3.1238 \(\int \frac{x^m (a+b \tan ^{-1}(c x))}{(d+e x^2)^{5/2}} \, dx\)

Optimal. Leaf size=75 \[ \frac{a x^{m+1} \text{Hypergeometric2F1}\left (1,\frac{m-2}{2},\frac{m+3}{2},-\frac{e x^2}{d}\right )}{d (m+1) \left (d+e x^2\right )^{3/2}}+b \text{Unintegrable}\left (\frac{x^m \tan ^{-1}(c x)}{\left (d+e x^2\right )^{5/2}},x\right ) \]

[Out]

(a*x^(1 + m)*Hypergeometric2F1[1, (-2 + m)/2, (3 + m)/2, -((e*x^2)/d)])/(d*(1 + m)*(d + e*x^2)^(3/2)) + b*Unin
tegrable[(x^m*ArcTan[c*x])/(d + e*x^2)^(5/2), x]

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Rubi [A]  time = 0.176407, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{x^m \left (a+b \tan ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

(a*x^(1 + m)*Sqrt[1 + (e*x^2)/d]*Hypergeometric2F1[5/2, (1 + m)/2, (3 + m)/2, -((e*x^2)/d)])/(d^2*(1 + m)*Sqrt
[d + e*x^2]) + b*Defer[Int][(x^m*ArcTan[c*x])/(d + e*x^2)^(5/2), x]

Rubi steps

\begin{align*} \int \frac{x^m \left (a+b \tan ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx &=a \int \frac{x^m}{\left (d+e x^2\right )^{5/2}} \, dx+b \int \frac{x^m \tan ^{-1}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx\\ &=b \int \frac{x^m \tan ^{-1}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx+\frac{\left (a \sqrt{1+\frac{e x^2}{d}}\right ) \int \frac{x^m}{\left (1+\frac{e x^2}{d}\right )^{5/2}} \, dx}{d^2 \sqrt{d+e x^2}}\\ &=\frac{a x^{1+m} \sqrt{1+\frac{e x^2}{d}} \, _2F_1\left (\frac{5}{2},\frac{1+m}{2};\frac{3+m}{2};-\frac{e x^2}{d}\right )}{d^2 (1+m) \sqrt{d+e x^2}}+b \int \frac{x^m \tan ^{-1}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx\\ \end{align*}

Mathematica [A]  time = 5.77045, size = 0, normalized size = 0. \[ \int \frac{x^m \left (a+b \tan ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.606, size = 0, normalized size = 0. \begin{align*} \int{{x}^{m} \left ( a+b\arctan \left ( cx \right ) \right ) \left ( e{x}^{2}+d \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \arctan \left (c x\right ) + a\right )} x^{m}}{{\left (e x^{2} + d\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{e x^{2} + d}{\left (b \arctan \left (c x\right ) + a\right )} x^{m}}{e^{3} x^{6} + 3 \, d e^{2} x^{4} + 3 \, d^{2} e x^{2} + d^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(e*x^2 + d)*(b*arctan(c*x) + a)*x^m/(e^3*x^6 + 3*d*e^2*x^4 + 3*d^2*e*x^2 + d^3), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m*(a+b*atan(c*x))/(e*x**2+d)**(5/2),x)

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \arctan \left (c x\right ) + a\right )} x^{m}}{{\left (e x^{2} + d\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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