3.1236 \(\int \frac {x^m (a+b \tan ^{-1}(c x))}{\sqrt {d+e x^2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.16, 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 = {} \[ \int \frac {x^m \left (a+b \tan ^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 3.66, size = 0, normalized size = 0.00 \[ \int \frac {x^m \left (a+b \tan ^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b \arctan \left (c x\right ) + a\right )} x^{m}}{\sqrt {e x^{2} + d}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(a+b*arctan(c*x))/(e*x^2+d)^(1/2),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(a+b*arctan(c*x))/(e*x^2+d)^(1/2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 1.10, size = 0, normalized size = 0.00 \[ \int \frac {x^{m} \left (a +b \arctan \left (c x \right )\right )}{\sqrt {e \,x^{2}+d}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*(a+b*arctan(c*x))/(e*x^2+d)^(1/2),x)

[Out]

int(x^m*(a+b*arctan(c*x))/(e*x^2+d)^(1/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(a+b*arctan(c*x))/(e*x^2+d)^(1/2),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^m*(a + b*atan(c*x)))/(d + e*x^2)^(1/2),x)

[Out]

int((x^m*(a + b*atan(c*x)))/(d + e*x^2)^(1/2), x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{m} \left (a + b \operatorname {atan}{\left (c x \right )}\right )}{\sqrt {d + e x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m*(a+b*atan(c*x))/(e*x**2+d)**(1/2),x)

[Out]

Integral(x**m*(a + b*atan(c*x))/sqrt(d + e*x**2), x)

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