3.1235 \(\int x^m \sqrt {d+e x^2} (a+b \tan ^{-1}(c x)) \, dx\)

Optimal. Leaf size=76 \[ b \text {Int}\left (x^m \tan ^{-1}(c x) \sqrt {d+e x^2},x\right )+\frac {a x^{m+1} \left (d+e x^2\right )^{3/2} \, _2F_1\left (1,\frac {m+4}{2};\frac {m+3}{2};-\frac {e x^2}{d}\right )}{d (m+1)} \]

[Out]

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

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Rubi [A]  time = 0.15, 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 = {} \[ \int x^m \sqrt {d+e x^2} \left (a+b \tan ^{-1}(c x)\right ) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.11, size = 0, normalized size = 0.00 \[ \int x^m \sqrt {d+e x^2} \left (a+b \tan ^{-1}(c x)\right ) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 1.11, size = 0, normalized size = 0.00 \[ \int x^{m} \sqrt {e \,x^{2}+d}\, \left (a +b \arctan \left (c x \right )\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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 \[ \int x^{m} \left (a + b \operatorname {atan}{\left (c x \right )}\right ) \sqrt {d + e x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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