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{7} + 294525 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{6} + 696150 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{5} + 425425 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 109395 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(3/2)*(B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")

[Out]

2/765765*(45045*B*b^5*x^8 + 153153*A*a^5*x^2 + 51051*(5*B*a*b^4 + A*b^5)*x^7 + 294525*(2*B*a^2*b^3 + A*a*b^4)*
x^6 + 696150*(B*a^3*b^2 + A*a^2*b^3)*x^5 + 425425*(B*a^4*b + 2*A*a^3*b^2)*x^4 + 109395*(B*a^5 + 5*A*a^4*b)*x^3
)*sqrt(x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(3/2)*(B*x+A)*(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Timed out

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Giac [A]  time = 1.17518, size = 266, normalized size = 0.83 \begin{align*} \frac{2}{17} \, B b^{5} x^{\frac{17}{2}} \mathrm{sgn}\left (b x + a\right ) + \frac{2}{3} \, B a b^{4} x^{\frac{15}{2}} \mathrm{sgn}\left (b x + a\right ) + \frac{2}{15} \, A b^{5} x^{\frac{15}{2}} \mathrm{sgn}\left (b x + a\right ) + \frac{20}{13} \, B a^{2} b^{3} x^{\frac{13}{2}} \mathrm{sgn}\left (b x + a\right ) + \frac{10}{13} \, A a b^{4} x^{\frac{13}{2}} \mathrm{sgn}\left (b x + a\right ) + \frac{20}{11} \, B a^{3} b^{2} x^{\frac{11}{2}} \mathrm{sgn}\left (b x + a\right ) + \frac{20}{11} \, A a^{2} b^{3} x^{\frac{11}{2}} \mathrm{sgn}\left (b x + a\right ) + \frac{10}{9} \, B a^{4} b x^{\frac{9}{2}} \mathrm{sgn}\left (b x + a\right ) + \frac{20}{9} \, A a^{3} b^{2} x^{\frac{9}{2}} \mathrm{sgn}\left (b x + a\right ) + \frac{2}{7} \, B a^{5} x^{\frac{7}{2}} \mathrm{sgn}\left (b x + a\right ) + \frac{10}{7} \, A a^{4} b x^{\frac{7}{2}} \mathrm{sgn}\left (b x + a\right ) + \frac{2}{5} \, A a^{5} x^{\frac{5}{2}} \mathrm{sgn}\left (b x + a\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(3/2)*(B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")

[Out]

2/17*B*b^5*x^(17/2)*sgn(b*x + a) + 2/3*B*a*b^4*x^(15/2)*sgn(b*x + a) + 2/15*A*b^5*x^(15/2)*sgn(b*x + a) + 20/1
3*B*a^2*b^3*x^(13/2)*sgn(b*x + a) + 10/13*A*a*b^4*x^(13/2)*sgn(b*x + a) + 20/11*B*a^3*b^2*x^(11/2)*sgn(b*x + a
) + 20/11*A*a^2*b^3*x^(11/2)*sgn(b*x + a) + 10/9*B*a^4*b*x^(9/2)*sgn(b*x + a) + 20/9*A*a^3*b^2*x^(9/2)*sgn(b*x
+ a) + 2/7*B*a^5*x^(7/2)*sgn(b*x + a) + 10/7*A*a^4*b*x^(7/2)*sgn(b*x + a) + 2/5*A*a^5*x^(5/2)*sgn(b*x + a)