3.13.37 \(\int \frac {x^m (a+b \text {ArcTan}(c x))}{(d+e x^2)^{3/2}} \, dx\) [1237]

Optimal. Leaf size=74 \[ \frac {a x^{1+m} \, _2F_1\left (1,\frac {m}{2};\frac {3+m}{2};-\frac {e x^2}{d}\right )}{d (1+m) \sqrt {d+e x^2}}+b \text {Int}\left (\frac {x^m \text {ArcTan}(c x)}{\left (d+e x^2\right )^{3/2}},x\right ) \]

[Out]

a*x^(1+m)*hypergeom([1, 1/2*m],[3/2+1/2*m],-e*x^2/d)/d/(1+m)/(e*x^2+d)^(1/2)+b*Unintegrable(x^m*arctan(c*x)/(e
*x^2+d)^(3/2),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 = {} \begin {gather*} \int \frac {x^m (a+b \text {ArcTan}(c x))}{\left (d+e x^2\right )^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 3.65, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^m (a+b \text {ArcTan}(c x))}{\left (d+e x^2\right )^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.43, size = 0, normalized size = 0.00 \[\int \frac {x^{m} \left (a +b \arctan \left (c x \right )\right )}{\left (e \,x^{2}+d \right )^{\frac {3}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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