3.1239 \(\int x^m (d+e x^2)^p (a+b \tan ^{-1}(c x)) \, dx\)

Optimal. Leaf size=76 \[ \frac{a x^{m+1} \left (d+e x^2\right )^{p+1} \text{Hypergeometric2F1}\left (1,\frac{1}{2} (m+2 p+3),\frac{m+3}{2},-\frac{e x^2}{d}\right )}{d (m+1)}+b \text{Unintegrable}\left (x^m \tan ^{-1}(c x) \left (d+e x^2\right )^p,x\right ) \]

[Out]

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

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Rubi [A]  time = 0.11844, 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 = {} \[ \int x^m \left (d+e x^2\right )^p \left (a+b \tan ^{-1}(c x)\right ) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 3.05692, size = 0, normalized size = 0. \[ \int x^m \left (d+e x^2\right )^p \left (a+b \tan ^{-1}(c x)\right ) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.886, size = 0, normalized size = 0. \begin{align*} \int{x}^{m} \left ( e{x}^{2}+d \right ) ^{p} \left ( a+b\arctan \left ( cx \right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*arctan(c*x) + a)*(e*x^2 + d)^p*x^m, 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*(e*x**2+d)**p*(a+b*atan(c*x)),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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