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

Optimal. Leaf size=77 \[ b \text {Int}\left (x^m \tan ^{-1}(c x) \left (d+e x^2\right )^p,x\right )+\frac {a x^{m+1} \left (d+e x^2\right )^{p+1} \, _2F_1\left (1,\frac {1}{2} (m+2 p+3);\frac {m+3}{2};-\frac {e x^2}{d}\right )}{d (m+1)} \]

[Out]

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

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Rubi [A]  time = 0.12, 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 = {} \[ \int x^m \left (d+e x^2\right )^p \left (a+b \tan ^{-1}(c x)\right ) \, dx \]

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*}

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Mathematica [A]  time = 3.35, size = 0, normalized size = 0.00 \[ \int x^m \left (d+e x^2\right )^p \left (a+b \tan ^{-1}(c x)\right ) \, dx \]

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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fricas [A]  time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \arctan \left (c x\right ) + a\right )} {\left (e x^{2} + d\right )}^{p} x^{m}, x\right ) \]

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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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \arctan \left (c x\right ) + a\right )} {\left (e x^{2} + d\right )}^{p} x^{m}\,{d x} \]

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)

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maple [A]  time = 1.47, size = 0, normalized size = 0.00 \[ \int x^{m} \left (e \,x^{2}+d \right )^{p} \left (a +b \arctan \left (c x \right )\right )\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \arctan \left (c x\right ) + a\right )} {\left (e x^{2} + d\right )}^{p} x^{m}\,{d x} \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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