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

Optimal. Leaf size=216 \[ -\frac{2 a^2 \sqrt{a^2+2 a b x+b^2 x^2} (a B+3 A b)}{\sqrt{x} (a+b x)}+\frac{6 a b \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{a+b x}+\frac{2 b^2 x^{3/2} \sqrt{a^2+2 a b x+b^2 x^2} (3 a B+A b)}{3 (a+b x)}+\frac{2 b^3 B x^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)}-\frac{2 a^3 A \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^{3/2} (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.251799, antiderivative size = 216, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ -\frac{2 a^2 \sqrt{a^2+2 a b x+b^2 x^2} (a B+3 A b)}{\sqrt{x} (a+b x)}+\frac{6 a b \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{a+b x}+\frac{2 b^2 x^{3/2} \sqrt{a^2+2 a b x+b^2 x^2} (3 a B+A b)}{3 (a+b x)}+\frac{2 b^3 B x^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)}-\frac{2 a^3 A \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^{3/2} (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 23.9066, size = 216, normalized size = 1. \[ - \frac{A \left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{3 a x^{\frac{3}{2}}} + \frac{32 a b \sqrt{x} \left (5 A b + 3 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{15 \left (a + b x\right )} + \frac{16 b \sqrt{x} \left (5 A b + 3 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{15} + \frac{4 b \sqrt{x} \left (a + b x\right ) \left (5 A b + 3 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{5 a} - \frac{2 \left (5 A b + 3 B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{3 a \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A*(2*a + 2*b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(3*a*x**(3/2)) + 32*a*b*sqr
t(x)*(5*A*b + 3*B*a)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(15*(a + b*x)) + 16*b*sqrt
(x)*(5*A*b + 3*B*a)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/15 + 4*b*sqrt(x)*(a + b*x)*
(5*A*b + 3*B*a)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(5*a) - 2*(5*A*b + 3*B*a)*(a**2
+ 2*a*b*x + b**2*x**2)**(3/2)/(3*a*sqrt(x))

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Mathematica [A]  time = 0.0705681, size = 84, normalized size = 0.39 \[ -\frac{2 \sqrt{(a+b x)^2} \left (5 a^3 (A+3 B x)+45 a^2 b x (A-B x)-15 a b^2 x^2 (3 A+B x)-b^3 x^3 (5 A+3 B x)\right )}{15 x^{3/2} (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[(a + b*x)^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)*(a + b*x))

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Maple [A]  time = 0.01, size = 92, normalized size = 0.4 \[ -{\frac{-6\,B{x}^{4}{b}^{3}-10\,A{b}^{3}{x}^{3}-30\,B{x}^{3}a{b}^{2}-90\,A{x}^{2}a{b}^{2}-90\,B{x}^{2}{a}^{2}b+90\,A{a}^{2}bx+30\,{a}^{3}Bx+10\,A{a}^{3}}{15\, \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}{x}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

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Maxima [A]  time = 0.706211, size = 176, normalized size = 0.81 \[ \frac{2}{15} \,{\left ({\left (3 \, b^{3} x^{2} + 5 \, a b^{2} x\right )} \sqrt{x} + \frac{10 \,{\left (a b^{2} x^{2} + 3 \, a^{2} b x\right )}}{\sqrt{x}} + \frac{15 \,{\left (a^{2} b x^{2} - a^{3} x\right )}}{x^{\frac{3}{2}}}\right )} B + \frac{2}{3} \, A{\left (\frac{b^{3} x^{2} + 3 \, a b^{2} x}{\sqrt{x}} + \frac{6 \,{\left (a b^{2} x^{2} - a^{2} b x\right )}}{x^{\frac{3}{2}}} - \frac{3 \, a^{2} b x^{2} + a^{3} x}{x^{\frac{5}{2}}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*((3*b^3*x^2 + 5*a*b^2*x)*sqrt(x) + 10*(a*b^2*x^2 + 3*a^2*b*x)/sqrt(x) + 15*
(a^2*b*x^2 - a^3*x)/x^(3/2))*B + 2/3*A*((b^3*x^2 + 3*a*b^2*x)/sqrt(x) + 6*(a*b^2
*x^2 - a^2*b*x)/x^(3/2) - (3*a^2*b*x^2 + a^3*x)/x^(5/2))

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Fricas [A]  time = 0.309936, size = 99, normalized size = 0.46 \[ \frac{2 \,{\left (3 \, B b^{3} x^{4} - 5 \, A a^{3} + 5 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 45 \,{\left (B a^{2} b + A a b^{2}\right )} x^{2} - 15 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x\right )}}{15 \, x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(B*x + A)/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 [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}}{x^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((A + B*x)*((a + b*x)**2)**(3/2)/x**(5/2), x)

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GIAC/XCAS [A]  time = 0.271675, size = 166, normalized size = 0.77 \[ \frac{2}{5} \, B b^{3} x^{\frac{5}{2}}{\rm sign}\left (b x + a\right ) + 2 \, B a b^{2} x^{\frac{3}{2}}{\rm sign}\left (b x + a\right ) + \frac{2}{3} \, A b^{3} x^{\frac{3}{2}}{\rm sign}\left (b x + a\right ) + 6 \, B a^{2} b \sqrt{x}{\rm sign}\left (b x + a\right ) + 6 \, A a b^{2} \sqrt{x}{\rm sign}\left (b x + a\right ) - \frac{2 \,{\left (3 \, B a^{3} x{\rm sign}\left (b x + a\right ) + 9 \, A a^{2} b x{\rm sign}\left (b x + a\right ) + A a^{3}{\rm sign}\left (b x + a\right )\right )}}{3 \, x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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