3.693 \(\int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^4} \, dx\)

Optimal. Leaf size=297 \[ \frac{10 a^2 b^2 \log (x) \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{a+b x}+\frac{b^4 x^2 \sqrt{a^2+2 a b x+b^2 x^2} (5 a B+A b)}{2 (a+b x)}+\frac{5 a b^3 x \sqrt{a^2+2 a b x+b^2 x^2} (2 a B+A b)}{a+b x}+\frac{b^5 B x^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 (a+b x)}-\frac{a^5 A \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)}-\frac{a^4 \sqrt{a^2+2 a b x+b^2 x^2} (a B+5 A b)}{2 x^2 (a+b x)}-\frac{5 a^3 b \sqrt{a^2+2 a b x+b^2 x^2} (a B+2 A b)}{x (a+b x)} \]

[Out]

-(a^5*A*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*x^3*(a + b*x)) - (a^4*(5*A*b + a*B)*Sq
rt[a^2 + 2*a*b*x + b^2*x^2])/(2*x^2*(a + b*x)) - (5*a^3*b*(2*A*b + a*B)*Sqrt[a^2
+ 2*a*b*x + b^2*x^2])/(x*(a + b*x)) + (5*a*b^3*(A*b + 2*a*B)*x*Sqrt[a^2 + 2*a*b
*x + b^2*x^2])/(a + b*x) + (b^4*(A*b + 5*a*B)*x^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])
/(2*(a + b*x)) + (b^5*B*x^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*(a + b*x)) + (10*a
^2*b^2*(A*b + a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Log[x])/(a + b*x)

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Rubi [A]  time = 0.363446, antiderivative size = 297, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ \frac{10 a^2 b^2 \log (x) \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{a+b x}+\frac{b^4 x^2 \sqrt{a^2+2 a b x+b^2 x^2} (5 a B+A b)}{2 (a+b x)}+\frac{5 a b^3 x \sqrt{a^2+2 a b x+b^2 x^2} (2 a B+A b)}{a+b x}+\frac{b^5 B x^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 (a+b x)}-\frac{a^5 A \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)}-\frac{a^4 \sqrt{a^2+2 a b x+b^2 x^2} (a B+5 A b)}{2 x^2 (a+b x)}-\frac{5 a^3 b \sqrt{a^2+2 a b x+b^2 x^2} (a B+2 A b)}{x (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/x^4,x]

[Out]

-(a^5*A*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*x^3*(a + b*x)) - (a^4*(5*A*b + a*B)*Sq
rt[a^2 + 2*a*b*x + b^2*x^2])/(2*x^2*(a + b*x)) - (5*a^3*b*(2*A*b + a*B)*Sqrt[a^2
+ 2*a*b*x + b^2*x^2])/(x*(a + b*x)) + (5*a*b^3*(A*b + 2*a*B)*x*Sqrt[a^2 + 2*a*b
*x + b^2*x^2])/(a + b*x) + (b^4*(A*b + 5*a*B)*x^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])
/(2*(a + b*x)) + (b^5*B*x^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*(a + b*x)) + (10*a
^2*b^2*(A*b + a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Log[x])/(a + b*x)

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Rubi in Sympy [A]  time = 35.7111, size = 267, normalized size = 0.9 \[ - \frac{A \left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{6 a x^{3}} + \frac{10 a^{2} b^{2} \left (A b + B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (x \right )}}{a + b x} + 10 a b^{2} \left (A b + B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} + \frac{5 b^{2} \left (3 a + 3 b x\right ) \left (A b + B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{3} + \frac{10 b^{2} \left (A b + B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{3 a} - \frac{5 b \left (a + b x\right ) \left (A b + B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{2 a x} - \frac{\left (A b + B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{2 a x^{2}} \]

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)**(5/2)/x**4,x)

[Out]

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

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Mathematica [A]  time = 0.106476, size = 127, normalized size = 0.43 \[ \frac{\sqrt{(a+b x)^2} \left (a^5 (-(2 A+3 B x))-15 a^4 b x (A+2 B x)-60 a^3 A b^2 x^2+60 a^2 b^2 x^3 \log (x) (a B+A b)+60 a^2 b^3 B x^4+15 a b^4 x^4 (2 A+B x)+b^5 x^5 (3 A+2 B x)\right )}{6 x^3 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/x^4,x]

[Out]

(Sqrt[(a + b*x)^2]*(-60*a^3*A*b^2*x^2 + 60*a^2*b^3*B*x^4 + 15*a*b^4*x^4*(2*A + B
*x) - 15*a^4*b*x*(A + 2*B*x) + b^5*x^5*(3*A + 2*B*x) - a^5*(2*A + 3*B*x) + 60*a^
2*b^2*(A*b + a*B)*x^3*Log[x]))/(6*x^3*(a + b*x))

