3.1189 \(\int \frac {(d+e x^2)^{3/2} (a+b \tan ^{-1}(c x))}{x^3} \, dx\)

Optimal. Leaf size=90 \[ b \text {Int}\left (\frac {\tan ^{-1}(c x) \left (d+e x^2\right )^{3/2}}{x^3},x\right )-\frac {a \left (d+e x^2\right )^{3/2}}{2 x^2}+\frac {3}{2} a e \sqrt {d+e x^2}-\frac {3}{2} a \sqrt {d} e \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right ) \]

[Out]

-1/2*a*(e*x^2+d)^(3/2)/x^2-3/2*a*e*arctanh((e*x^2+d)^(1/2)/d^(1/2))*d^(1/2)+3/2*a*e*(e*x^2+d)^(1/2)+b*Unintegr
able((e*x^2+d)^(3/2)*arctan(c*x)/x^3,x)

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Rubi [A]  time = 0.20, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]

Int[((d + e*x^2)^(3/2)*(a + b*ArcTan[c*x]))/x^3,x]

[Out]

(3*a*e*Sqrt[d + e*x^2])/2 - (a*(d + e*x^2)^(3/2))/(2*x^2) - (3*a*Sqrt[d]*e*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/2
+ b*Defer[Int][((d + e*x^2)^(3/2)*ArcTan[c*x])/x^3, x]

Rubi steps

\begin {align*} \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{x^3} \, dx &=a \int \frac {\left (d+e x^2\right )^{3/2}}{x^3} \, dx+b \int \frac {\left (d+e x^2\right )^{3/2} \tan ^{-1}(c x)}{x^3} \, dx\\ &=\frac {1}{2} a \operatorname {Subst}\left (\int \frac {(d+e x)^{3/2}}{x^2} \, dx,x,x^2\right )+b \int \frac {\left (d+e x^2\right )^{3/2} \tan ^{-1}(c x)}{x^3} \, dx\\ &=-\frac {a \left (d+e x^2\right )^{3/2}}{2 x^2}+b \int \frac {\left (d+e x^2\right )^{3/2} \tan ^{-1}(c x)}{x^3} \, dx+\frac {1}{4} (3 a e) \operatorname {Subst}\left (\int \frac {\sqrt {d+e x}}{x} \, dx,x,x^2\right )\\ &=\frac {3}{2} a e \sqrt {d+e x^2}-\frac {a \left (d+e x^2\right )^{3/2}}{2 x^2}+b \int \frac {\left (d+e x^2\right )^{3/2} \tan ^{-1}(c x)}{x^3} \, dx+\frac {1}{4} (3 a d e) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d+e x}} \, dx,x,x^2\right )\\ &=\frac {3}{2} a e \sqrt {d+e x^2}-\frac {a \left (d+e x^2\right )^{3/2}}{2 x^2}+b \int \frac {\left (d+e x^2\right )^{3/2} \tan ^{-1}(c x)}{x^3} \, dx+\frac {1}{2} (3 a d) \operatorname {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x^2}\right )\\ &=\frac {3}{2} a e \sqrt {d+e x^2}-\frac {a \left (d+e x^2\right )^{3/2}}{2 x^2}-\frac {3}{2} a \sqrt {d} e \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )+b \int \frac {\left (d+e x^2\right )^{3/2} \tan ^{-1}(c x)}{x^3} \, dx\\ \end {align*}

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Mathematica [A]  time = 55.09, size = 0, normalized size = 0.00 \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[((d + e*x^2)^(3/2)*(a + b*ArcTan[c*x]))/x^3,x]

[Out]

Integrate[((d + e*x^2)^(3/2)*(a + b*ArcTan[c*x]))/x^3, x]

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fricas [A]  time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a e x^{2} + a d + {\left (b e x^{2} + b d\right )} \arctan \left (c x\right )\right )} \sqrt {e x^{2} + d}}{x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^(3/2)*(a+b*arctan(c*x))/x^3,x, algorithm="fricas")

[Out]

integral((a*e*x^2 + a*d + (b*e*x^2 + b*d)*arctan(c*x))*sqrt(e*x^2 + d)/x^3, x)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^(3/2)*(a+b*arctan(c*x))/x^3,x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 1.04, size = 0, normalized size = 0.00 \[ \int \frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \arctan \left (c x \right )\right )}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d)^(3/2)*(a+b*arctan(c*x))/x^3,x)

[Out]

int((e*x^2+d)^(3/2)*(a+b*arctan(c*x))/x^3,x)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{2} \, {\left (3 \, \sqrt {d} e \operatorname {arsinh}\left (\frac {d}{\sqrt {d e} {\left | x \right |}}\right ) - 3 \, \sqrt {e x^{2} + d} e - \frac {{\left (e x^{2} + d\right )}^{\frac {3}{2}} e}{d} + \frac {{\left (e x^{2} + d\right )}^{\frac {5}{2}}}{d x^{2}}\right )} a + \frac {1}{2} \, b \int \frac {2 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} \arctan \left (c x\right )}{x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^(3/2)*(a+b*arctan(c*x))/x^3,x, algorithm="maxima")

[Out]

-1/2*(3*sqrt(d)*e*arcsinh(d/(sqrt(d*e)*abs(x))) - 3*sqrt(e*x^2 + d)*e - (e*x^2 + d)^(3/2)*e/d + (e*x^2 + d)^(5
/2)/(d*x^2))*a + 1/2*b*integrate(2*(e*x^2 + d)^(3/2)*arctan(c*x)/x^3, x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^{3/2}}{x^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*atan(c*x))*(d + e*x^2)^(3/2))/x^3,x)

[Out]

int(((a + b*atan(c*x))*(d + e*x^2)^(3/2))/x^3, x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {atan}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{\frac {3}{2}}}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x**2+d)**(3/2)*(a+b*atan(c*x))/x**3,x)

[Out]

Integral((a + b*atan(c*x))*(d + e*x**2)**(3/2)/x**3, x)

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