3.498 \(\int \frac{(a+b x)^{5/2} (A+B x)}{x^{13/2}} \, dx\)

Optimal. Leaf size=84 \[ -\frac{4 b (a+b x)^{7/2} (4 A b-11 a B)}{693 a^3 x^{7/2}}+\frac{2 (a+b x)^{7/2} (4 A b-11 a B)}{99 a^2 x^{9/2}}-\frac{2 A (a+b x)^{7/2}}{11 a x^{11/2}} \]

[Out]

(-2*A*(a + b*x)^(7/2))/(11*a*x^(11/2)) + (2*(4*A*b - 11*a*B)*(a + b*x)^(7/2))/(9
9*a^2*x^(9/2)) - (4*b*(4*A*b - 11*a*B)*(a + b*x)^(7/2))/(693*a^3*x^(7/2))

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Rubi [A]  time = 0.100359, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ -\frac{4 b (a+b x)^{7/2} (4 A b-11 a B)}{693 a^3 x^{7/2}}+\frac{2 (a+b x)^{7/2} (4 A b-11 a B)}{99 a^2 x^{9/2}}-\frac{2 A (a+b x)^{7/2}}{11 a x^{11/2}} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^(5/2)*(A + B*x))/x^(13/2),x]

[Out]

(-2*A*(a + b*x)^(7/2))/(11*a*x^(11/2)) + (2*(4*A*b - 11*a*B)*(a + b*x)^(7/2))/(9
9*a^2*x^(9/2)) - (4*b*(4*A*b - 11*a*B)*(a + b*x)^(7/2))/(693*a^3*x^(7/2))

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Rubi in Sympy [A]  time = 7.72436, size = 82, normalized size = 0.98 \[ - \frac{2 A \left (a + b x\right )^{\frac{7}{2}}}{11 a x^{\frac{11}{2}}} + \frac{2 \left (a + b x\right )^{\frac{7}{2}} \left (4 A b - 11 B a\right )}{99 a^{2} x^{\frac{9}{2}}} - \frac{4 b \left (a + b x\right )^{\frac{7}{2}} \left (4 A b - 11 B a\right )}{693 a^{3} x^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(5/2)*(B*x+A)/x**(13/2),x)

[Out]

-2*A*(a + b*x)**(7/2)/(11*a*x**(11/2)) + 2*(a + b*x)**(7/2)*(4*A*b - 11*B*a)/(99
*a**2*x**(9/2)) - 4*b*(a + b*x)**(7/2)*(4*A*b - 11*B*a)/(693*a**3*x**(7/2))

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Mathematica [A]  time = 0.107055, size = 57, normalized size = 0.68 \[ -\frac{2 (a+b x)^{7/2} \left (7 a^2 (9 A+11 B x)-2 a b x (14 A+11 B x)+8 A b^2 x^2\right )}{693 a^3 x^{11/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^(5/2)*(A + B*x))/x^(13/2),x]

[Out]

(-2*(a + b*x)^(7/2)*(8*A*b^2*x^2 + 7*a^2*(9*A + 11*B*x) - 2*a*b*x*(14*A + 11*B*x
)))/(693*a^3*x^(11/2))

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Maple [A]  time = 0.009, size = 53, normalized size = 0.6 \[ -{\frac{16\,A{b}^{2}{x}^{2}-44\,B{x}^{2}ab-56\,aAbx+154\,{a}^{2}Bx+126\,A{a}^{2}}{693\,{a}^{3}} \left ( bx+a \right ) ^{{\frac{7}{2}}}{x}^{-{\frac{11}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(5/2)*(B*x+A)/x^(13/2),x)

[Out]

-2/693*(b*x+a)^(7/2)*(8*A*b^2*x^2-22*B*a*b*x^2-28*A*a*b*x+77*B*a^2*x+63*A*a^2)/x
^(11/2)/a^3

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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^(13/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.227373, size = 166, normalized size = 1.98 \[ -\frac{2 \,{\left (63 \, A a^{5} - 2 \,{\left (11 \, B a b^{4} - 4 \, A b^{5}\right )} x^{5} +{\left (11 \, B a^{2} b^{3} - 4 \, A a b^{4}\right )} x^{4} + 3 \,{\left (55 \, B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} +{\left (209 \, B a^{4} b + 113 \, A a^{3} b^{2}\right )} x^{2} + 7 \,{\left (11 \, B a^{5} + 23 \, A a^{4} b\right )} x\right )} \sqrt{b x + a}}{693 \, a^{3} x^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(5/2)/x^(13/2),x, algorithm="fricas")

[Out]

-2/693*(63*A*a^5 - 2*(11*B*a*b^4 - 4*A*b^5)*x^5 + (11*B*a^2*b^3 - 4*A*a*b^4)*x^4
+ 3*(55*B*a^3*b^2 + A*a^2*b^3)*x^3 + (209*B*a^4*b + 113*A*a^3*b^2)*x^2 + 7*(11*
B*a^5 + 23*A*a^4*b)*x)*sqrt(b*x + a)/(a^3*x^(11/2))

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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**(13/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.246905, size = 163, normalized size = 1.94 \[ -\frac{{\left (b x + a\right )}^{\frac{7}{2}}{\left ({\left (b x + a\right )}{\left (\frac{2 \,{\left (11 \, B a^{3} b^{10} - 4 \, A a^{2} b^{11}\right )}{\left (b x + a\right )}}{a^{6} b^{18}} - \frac{11 \,{\left (11 \, B a^{4} b^{10} - 4 \, A a^{3} b^{11}\right )}}{a^{6} b^{18}}\right )} + \frac{99 \,{\left (B a^{5} b^{10} - A a^{4} b^{11}\right )}}{a^{6} b^{18}}\right )} b}{2838528 \,{\left ({\left (b x + a\right )} b - a b\right )}^{\frac{11}{2}}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(5/2)/x^(13/2),x, algorithm="giac")

[Out]

-1/2838528*(b*x + a)^(7/2)*((b*x + a)*(2*(11*B*a^3*b^10 - 4*A*a^2*b^11)*(b*x + a
)/(a^6*b^18) - 11*(11*B*a^4*b^10 - 4*A*a^3*b^11)/(a^6*b^18)) + 99*(B*a^5*b^10 -
A*a^4*b^11)/(a^6*b^18))*b/(((b*x + a)*b - a*b)^(11/2)*abs(b))