3.497 \(\int \frac{(a+b x)^{5/2} (A+B x)}{x^{11/2}} \, dx\)

Optimal. Leaf size=53 \[ \frac{2 (a+b x)^{7/2} (2 A b-9 a B)}{63 a^2 x^{7/2}}-\frac{2 A (a+b x)^{7/2}}{9 a x^{9/2}} \]

[Out]

(-2*A*(a + b*x)^(7/2))/(9*a*x^(9/2)) + (2*(2*A*b - 9*a*B)*(a + b*x)^(7/2))/(63*a
^2*x^(7/2))

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Rubi [A]  time = 0.0654023, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{2 (a+b x)^{7/2} (2 A b-9 a B)}{63 a^2 x^{7/2}}-\frac{2 A (a+b x)^{7/2}}{9 a x^{9/2}} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^(5/2)*(A + B*x))/x^(11/2),x]

[Out]

(-2*A*(a + b*x)^(7/2))/(9*a*x^(9/2)) + (2*(2*A*b - 9*a*B)*(a + b*x)^(7/2))/(63*a
^2*x^(7/2))

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Rubi in Sympy [A]  time = 5.31531, size = 49, normalized size = 0.92 \[ - \frac{2 A \left (a + b x\right )^{\frac{7}{2}}}{9 a x^{\frac{9}{2}}} + \frac{4 \left (a + b x\right )^{\frac{7}{2}} \left (A b - \frac{9 B a}{2}\right )}{63 a^{2} x^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(5/2)*(B*x+A)/x**(11/2),x)

[Out]

-2*A*(a + b*x)**(7/2)/(9*a*x**(9/2)) + 4*(a + b*x)**(7/2)*(A*b - 9*B*a/2)/(63*a*
*2*x**(7/2))

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Mathematica [A]  time = 0.0923177, size = 36, normalized size = 0.68 \[ -\frac{2 (a+b x)^{7/2} (7 a A+9 a B x-2 A b x)}{63 a^2 x^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^(5/2)*(A + B*x))/x^(11/2),x]

[Out]

(-2*(a + b*x)^(7/2)*(7*a*A - 2*A*b*x + 9*a*B*x))/(63*a^2*x^(9/2))

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Maple [A]  time = 0.008, size = 31, normalized size = 0.6 \[ -{\frac{-4\,Abx+18\,Bax+14\,Aa}{63\,{a}^{2}} \left ( bx+a \right ) ^{{\frac{7}{2}}}{x}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(5/2)*(B*x+A)/x^(11/2),x)

[Out]

-2/63*(b*x+a)^(7/2)*(-2*A*b*x+9*B*a*x+7*A*a)/x^(9/2)/a^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(5/2)/x^(11/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.230139, size = 132, normalized size = 2.49 \[ -\frac{2 \,{\left (7 \, A a^{4} +{\left (9 \, B a b^{3} - 2 \, A b^{4}\right )} x^{4} +{\left (27 \, B a^{2} b^{2} + A a b^{3}\right )} x^{3} + 3 \,{\left (9 \, B a^{3} b + 5 \, A a^{2} b^{2}\right )} x^{2} +{\left (9 \, B a^{4} + 19 \, A a^{3} b\right )} x\right )} \sqrt{b x + a}}{63 \, a^{2} x^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(5/2)/x^(11/2),x, algorithm="fricas")

[Out]

-2/63*(7*A*a^4 + (9*B*a*b^3 - 2*A*b^4)*x^4 + (27*B*a^2*b^2 + A*a*b^3)*x^3 + 3*(9
*B*a^3*b + 5*A*a^2*b^2)*x^2 + (9*B*a^4 + 19*A*a^3*b)*x)*sqrt(b*x + a)/(a^2*x^(9/
2))

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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((b*x+a)**(5/2)*(B*x+A)/x**(11/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.246361, size = 116, normalized size = 2.19 \[ \frac{{\left (b x + a\right )}^{\frac{7}{2}} b{\left (\frac{{\left (9 \, B a^{3} b^{8} - 2 \, A a^{2} b^{9}\right )}{\left (b x + a\right )}}{a^{5} b^{15}} - \frac{9 \,{\left (B a^{4} b^{8} - A a^{3} b^{9}\right )}}{a^{5} b^{15}}\right )}}{64512 \,{\left ({\left (b x + a\right )} b - a b\right )}^{\frac{9}{2}}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(5/2)/x^(11/2),x, algorithm="giac")

[Out]

1/64512*(b*x + a)^(7/2)*b*((9*B*a^3*b^8 - 2*A*a^2*b^9)*(b*x + a)/(a^5*b^15) - 9*
(B*a^4*b^8 - A*a^3*b^9)/(a^5*b^15))/(((b*x + a)*b - a*b)^(9/2)*abs(b))