3.486 \(\int \frac{(a+b x)^{3/2} (A+B x)}{x^{11/2}} \, dx\)

Optimal. Leaf size=84 \[ -\frac{4 b (a+b x)^{5/2} (4 A b-9 a B)}{315 a^3 x^{5/2}}+\frac{2 (a+b x)^{5/2} (4 A b-9 a B)}{63 a^2 x^{7/2}}-\frac{2 A (a+b x)^{5/2}}{9 a x^{9/2}} \]

[Out]

(-2*A*(a + b*x)^(5/2))/(9*a*x^(9/2)) + (2*(4*A*b - 9*a*B)*(a + b*x)^(5/2))/(63*a
^2*x^(7/2)) - (4*b*(4*A*b - 9*a*B)*(a + b*x)^(5/2))/(315*a^3*x^(5/2))

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Rubi [A]  time = 0.100851, 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 \[ -\frac{4 b (a+b x)^{5/2} (4 A b-9 a B)}{315 a^3 x^{5/2}}+\frac{2 (a+b x)^{5/2} (4 A b-9 a B)}{63 a^2 x^{7/2}}-\frac{2 A (a+b x)^{5/2}}{9 a x^{9/2}} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^(3/2)*(A + B*x))/x^(11/2),x]

[Out]

(-2*A*(a + b*x)^(5/2))/(9*a*x^(9/2)) + (2*(4*A*b - 9*a*B)*(a + b*x)^(5/2))/(63*a
^2*x^(7/2)) - (4*b*(4*A*b - 9*a*B)*(a + b*x)^(5/2))/(315*a^3*x^(5/2))

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Rubi in Sympy [A]  time = 7.62753, size = 82, normalized size = 0.98 \[ - \frac{2 A \left (a + b x\right )^{\frac{5}{2}}}{9 a x^{\frac{9}{2}}} + \frac{2 \left (a + b x\right )^{\frac{5}{2}} \left (4 A b - 9 B a\right )}{63 a^{2} x^{\frac{7}{2}}} - \frac{4 b \left (a + b x\right )^{\frac{5}{2}} \left (4 A b - 9 B a\right )}{315 a^{3} x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(3/2)*(B*x+A)/x**(11/2),x)

[Out]

-2*A*(a + b*x)**(5/2)/(9*a*x**(9/2)) + 2*(a + b*x)**(5/2)*(4*A*b - 9*B*a)/(63*a*
*2*x**(7/2)) - 4*b*(a + b*x)**(5/2)*(4*A*b - 9*B*a)/(315*a**3*x**(5/2))

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Mathematica [A]  time = 0.0857647, size = 57, normalized size = 0.68 \[ -\frac{2 (a+b x)^{5/2} \left (5 a^2 (7 A+9 B x)-2 a b x (10 A+9 B x)+8 A b^2 x^2\right )}{315 a^3 x^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^(3/2)*(A + B*x))/x^(11/2),x]

[Out]

(-2*(a + b*x)^(5/2)*(8*A*b^2*x^2 + 5*a^2*(7*A + 9*B*x) - 2*a*b*x*(10*A + 9*B*x))
)/(315*a^3*x^(9/2))

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Maple [A]  time = 0.007, size = 53, normalized size = 0.6 \[ -{\frac{16\,A{b}^{2}{x}^{2}-36\,B{x}^{2}ab-40\,aAbx+90\,{a}^{2}Bx+70\,A{a}^{2}}{315\,{a}^{3}} \left ( bx+a \right ) ^{{\frac{5}{2}}}{x}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(3/2)*(B*x+A)/x^(11/2),x)

[Out]

-2/315*(b*x+a)^(5/2)*(8*A*b^2*x^2-18*B*a*b*x^2-20*A*a*b*x+45*B*a^2*x+35*A*a^2)/x
^(9/2)/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)^(3/2)/x^(11/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.225403, size = 135, normalized size = 1.61 \[ -\frac{2 \,{\left (35 \, A a^{4} - 2 \,{\left (9 \, B a b^{3} - 4 \, A b^{4}\right )} x^{4} +{\left (9 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{3} + 3 \,{\left (24 \, B a^{3} b + A a^{2} b^{2}\right )} x^{2} + 5 \,{\left (9 \, B a^{4} + 10 \, A a^{3} b\right )} x\right )} \sqrt{b x + a}}{315 \, a^{3} x^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(3/2)/x^(11/2),x, algorithm="fricas")

[Out]

-2/315*(35*A*a^4 - 2*(9*B*a*b^3 - 4*A*b^4)*x^4 + (9*B*a^2*b^2 - 4*A*a*b^3)*x^3 +
3*(24*B*a^3*b + A*a^2*b^2)*x^2 + 5*(9*B*a^4 + 10*A*a^3*b)*x)*sqrt(b*x + a)/(a^3
*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)**(3/2)*(B*x+A)/x**(11/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.230529, size = 161, normalized size = 1.92 \[ -\frac{{\left (b x + a\right )}^{\frac{5}{2}}{\left ({\left (b x + a\right )}{\left (\frac{2 \,{\left (9 \, B a^{2} b^{8} - 4 \, A a b^{9}\right )}{\left (b x + a\right )}}{a^{5} b^{15}} - \frac{9 \,{\left (9 \, B a^{3} b^{8} - 4 \, A a^{2} b^{9}\right )}}{a^{5} b^{15}}\right )} + \frac{63 \,{\left (B a^{4} b^{8} - A a^{3} b^{9}\right )}}{a^{5} b^{15}}\right )} b}{322560 \,{\left ({\left (b x + a\right )} b - a b\right )}^{\frac{9}{2}}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(3/2)/x^(11/2),x, algorithm="giac")

[Out]

-1/322560*(b*x + a)^(5/2)*((b*x + a)*(2*(9*B*a^2*b^8 - 4*A*a*b^9)*(b*x + a)/(a^5
*b^15) - 9*(9*B*a^3*b^8 - 4*A*a^2*b^9)/(a^5*b^15)) + 63*(B*a^4*b^8 - A*a^3*b^9)/
(a^5*b^15))*b/(((b*x + a)*b - a*b)^(9/2)*abs(b))