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

Optimal. Leaf size=95 \[ -\frac{2 A \left (b x+c x^2\right )^{5/2}}{5 b x^5}+2 B c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )-\frac{2 B c \sqrt{b x+c x^2}}{x}-\frac{2 B \left (b x+c x^2\right )^{3/2}}{3 x^3} \]

[Out]

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

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Rubi [A]  time = 0.235863, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{2 A \left (b x+c x^2\right )^{5/2}}{5 b x^5}+2 B c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )-\frac{2 B c \sqrt{b x+c x^2}}{x}-\frac{2 B \left (b x+c x^2\right )^{3/2}}{3 x^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 13.6365, size = 88, normalized size = 0.93 \[ - \frac{2 A \left (b x + c x^{2}\right )^{\frac{5}{2}}}{5 b x^{5}} + 2 B c^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )} - \frac{2 B c \sqrt{b x + c x^{2}}}{x} - \frac{2 B \left (b x + c x^{2}\right )^{\frac{3}{2}}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A*(b*x + c*x**2)**(5/2)/(5*b*x**5) + 2*B*c**(3/2)*atanh(sqrt(c)*x/sqrt(b*x +
c*x**2)) - 2*B*c*sqrt(b*x + c*x**2)/x - 2*B*(b*x + c*x**2)**(3/2)/(3*x**3)

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Mathematica [A]  time = 0.277688, size = 88, normalized size = 0.93 \[ \frac{2 \sqrt{x (b+c x)} \left (-\frac{3 A (b+c x)^2}{b}+\frac{15 B c^{3/2} x^{5/2} \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{\sqrt{b+c x}}-5 B x (b+4 c x)\right )}{15 x^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.014, size = 176, normalized size = 1.9 \[ -{\frac{2\,A}{5\,b{x}^{5}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}-{\frac{2\,B}{3\,b{x}^{4}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}-{\frac{4\,Bc}{3\,{b}^{2}{x}^{3}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}+{\frac{16\,B{c}^{2}}{3\,{b}^{3}{x}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}-{\frac{16\,B{c}^{3}}{3\,{b}^{3}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-4\,{\frac{B{c}^{3}\sqrt{c{x}^{2}+bx}x}{{b}^{2}}}-2\,{\frac{B{c}^{2}\sqrt{c{x}^{2}+bx}}{b}}+B{c}^{{\frac{3}{2}}}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/5*A*(c*x^2+b*x)^(5/2)/b/x^5-2/3*B/b/x^4*(c*x^2+b*x)^(5/2)-4/3*B*c/b^2/x^3*(c*
x^2+b*x)^(5/2)+16/3*B*c^2/b^3/x^2*(c*x^2+b*x)^(5/2)-16/3*B*c^3/b^3*(c*x^2+b*x)^(
3/2)-4*B*c^3/b^2*(c*x^2+b*x)^(1/2)*x-2*B*c^2/b*(c*x^2+b*x)^(1/2)+B*c^(3/2)*ln((1
/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))

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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((c*x^2 + b*x)^(3/2)*(B*x + A)/x^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.296303, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, B b c^{\frac{3}{2}} x^{3} \log \left (2 \, c x + b + 2 \, \sqrt{c x^{2} + b x} \sqrt{c}\right ) - 2 \,{\left (3 \, A b^{2} +{\left (20 \, B b c + 3 \, A c^{2}\right )} x^{2} +{\left (5 \, B b^{2} + 6 \, A b c\right )} x\right )} \sqrt{c x^{2} + b x}}{15 \, b x^{3}}, \frac{2 \,{\left (15 \, B b \sqrt{-c} c x^{3} \arctan \left (\frac{\sqrt{c x^{2} + b x}}{\sqrt{-c} x}\right ) -{\left (3 \, A b^{2} +{\left (20 \, B b c + 3 \, A c^{2}\right )} x^{2} +{\left (5 \, B b^{2} + 6 \, A b c\right )} x\right )} \sqrt{c x^{2} + b x}\right )}}{15 \, b x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/15*(15*B*b*c^(3/2)*x^3*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c)) - 2*(3*A*
b^2 + (20*B*b*c + 3*A*c^2)*x^2 + (5*B*b^2 + 6*A*b*c)*x)*sqrt(c*x^2 + b*x))/(b*x^
3), 2/15*(15*B*b*sqrt(-c)*c*x^3*arctan(sqrt(c*x^2 + b*x)/(sqrt(-c)*x)) - (3*A*b^
2 + (20*B*b*c + 3*A*c^2)*x^2 + (5*B*b^2 + 6*A*b*c)*x)*sqrt(c*x^2 + b*x))/(b*x^3)
]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.290269, size = 365, normalized size = 3.84 \[ -B c^{\frac{3}{2}}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right ) + \frac{2 \,{\left (30 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{4} B b c^{\frac{3}{2}} + 15 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{4} A c^{\frac{5}{2}} + 15 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} B b^{2} c + 30 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} A b c^{2} + 5 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} B b^{3} \sqrt{c} + 30 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} A b^{2} c^{\frac{3}{2}} + 15 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} A b^{3} c + 3 \, A b^{4} \sqrt{c}\right )}}{15 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{5} \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-B*c^(3/2)*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) - b)) + 2/15*(30*(s
qrt(c)*x - sqrt(c*x^2 + b*x))^4*B*b*c^(3/2) + 15*(sqrt(c)*x - sqrt(c*x^2 + b*x))
^4*A*c^(5/2) + 15*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b^2*c + 30*(sqrt(c)*x - sq
rt(c*x^2 + b*x))^3*A*b*c^2 + 5*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^3*sqrt(c) +
30*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*b^2*c^(3/2) + 15*(sqrt(c)*x - sqrt(c*x^2
+ b*x))*A*b^3*c + 3*A*b^4*sqrt(c))/((sqrt(c)*x - sqrt(c*x^2 + b*x))^5*sqrt(c))