3.243 \(\int \frac{\left (b x^2+c x^4\right )^{3/2}}{x^7} \, dx\)

Optimal. Leaf size=75 \[ c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )-\frac{c \sqrt{b x^2+c x^4}}{x^2}-\frac{\left (b x^2+c x^4\right )^{3/2}}{3 x^6} \]

[Out]

-((c*Sqrt[b*x^2 + c*x^4])/x^2) - (b*x^2 + c*x^4)^(3/2)/(3*x^6) + c^(3/2)*ArcTanh
[(Sqrt[c]*x^2)/Sqrt[b*x^2 + c*x^4]]

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Rubi [A]  time = 0.186365, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )-\frac{c \sqrt{b x^2+c x^4}}{x^2}-\frac{\left (b x^2+c x^4\right )^{3/2}}{3 x^6} \]

Antiderivative was successfully verified.

[In]  Int[(b*x^2 + c*x^4)^(3/2)/x^7,x]

[Out]

-((c*Sqrt[b*x^2 + c*x^4])/x^2) - (b*x^2 + c*x^4)^(3/2)/(3*x^6) + c^(3/2)*ArcTanh
[(Sqrt[c]*x^2)/Sqrt[b*x^2 + c*x^4]]

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Rubi in Sympy [A]  time = 16.5436, size = 65, normalized size = 0.87 \[ c^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{b x^{2} + c x^{4}}} \right )} - \frac{c \sqrt{b x^{2} + c x^{4}}}{x^{2}} - \frac{\left (b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{3 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**4+b*x**2)**(3/2)/x**7,x)

[Out]

c**(3/2)*atanh(sqrt(c)*x**2/sqrt(b*x**2 + c*x**4)) - c*sqrt(b*x**2 + c*x**4)/x**
2 - (b*x**2 + c*x**4)**(3/2)/(3*x**6)

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Mathematica [A]  time = 0.0656077, size = 87, normalized size = 1.16 \[ \frac{\sqrt{x^2 \left (b+c x^2\right )} \left (3 c^{3/2} x^3 \log \left (\sqrt{c} \sqrt{b+c x^2}+c x\right )-\sqrt{b+c x^2} \left (b+4 c x^2\right )\right )}{3 x^4 \sqrt{b+c x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x^2 + c*x^4)^(3/2)/x^7,x]

[Out]

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

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Maple [B]  time = 0.009, size = 129, normalized size = 1.7 \[{\frac{1}{3\,{b}^{2}{x}^{6}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( 2\, \left ( c{x}^{2}+b \right ) ^{3/2}{c}^{5/2}{x}^{4}-2\, \left ( c{x}^{2}+b \right ) ^{5/2}{c}^{3/2}{x}^{2}+3\,\sqrt{c{x}^{2}+b}{c}^{5/2}{x}^{4}b+3\,\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+b} \right ){x}^{3}{b}^{2}{c}^{2}- \left ( c{x}^{2}+b \right ) ^{{\frac{5}{2}}}b\sqrt{c} \right ) \left ( c{x}^{2}+b \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+b*x^2)^(3/2)/x^7,x)

[Out]

1/3*(c*x^4+b*x^2)^(3/2)*(2*(c*x^2+b)^(3/2)*c^(5/2)*x^4-2*(c*x^2+b)^(5/2)*c^(3/2)
*x^2+3*(c*x^2+b)^(1/2)*c^(5/2)*x^4*b+3*ln(x*c^(1/2)+(c*x^2+b)^(1/2))*x^3*b^2*c^2
-(c*x^2+b)^(5/2)*b*c^(1/2))/x^6/(c*x^2+b)^(3/2)/b^2/c^(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^4 + b*x^2)^(3/2)/x^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.27462, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, c^{\frac{3}{2}} x^{4} \log \left (-2 \, c x^{2} - b - 2 \, \sqrt{c x^{4} + b x^{2}} \sqrt{c}\right ) - 2 \, \sqrt{c x^{4} + b x^{2}}{\left (4 \, c x^{2} + b\right )}}{6 \, x^{4}}, \frac{3 \, \sqrt{-c} c x^{4} \arctan \left (\frac{c x^{2}}{\sqrt{c x^{4} + b x^{2}} \sqrt{-c}}\right ) - \sqrt{c x^{4} + b x^{2}}{\left (4 \, c x^{2} + b\right )}}{3 \, x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)^(3/2)/x^7,x, algorithm="fricas")

[Out]

[1/6*(3*c^(3/2)*x^4*log(-2*c*x^2 - b - 2*sqrt(c*x^4 + b*x^2)*sqrt(c)) - 2*sqrt(c
*x^4 + b*x^2)*(4*c*x^2 + b))/x^4, 1/3*(3*sqrt(-c)*c*x^4*arctan(c*x^2/(sqrt(c*x^4
+ b*x^2)*sqrt(-c))) - sqrt(c*x^4 + b*x^2)*(4*c*x^2 + b))/x^4]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac{3}{2}}}{x^{7}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**4+b*x**2)**(3/2)/x**7,x)

[Out]

Integral((x**2*(b + c*x**2))**(3/2)/x**7, x)

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GIAC/XCAS [A]  time = 0.370355, size = 165, normalized size = 2.2 \[ -\frac{1}{2} \, c^{\frac{3}{2}}{\rm ln}\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{2}\right ){\rm sign}\left (x\right ) + \frac{4 \,{\left (3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{4} b c^{\frac{3}{2}}{\rm sign}\left (x\right ) - 3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{2} b^{2} c^{\frac{3}{2}}{\rm sign}\left (x\right ) + 2 \, b^{3} c^{\frac{3}{2}}{\rm sign}\left (x\right )\right )}}{3 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{2} - b\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)^(3/2)/x^7,x, algorithm="giac")

[Out]

-1/2*c^(3/2)*ln((sqrt(c)*x - sqrt(c*x^2 + b))^2)*sign(x) + 4/3*(3*(sqrt(c)*x - s
qrt(c*x^2 + b))^4*b*c^(3/2)*sign(x) - 3*(sqrt(c)*x - sqrt(c*x^2 + b))^2*b^2*c^(3
/2)*sign(x) + 2*b^3*c^(3/2)*sign(x))/((sqrt(c)*x - sqrt(c*x^2 + b))^2 - b)^3