3.311 \(\int \frac{(a+b x)^3 (A+B x)}{x^{5/2}} \, dx\)

Optimal. Leaf size=81 \[ -\frac{2 a^3 A}{3 x^{3/2}}-\frac{2 a^2 (a B+3 A b)}{\sqrt{x}}+\frac{2}{3} b^2 x^{3/2} (3 a B+A b)+6 a b \sqrt{x} (a B+A b)+\frac{2}{5} b^3 B x^{5/2} \]

[Out]

(-2*a^3*A)/(3*x^(3/2)) - (2*a^2*(3*A*b + a*B))/Sqrt[x] + 6*a*b*(A*b + a*B)*Sqrt[
x] + (2*b^2*(A*b + 3*a*B)*x^(3/2))/3 + (2*b^3*B*x^(5/2))/5

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Rubi [A]  time = 0.106121, antiderivative size = 81, 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 \[ -\frac{2 a^3 A}{3 x^{3/2}}-\frac{2 a^2 (a B+3 A b)}{\sqrt{x}}+\frac{2}{3} b^2 x^{3/2} (3 a B+A b)+6 a b \sqrt{x} (a B+A b)+\frac{2}{5} b^3 B x^{5/2} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^3*(A + B*x))/x^(5/2),x]

[Out]

(-2*a^3*A)/(3*x^(3/2)) - (2*a^2*(3*A*b + a*B))/Sqrt[x] + 6*a*b*(A*b + a*B)*Sqrt[
x] + (2*b^2*(A*b + 3*a*B)*x^(3/2))/3 + (2*b^3*B*x^(5/2))/5

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Rubi in Sympy [A]  time = 12.4531, size = 80, normalized size = 0.99 \[ - \frac{2 A a^{3}}{3 x^{\frac{3}{2}}} + \frac{2 B b^{3} x^{\frac{5}{2}}}{5} - \frac{2 a^{2} \left (3 A b + B a\right )}{\sqrt{x}} + 6 a b \sqrt{x} \left (A b + B a\right ) + 2 b^{2} x^{\frac{3}{2}} \left (\frac{A b}{3} + B a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**3*(B*x+A)/x**(5/2),x)

[Out]

-2*A*a**3/(3*x**(3/2)) + 2*B*b**3*x**(5/2)/5 - 2*a**2*(3*A*b + B*a)/sqrt(x) + 6*
a*b*sqrt(x)*(A*b + B*a) + 2*b**2*x**(3/2)*(A*b/3 + B*a)

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Mathematica [A]  time = 0.0341342, size = 66, normalized size = 0.81 \[ \frac{2 \left (-5 a^3 (A+3 B x)+45 a^2 b x (B x-A)+15 a b^2 x^2 (3 A+B x)+b^3 x^3 (5 A+3 B x)\right )}{15 x^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^3*(A + B*x))/x^(5/2),x]

[Out]

(2*(45*a^2*b*x*(-A + B*x) + 15*a*b^2*x^2*(3*A + B*x) - 5*a^3*(A + 3*B*x) + b^3*x
^3*(5*A + 3*B*x)))/(15*x^(3/2))

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Maple [A]  time = 0.009, size = 76, normalized size = 0.9 \[ -{\frac{-6\,B{b}^{3}{x}^{4}-10\,A{b}^{3}{x}^{3}-30\,B{x}^{3}a{b}^{2}-90\,aA{b}^{2}{x}^{2}-90\,B{x}^{2}{a}^{2}b+90\,{a}^{2}Abx+30\,{a}^{3}Bx+10\,{a}^{3}A}{15}{x}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^3*(B*x+A)/x^(5/2),x)

[Out]

-2/15*(-3*B*b^3*x^4-5*A*b^3*x^3-15*B*a*b^2*x^3-45*A*a*b^2*x^2-45*B*a^2*b*x^2+45*
A*a^2*b*x+15*B*a^3*x+5*A*a^3)/x^(3/2)

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Maxima [A]  time = 1.32527, size = 99, normalized size = 1.22 \[ \frac{2}{5} \, B b^{3} x^{\frac{5}{2}} + \frac{2}{3} \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{\frac{3}{2}} + 6 \,{\left (B a^{2} b + A a b^{2}\right )} \sqrt{x} - \frac{2 \,{\left (A a^{3} + 3 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x\right )}}{3 \, x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^3/x^(5/2),x, algorithm="maxima")

[Out]

2/5*B*b^3*x^(5/2) + 2/3*(3*B*a*b^2 + A*b^3)*x^(3/2) + 6*(B*a^2*b + A*a*b^2)*sqrt
(x) - 2/3*(A*a^3 + 3*(B*a^3 + 3*A*a^2*b)*x)/x^(3/2)

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Fricas [A]  time = 0.210056, size = 99, normalized size = 1.22 \[ \frac{2 \,{\left (3 \, B b^{3} x^{4} - 5 \, A a^{3} + 5 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 45 \,{\left (B a^{2} b + A a b^{2}\right )} x^{2} - 15 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x\right )}}{15 \, x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^3/x^(5/2),x, algorithm="fricas")

[Out]

2/15*(3*B*b^3*x^4 - 5*A*a^3 + 5*(3*B*a*b^2 + A*b^3)*x^3 + 45*(B*a^2*b + A*a*b^2)
*x^2 - 15*(B*a^3 + 3*A*a^2*b)*x)/x^(3/2)

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Sympy [A]  time = 10.9619, size = 105, normalized size = 1.3 \[ - \frac{2 A a^{3}}{3 x^{\frac{3}{2}}} - \frac{6 A a^{2} b}{\sqrt{x}} + 6 A a b^{2} \sqrt{x} + \frac{2 A b^{3} x^{\frac{3}{2}}}{3} - \frac{2 B a^{3}}{\sqrt{x}} + 6 B a^{2} b \sqrt{x} + 2 B a b^{2} x^{\frac{3}{2}} + \frac{2 B b^{3} x^{\frac{5}{2}}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**3*(B*x+A)/x**(5/2),x)

[Out]

-2*A*a**3/(3*x**(3/2)) - 6*A*a**2*b/sqrt(x) + 6*A*a*b**2*sqrt(x) + 2*A*b**3*x**(
3/2)/3 - 2*B*a**3/sqrt(x) + 6*B*a**2*b*sqrt(x) + 2*B*a*b**2*x**(3/2) + 2*B*b**3*
x**(5/2)/5

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GIAC/XCAS [A]  time = 0.254932, size = 101, normalized size = 1.25 \[ \frac{2}{5} \, B b^{3} x^{\frac{5}{2}} + 2 \, B a b^{2} x^{\frac{3}{2}} + \frac{2}{3} \, A b^{3} x^{\frac{3}{2}} + 6 \, B a^{2} b \sqrt{x} + 6 \, A a b^{2} \sqrt{x} - \frac{2 \,{\left (3 \, B a^{3} x + 9 \, A a^{2} b x + A a^{3}\right )}}{3 \, x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^3/x^(5/2),x, algorithm="giac")

[Out]

2/5*B*b^3*x^(5/2) + 2*B*a*b^2*x^(3/2) + 2/3*A*b^3*x^(3/2) + 6*B*a^2*b*sqrt(x) +
6*A*a*b^2*sqrt(x) - 2/3*(3*B*a^3*x + 9*A*a^2*b*x + A*a^3)/x^(3/2)