3.297 \(\int x^{7/2} (a+b x)^2 (A+B x) \, dx\)

Optimal. Leaf size=63 \[ \frac{2}{9} a^2 A x^{9/2}+\frac{2}{13} b x^{13/2} (2 a B+A b)+\frac{2}{11} a x^{11/2} (a B+2 A b)+\frac{2}{15} b^2 B x^{15/2} \]

[Out]

(2*a^2*A*x^(9/2))/9 + (2*a*(2*A*b + a*B)*x^(11/2))/11 + (2*b*(A*b + 2*a*B)*x^(13
/2))/13 + (2*b^2*B*x^(15/2))/15

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Rubi [A]  time = 0.0884939, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ \frac{2}{9} a^2 A x^{9/2}+\frac{2}{13} b x^{13/2} (2 a B+A b)+\frac{2}{11} a x^{11/2} (a B+2 A b)+\frac{2}{15} b^2 B x^{15/2} \]

Antiderivative was successfully verified.

[In]  Int[x^(7/2)*(a + b*x)^2*(A + B*x),x]

[Out]

(2*a^2*A*x^(9/2))/9 + (2*a*(2*A*b + a*B)*x^(11/2))/11 + (2*b*(A*b + 2*a*B)*x^(13
/2))/13 + (2*b^2*B*x^(15/2))/15

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Rubi in Sympy [A]  time = 9.28176, size = 63, normalized size = 1. \[ \frac{2 A a^{2} x^{\frac{9}{2}}}{9} + \frac{2 B b^{2} x^{\frac{15}{2}}}{15} + \frac{2 a x^{\frac{11}{2}} \left (2 A b + B a\right )}{11} + \frac{2 b x^{\frac{13}{2}} \left (A b + 2 B a\right )}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(7/2)*(b*x+a)**2*(B*x+A),x)

[Out]

2*A*a**2*x**(9/2)/9 + 2*B*b**2*x**(15/2)/15 + 2*a*x**(11/2)*(2*A*b + B*a)/11 + 2
*b*x**(13/2)*(A*b + 2*B*a)/13

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Mathematica [A]  time = 0.0316591, size = 51, normalized size = 0.81 \[ \frac{2 x^{9/2} \left (715 a^2 A+495 b x^2 (2 a B+A b)+585 a x (a B+2 A b)+429 b^2 B x^3\right )}{6435} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(7/2)*(a + b*x)^2*(A + B*x),x]

[Out]

(2*x^(9/2)*(715*a^2*A + 585*a*(2*A*b + a*B)*x + 495*b*(A*b + 2*a*B)*x^2 + 429*b^
2*B*x^3))/6435

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Maple [A]  time = 0.007, size = 52, normalized size = 0.8 \[{\frac{858\,B{b}^{2}{x}^{3}+990\,A{b}^{2}{x}^{2}+1980\,B{x}^{2}ab+2340\,aAbx+1170\,{a}^{2}Bx+1430\,{a}^{2}A}{6435}{x}^{{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(7/2)*(b*x+a)^2*(B*x+A),x)

[Out]

2/6435*x^(9/2)*(429*B*b^2*x^3+495*A*b^2*x^2+990*B*a*b*x^2+1170*A*a*b*x+585*B*a^2
*x+715*A*a^2)

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Maxima [A]  time = 1.339, size = 69, normalized size = 1.1 \[ \frac{2}{15} \, B b^{2} x^{\frac{15}{2}} + \frac{2}{9} \, A a^{2} x^{\frac{9}{2}} + \frac{2}{13} \,{\left (2 \, B a b + A b^{2}\right )} x^{\frac{13}{2}} + \frac{2}{11} \,{\left (B a^{2} + 2 \, A a b\right )} x^{\frac{11}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^2*x^(7/2),x, algorithm="maxima")

[Out]

2/15*B*b^2*x^(15/2) + 2/9*A*a^2*x^(9/2) + 2/13*(2*B*a*b + A*b^2)*x^(13/2) + 2/11
*(B*a^2 + 2*A*a*b)*x^(11/2)

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Fricas [A]  time = 0.205497, size = 76, normalized size = 1.21 \[ \frac{2}{6435} \,{\left (429 \, B b^{2} x^{7} + 715 \, A a^{2} x^{4} + 495 \,{\left (2 \, B a b + A b^{2}\right )} x^{6} + 585 \,{\left (B a^{2} + 2 \, A a b\right )} x^{5}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^2*x^(7/2),x, algorithm="fricas")

[Out]

2/6435*(429*B*b^2*x^7 + 715*A*a^2*x^4 + 495*(2*B*a*b + A*b^2)*x^6 + 585*(B*a^2 +
2*A*a*b)*x^5)*sqrt(x)

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Sympy [A]  time = 22.6499, size = 80, normalized size = 1.27 \[ \frac{2 A a^{2} x^{\frac{9}{2}}}{9} + \frac{4 A a b x^{\frac{11}{2}}}{11} + \frac{2 A b^{2} x^{\frac{13}{2}}}{13} + \frac{2 B a^{2} x^{\frac{11}{2}}}{11} + \frac{4 B a b x^{\frac{13}{2}}}{13} + \frac{2 B b^{2} x^{\frac{15}{2}}}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(7/2)*(b*x+a)**2*(B*x+A),x)

[Out]

2*A*a**2*x**(9/2)/9 + 4*A*a*b*x**(11/2)/11 + 2*A*b**2*x**(13/2)/13 + 2*B*a**2*x*
*(11/2)/11 + 4*B*a*b*x**(13/2)/13 + 2*B*b**2*x**(15/2)/15

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GIAC/XCAS [A]  time = 0.253962, size = 72, normalized size = 1.14 \[ \frac{2}{15} \, B b^{2} x^{\frac{15}{2}} + \frac{4}{13} \, B a b x^{\frac{13}{2}} + \frac{2}{13} \, A b^{2} x^{\frac{13}{2}} + \frac{2}{11} \, B a^{2} x^{\frac{11}{2}} + \frac{4}{11} \, A a b x^{\frac{11}{2}} + \frac{2}{9} \, A a^{2} x^{\frac{9}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^2*x^(7/2),x, algorithm="giac")

[Out]

2/15*B*b^2*x^(15/2) + 4/13*B*a*b*x^(13/2) + 2/13*A*b^2*x^(13/2) + 2/11*B*a^2*x^(
11/2) + 4/11*A*a*b*x^(11/2) + 2/9*A*a^2*x^(9/2)