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Maple [A]  time = 0.02, size = 144, normalized size = 0.5 \[{\frac{2\,B{b}^{5}{x}^{6}+3\,A{x}^{5}{b}^{5}+15\,B{x}^{5}a{b}^{4}+60\,A\ln \left ( x \right ){x}^{3}{a}^{2}{b}^{3}+30\,A{x}^{4}a{b}^{4}+60\,B\ln \left ( x \right ){x}^{3}{a}^{3}{b}^{2}+60\,B{x}^{4}{a}^{2}{b}^{3}-60\,A{x}^{2}{a}^{3}{b}^{2}-30\,B{x}^{2}{a}^{4}b-15\,Ax{a}^{4}b-3\,Bx{a}^{5}-2\,A{a}^{5}}{6\, \left ( bx+a \right ) ^{5}{x}^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

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)^(5/2)/x^4,x)

[Out]

1/6*((b*x+a)^2)^(5/2)*(2*B*b^5*x^6+3*A*x^5*b^5+15*B*x^5*a*b^4+60*A*ln(x)*x^3*a^2
*b^3+30*A*x^4*a*b^4+60*B*ln(x)*x^3*a^3*b^2+60*B*x^4*a^2*b^3-60*A*x^2*a^3*b^2-30*
B*x^2*a^4*b-15*A*x*a^4*b-3*B*x*a^5-2*A*a^5)/(b*x+a)^5/x^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^2*x^2 + 2*a*b*x + a^2)^(5/2)*(B*x + A)/x^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.319161, size = 163, normalized size = 0.55 \[ \frac{2 \, B b^{5} x^{6} - 2 \, A a^{5} + 3 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 30 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 60 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} \log \left (x\right ) - 30 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} - 3 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(B*x + A)/x^4,x, algorithm="fricas")

[Out]

1/6*(2*B*b^5*x^6 - 2*A*a^5 + 3*(5*B*a*b^4 + A*b^5)*x^5 + 30*(2*B*a^2*b^3 + A*a*b
^4)*x^4 + 60*(B*a^3*b^2 + A*a^2*b^3)*x^3*log(x) - 30*(B*a^4*b + 2*A*a^3*b^2)*x^2
- 3*(B*a^5 + 5*A*a^4*b)*x)/x^3

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}}{x^{4}}\, dx \]

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)**(5/2)/x**4,x)

[Out]

Integral((A + B*x)*((a + b*x)**2)**(5/2)/x**4, x)

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GIAC/XCAS [A]  time = 0.276179, size = 257, normalized size = 0.87 \[ \frac{1}{3} \, B b^{5} x^{3}{\rm sign}\left (b x + a\right ) + \frac{5}{2} \, B a b^{4} x^{2}{\rm sign}\left (b x + a\right ) + \frac{1}{2} \, A b^{5} x^{2}{\rm sign}\left (b x + a\right ) + 10 \, B a^{2} b^{3} x{\rm sign}\left (b x + a\right ) + 5 \, A a b^{4} x{\rm sign}\left (b x + a\right ) + 10 \,{\left (B a^{3} b^{2}{\rm sign}\left (b x + a\right ) + A a^{2} b^{3}{\rm sign}\left (b x + a\right )\right )}{\rm ln}\left ({\left | x \right |}\right ) - \frac{2 \, A a^{5}{\rm sign}\left (b x + a\right ) + 30 \,{\left (B a^{4} b{\rm sign}\left (b x + a\right ) + 2 \, A a^{3} b^{2}{\rm sign}\left (b x + a\right )\right )} x^{2} + 3 \,{\left (B a^{5}{\rm sign}\left (b x + a\right ) + 5 \, A a^{4} b{\rm sign}\left (b x + a\right )\right )} x}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(B*x + A)/x^4,x, algorithm="giac")

[Out]

1/3*B*b^5*x^3*sign(b*x + a) + 5/2*B*a*b^4*x^2*sign(b*x + a) + 1/2*A*b^5*x^2*sign
(b*x + a) + 10*B*a^2*b^3*x*sign(b*x + a) + 5*A*a*b^4*x*sign(b*x + a) + 10*(B*a^3
*b^2*sign(b*x + a) + A*a^2*b^3*sign(b*x + a))*ln(abs(x)) - 1/6*(2*A*a^5*sign(b*x
+ a) + 30*(B*a^4*b*sign(b*x + a) + 2*A*a^3*b^2*sign(b*x + a))*x^2 + 3*(B*a^5*si
gn(b*x + a) + 5*A*a^4*b*sign(b*x + a))*x)/x^3