3.801 \(\int x^{5/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=320 \[ \frac{20 a^2 b^2 x^{13/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{13 (a+b x)}+\frac{2 b^4 x^{17/2} \sqrt{a^2+2 a b x+b^2 x^2} (5 a B+A b)}{17 (a+b x)}+\frac{2 a b^3 x^{15/2} \sqrt{a^2+2 a b x+b^2 x^2} (2 a B+A b)}{3 (a+b x)}+\frac{2 b^5 B x^{19/2} \sqrt{a^2+2 a b x+b^2 x^2}}{19 (a+b x)}+\frac{2 a^5 A x^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}{7 (a+b x)}+\frac{2 a^4 x^{9/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+5 A b)}{9 (a+b x)}+\frac{10 a^3 b x^{11/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+2 A b)}{11 (a+b x)} \]

[Out]

(2*a^5*A*x^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*(a + b*x)) + (2*a^4*(5*A*b +
a*B)*x^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*(a + b*x)) + (10*a^3*b*(2*A*b + a
*B)*x^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*(a + b*x)) + (20*a^2*b^2*(A*b +
a*B)*x^(13/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*(a + b*x)) + (2*a*b^3*(A*b + 2*
a*B)*x^(15/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*(a + b*x)) + (2*b^4*(A*b + 5*a*B
)*x^(17/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(17*(a + b*x)) + (2*b^5*B*x^(19/2)*Sqr
t[a^2 + 2*a*b*x + b^2*x^2])/(19*(a + b*x))

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Rubi [A]  time = 0.343795, 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 \[ \frac{20 a^2 b^2 x^{13/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{13 (a+b x)}+\frac{2 b^4 x^{17/2} \sqrt{a^2+2 a b x+b^2 x^2} (5 a B+A b)}{17 (a+b x)}+\frac{2 a b^3 x^{15/2} \sqrt{a^2+2 a b x+b^2 x^2} (2 a B+A b)}{3 (a+b x)}+\frac{2 b^5 B x^{19/2} \sqrt{a^2+2 a b x+b^2 x^2}}{19 (a+b x)}+\frac{2 a^5 A x^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}{7 (a+b x)}+\frac{2 a^4 x^{9/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+5 A b)}{9 (a+b x)}+\frac{10 a^3 b x^{11/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+2 A b)}{11 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(2*a^5*A*x^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*(a + b*x)) + (2*a^4*(5*A*b +
a*B)*x^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*(a + b*x)) + (10*a^3*b*(2*A*b + a
*B)*x^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*(a + b*x)) + (20*a^2*b^2*(A*b +
a*B)*x^(13/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*(a + b*x)) + (2*a*b^3*(A*b + 2*
a*B)*x^(15/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*(a + b*x)) + (2*b^4*(A*b + 5*a*B
)*x^(17/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(17*(a + b*x)) + (2*b^5*B*x^(19/2)*Sqr
t[a^2 + 2*a*b*x + b^2*x^2])/(19*(a + b*x))

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Rubi in Sympy [A]  time = 35.4712, size = 320, normalized size = 1. \[ \frac{B x^{\frac{7}{2}} \left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{19 b} + \frac{512 a^{5} x^{\frac{7}{2}} \left (19 A b - 7 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{2909907 b \left (a + b x\right )} + \frac{256 a^{4} x^{\frac{7}{2}} \left (19 A b - 7 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{415701 b} + \frac{64 a^{3} x^{\frac{7}{2}} \left (3 a + 3 b x\right ) \left (19 A b - 7 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{138567 b} + \frac{32 a^{2} x^{\frac{7}{2}} \left (19 A b - 7 B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{12597 b} + \frac{4 a x^{\frac{7}{2}} \left (5 a + 5 b x\right ) \left (19 A b - 7 B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{4845 b} + \frac{2 x^{\frac{7}{2}} \left (19 A b - 7 B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{323 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(5/2)*(B*x+A)*(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

