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

Optimal. Leaf size=76 \[ b \text {Int}\left (\frac {x^m \tan ^{-1}(c x)}{\left (d+e x^2\right )^{5/2}},x\right )+\frac {a x^{m+1} \, _2F_1\left (1,\frac {m-2}{2};\frac {m+3}{2};-\frac {e x^2}{d}\right )}{d (m+1) \left (d+e x^2\right )^{3/2}} \]

[Out]

a*x^(1+m)*hypergeom([1, -1+1/2*m],[3/2+1/2*m],-e*x^2/d)/d/(1+m)/(e*x^2+d)^(3/2)+b*Unintegrable(x^m*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 = {} \[ \int \frac {x^m \left (a+b \tan ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx \]

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*}

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Mathematica [A]  time = 6.16, size = 0, normalized size = 0.00 \[ \int \frac {x^m \left (a+b \tan ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx \]

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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fricas [A]  time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e x^{2} + d} {\left (b \arctan \left (c x\right ) + a\right )} x^{m}}{e^{3} x^{6} + 3 \, d e^{2} x^{4} + 3 \, d^{2} e x^{2} + d^{3}}, x\right ) \]

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

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)

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maple [A]  time = 1.03, size = 0, normalized size = 0.00 \[ \int \frac {x^{m} \left (a +b \arctan \left (c x \right )\right )}{\left (e \,x^{2}+d \right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*(a+b*arctan(c*x))/(e*x^2+d)^(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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \arctan \left (c x\right ) + a\right )} x^{m}}{{\left (e x^{2} + d\right )}^{\frac {5}{2}}}\,{d x} \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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