3.1217 \(\int \frac {x^4 (a+b \tan ^{-1}(c x))}{(d+e x^2)^{5/2}} \, dx\)

Optimal. Leaf size=91 \[ b \text {Int}\left (\frac {x^4 \tan ^{-1}(c x)}{\left (d+e x^2\right )^{5/2}},x\right )+\frac {a \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{e^{5/2}}-\frac {a x}{e^2 \sqrt {d+e x^2}}-\frac {a x^3}{3 e \left (d+e x^2\right )^{3/2}} \]

[Out]

-1/3*a*x^3/e/(e*x^2+d)^(3/2)+a*arctanh(x*e^(1/2)/(e*x^2+d)^(1/2))/e^(5/2)-a*x/e^2/(e*x^2+d)^(1/2)+b*Unintegrab
le(x^4*arctan(c*x)/(e*x^2+d)^(5/2),x)

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Rubi [A]  time = 0.18, 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 = {} \[ \int \frac {x^4 \left (a+b \tan ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(x^4*(a + b*ArcTan[c*x]))/(d + e*x^2)^(5/2),x]

[Out]

-(a*x^3)/(3*e*(d + e*x^2)^(3/2)) - (a*x)/(e^2*Sqrt[d + e*x^2]) + (a*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/e^(5
/2) + b*Defer[Int][(x^4*ArcTan[c*x])/(d + e*x^2)^(5/2), x]

Rubi steps

\begin {align*} \int \frac {x^4 \left (a+b \tan ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx &=a \int \frac {x^4}{\left (d+e x^2\right )^{5/2}} \, dx+b \int \frac {x^4 \tan ^{-1}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx\\ &=-\frac {a x^3}{3 e \left (d+e x^2\right )^{3/2}}+b \int \frac {x^4 \tan ^{-1}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx+\frac {a \int \frac {x^2}{\left (d+e x^2\right )^{3/2}} \, dx}{e}\\ &=-\frac {a x^3}{3 e \left (d+e x^2\right )^{3/2}}-\frac {a x}{e^2 \sqrt {d+e x^2}}+b \int \frac {x^4 \tan ^{-1}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx+\frac {a \int \frac {1}{\sqrt {d+e x^2}} \, dx}{e^2}\\ &=-\frac {a x^3}{3 e \left (d+e x^2\right )^{3/2}}-\frac {a x}{e^2 \sqrt {d+e x^2}}+b \int \frac {x^4 \tan ^{-1}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx+\frac {a \operatorname {Subst}\left (\int \frac {1}{1-e x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )}{e^2}\\ &=-\frac {a x^3}{3 e \left (d+e x^2\right )^{3/2}}-\frac {a x}{e^2 \sqrt {d+e x^2}}+\frac {a \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{e^{5/2}}+b \int \frac {x^4 \tan ^{-1}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx\\ \end {align*}

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Mathematica [A]  time = 12.06, size = 0, normalized size = 0.00 \[ \int \frac {x^4 \left (a+b \tan ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(x^4*(a + b*ArcTan[c*x]))/(d + e*x^2)^(5/2),x]

[Out]

Integrate[(x^4*(a + b*ArcTan[c*x]))/(d + e*x^2)^(5/2), x]

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fricas [A]  time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{4} \arctan \left (c x\right ) + a x^{4}\right )} \sqrt {e x^{2} + d}}{e^{3} x^{6} + 3 \, d e^{2} x^{4} + 3 \, d^{2} e x^{2} + d^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*arctan(c*x))/(e*x^2+d)^(5/2),x, algorithm="fricas")

[Out]

integral((b*x^4*arctan(c*x) + a*x^4)*sqrt(e*x^2 + d)/(e^3*x^6 + 3*d*e^2*x^4 + 3*d^2*e*x^2 + d^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(x^4*(a+b*arctan(c*x))/(e*x^2+d)^(5/2),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 1.26, size = 0, normalized size = 0.00 \[ \int \frac {x^{4} \left (a +b \arctan \left (c x \right )\right )}{\left (e \,x^{2}+d \right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(a+b*arctan(c*x))/(e*x^2+d)^(5/2),x)

[Out]

int(x^4*(a+b*arctan(c*x))/(e*x^2+d)^(5/2),x)

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*arctan(c*x))/(e*x^2+d)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e-c^2*d>0)', see `assume?` for
more details)Is e-c^2*d positive or negative?

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mupad [A]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^4\,\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}{{\left (e\,x^2+d\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4*(a + b*atan(c*x)))/(d + e*x^2)^(5/2),x)

[Out]

int((x^4*(a + b*atan(c*x)))/(d + e*x^2)^(5/2), x)

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sympy [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(x**4*(a+b*atan(c*x))/(e*x**2+d)**(5/2),x)

[Out]

Timed out

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