B*x**(7/2)*(2*a + 2*b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(19*b) + 512*a**5*x
**(7/2)*(19*A*b - 7*B*a)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(2909907*b*(a + b*x))
+ 256*a**4*x**(7/2)*(19*A*b - 7*B*a)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(415701*b)
+ 64*a**3*x**(7/2)*(3*a + 3*b*x)*(19*A*b - 7*B*a)*sqrt(a**2 + 2*a*b*x + b**2*x*
*2)/(138567*b) + 32*a**2*x**(7/2)*(19*A*b - 7*B*a)*(a**2 + 2*a*b*x + b**2*x**2)*
*(3/2)/(12597*b) + 4*a*x**(7/2)*(5*a + 5*b*x)*(19*A*b - 7*B*a)*(a**2 + 2*a*b*x +
b**2*x**2)**(3/2)/(4845*b) + 2*x**(7/2)*(19*A*b - 7*B*a)*(a**2 + 2*a*b*x + b**2
*x**2)**(5/2)/(323*b)

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Mathematica [A]  time = 0.0990693, size = 127, normalized size = 0.4 \[ \frac{2 x^{7/2} \sqrt{(a+b x)^2} \left (46189 a^5 (9 A+7 B x)+146965 a^4 b x (11 A+9 B x)+203490 a^3 b^2 x^2 (13 A+11 B x)+149226 a^2 b^3 x^3 (15 A+13 B x)+57057 a b^4 x^4 (17 A+15 B x)+9009 b^5 x^5 (19 A+17 B x)\right )}{2909907 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(2*x^(7/2)*Sqrt[(a + b*x)^2]*(46189*a^5*(9*A + 7*B*x) + 146965*a^4*b*x*(11*A + 9
*B*x) + 203490*a^3*b^2*x^2*(13*A + 11*B*x) + 149226*a^2*b^3*x^3*(15*A + 13*B*x)
+ 57057*a*b^4*x^4*(17*A + 15*B*x) + 9009*b^5*x^5*(19*A + 17*B*x)))/(2909907*(a +
b*x))

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Maple [A]  time = 0.011, size = 140, normalized size = 0.4 \[{\frac{306306\,B{b}^{5}{x}^{6}+342342\,A{x}^{5}{b}^{5}+1711710\,B{x}^{5}a{b}^{4}+1939938\,A{x}^{4}a{b}^{4}+3879876\,B{x}^{4}{a}^{2}{b}^{3}+4476780\,A{x}^{3}{a}^{2}{b}^{3}+4476780\,B{x}^{3}{a}^{3}{b}^{2}+5290740\,A{x}^{2}{a}^{3}{b}^{2}+2645370\,B{x}^{2}{a}^{4}b+3233230\,Ax{a}^{4}b+646646\,Bx{a}^{5}+831402\,A{a}^{5}}{2909907\, \left ( bx+a \right ) ^{5}}{x}^{{\frac{7}{2}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/2909907*x^(7/2)*(153153*B*b^5*x^6+171171*A*b^5*x^5+855855*B*a*b^4*x^5+969969*A
*a*b^4*x^4+1939938*B*a^2*b^3*x^4+2238390*A*a^2*b^3*x^3+2238390*B*a^3*b^2*x^3+264
5370*A*a^3*b^2*x^2+1322685*B*a^4*b*x^2+1616615*A*a^4*b*x+323323*B*a^5*x+415701*A
*a^5)*((b*x+a)^2)^(5/2)/(b*x+a)^5

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Maxima [A]  time = 0.707189, size = 325, normalized size = 1.02 \[ \frac{2}{765765} \,{\left (3003 \,{\left (15 \, b^{5} x^{2} + 17 \, a b^{4} x\right )} x^{\frac{13}{2}} + 15708 \,{\left (13 \, a b^{4} x^{2} + 15 \, a^{2} b^{3} x\right )} x^{\frac{11}{2}} + 32130 \,{\left (11 \, a^{2} b^{3} x^{2} + 13 \, a^{3} b^{2} x\right )} x^{\frac{9}{2}} + 30940 \,{\left (9 \, a^{3} b^{2} x^{2} + 11 \, a^{4} b x\right )} x^{\frac{7}{2}} + 12155 \,{\left (7 \, a^{4} b x^{2} + 9 \, a^{5} x\right )} x^{\frac{5}{2}}\right )} A + \frac{2}{2078505} \,{\left (6435 \,{\left (17 \, b^{5} x^{2} + 19 \, a b^{4} x\right )} x^{\frac{15}{2}} + 32604 \,{\left (15 \, a b^{4} x^{2} + 17 \, a^{2} b^{3} x\right )} x^{\frac{13}{2}} + 63954 \,{\left (13 \, a^{2} b^{3} x^{2} + 15 \, a^{3} b^{2} x\right )} x^{\frac{11}{2}} + 58140 \,{\left (11 \, a^{3} b^{2} x^{2} + 13 \, a^{4} b x\right )} x^{\frac{9}{2}} + 20995 \,{\left (9 \, a^{4} b x^{2} + 11 \, a^{5} x\right )} x^{\frac{7}{2}}\right )} B \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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))*A + 2/207
8505*(6435*(17*b^5*x^2 + 19*a*b^4*x)*x^(15/2) + 32604*(15*a*b^4*x^2 + 17*a^2*b^3
*x)*x^(13/2) + 63954*(13*a^2*b^3*x^2 + 15*a^3*b^2*x)*x^(11/2) + 58140*(11*a^3*b^
2*x^2 + 13*a^4*b*x)*x^(9/2) + 20995*(9*a^4*b*x^2 + 11*a^5*x)*x^(7/2))*B

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Fricas [A]  time = 0.281698, size = 167, normalized size = 0.52 \[ \frac{2}{2909907} \,{\left (153153 \, B b^{5} x^{9} + 415701 \, A a^{5} x^{3} + 171171 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{8} + 969969 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{7} + 2238390 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 1322685 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{5} + 323323 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{4}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/2909907*(153153*B*b^5*x^9 + 415701*A*a^5*x^3 + 171171*(5*B*a*b^4 + A*b^5)*x^8
+ 969969*(2*B*a^2*b^3 + A*a*b^4)*x^7 + 2238390*(B*a^3*b^2 + A*a^2*b^3)*x^6 + 132
2685*(B*a^4*b + 2*A*a^3*b^2)*x^5 + 323323*(B*a^5 + 5*A*a^4*b)*x^4)*sqrt(x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(5/2)*(B*x+A)*(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.274964, size = 266, normalized size = 0.83 \[ \frac{2}{19} \, B b^{5} x^{\frac{19}{2}}{\rm sign}\left (b x + a\right ) + \frac{10}{17} \, B a b^{4} x^{\frac{17}{2}}{\rm sign}\left (b x + a\right ) + \frac{2}{17} \, A b^{5} x^{\frac{17}{2}}{\rm sign}\left (b x + a\right ) + \frac{4}{3} \, B a^{2} b^{3} x^{\frac{15}{2}}{\rm sign}\left (b x + a\right ) + \frac{2}{3} \, A a b^{4} x^{\frac{15}{2}}{\rm sign}\left (b x + a\right ) + \frac{20}{13} \, B a^{3} b^{2} x^{\frac{13}{2}}{\rm sign}\left (b x + a\right ) + \frac{20}{13} \, A a^{2} b^{3} x^{\frac{13}{2}}{\rm sign}\left (b x + a\right ) + \frac{10}{11} \, B a^{4} b x^{\frac{11}{2}}{\rm sign}\left (b x + a\right ) + \frac{20}{11} \, A a^{3} b^{2} x^{\frac{11}{2}}{\rm sign}\left (b x + a\right ) + \frac{2}{9} \, B a^{5} x^{\frac{9}{2}}{\rm sign}\left (b x + a\right ) + \frac{10}{9} \, A a^{4} b x^{\frac{9}{2}}{\rm sign}\left (b x + a\right ) + \frac{2}{7} \, A a^{5} x^{\frac{7}{2}}{\rm sign}\left (b x + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/19*B*b^5*x^(19/2)*sign(b*x + a) + 10/17*B*a*b^4*x^(17/2)*sign(b*x + a) + 2/17*
A*b^5*x^(17/2)*sign(b*x + a) + 4/3*B*a^2*b^3*x^(15/2)*sign(b*x + a) + 2/3*A*a*b^
4*x^(15/2)*sign(b*x + a) + 20/13*B*a^3*b^2*x^(13/2)*sign(b*x + a) + 20/13*A*a^2*
b^3*x^(13/2)*sign(b*x + a) + 10/11*B*a^4*b*x^(11/2)*sign(b*x + a) + 20/11*A*a^3*
b^2*x^(11/2)*sign(b*x + a) + 2/9*B*a^5*x^(9/2)*sign(b*x + a) + 10/9*A*a^4*b*x^(9
/2)*sign(b*x + a) + 2/7*A*a^5*x^(7/2)*sign(b*x + a)