3.255 \(\int x^m \sqrt{c+a^2 c x^2} \tan ^{-1}(a x) \, dx\)

Optimal. Leaf size=112 \[ -\frac{a x^{m+2} \sqrt{a^2 c x^2+c} \text{Hypergeometric2F1}\left (1,\frac{m+3}{2},\frac{m+4}{2},-a^2 x^2\right )}{(m+2)^2}+\frac{c \text{Unintegrable}\left (\frac{x^m \tan ^{-1}(a x)}{\sqrt{a^2 c x^2+c}},x\right )}{m+2}+\frac{x^{m+1} \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{m+2} \]

[Out]

(x^(1 + m)*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(2 + m) - (a*x^(2 + m)*Sqrt[c + a^2*c*x^2]*Hypergeometric2F1[1, (3
+ m)/2, (4 + m)/2, -(a^2*x^2)])/(2 + m)^2 + (c*Unintegrable[(x^m*ArcTan[a*x])/Sqrt[c + a^2*c*x^2], x])/(2 + m
)

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Rubi [A]  time = 0.17032, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int x^m \sqrt{c+a^2 c x^2} \tan ^{-1}(a x) \, dx \]

Verification is Not applicable to the result.

[In]

Int[x^m*Sqrt[c + a^2*c*x^2]*ArcTan[a*x],x]

[Out]

(x^(1 + m)*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(2 + m) - (a*c*x^(2 + m)*Sqrt[1 + a^2*x^2]*Hypergeometric2F1[1/2,
(2 + m)/2, (4 + m)/2, -(a^2*x^2)])/((2 + m)^2*Sqrt[c + a^2*c*x^2]) + (c*Defer[Int][(x^m*ArcTan[a*x])/Sqrt[c +
a^2*c*x^2], x])/(2 + m)

Rubi steps

\begin{align*} \int x^m \sqrt{c+a^2 c x^2} \tan ^{-1}(a x) \, dx &=\frac{x^{1+m} \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{2+m}+\frac{c \int \frac{x^m \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{2+m}-\frac{(a c) \int \frac{x^{1+m}}{\sqrt{c+a^2 c x^2}} \, dx}{2+m}\\ &=\frac{x^{1+m} \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{2+m}+\frac{c \int \frac{x^m \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{2+m}-\frac{\left (a c \sqrt{1+a^2 x^2}\right ) \int \frac{x^{1+m}}{\sqrt{1+a^2 x^2}} \, dx}{(2+m) \sqrt{c+a^2 c x^2}}\\ &=\frac{x^{1+m} \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{2+m}-\frac{a c x^{2+m} \sqrt{1+a^2 x^2} \, _2F_1\left (\frac{1}{2},\frac{2+m}{2};\frac{4+m}{2};-a^2 x^2\right )}{(2+m)^2 \sqrt{c+a^2 c x^2}}+\frac{c \int \frac{x^m \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{2+m}\\ \end{align*}

Mathematica [A]  time = 0.10003, size = 0, normalized size = 0. \[ \int x^m \sqrt{c+a^2 c x^2} \tan ^{-1}(a x) \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[x^m*Sqrt[c + a^2*c*x^2]*ArcTan[a*x],x]

[Out]

Integrate[x^m*Sqrt[c + a^2*c*x^2]*ArcTan[a*x], x]

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Maple [A]  time = 0.596, size = 0, normalized size = 0. \begin{align*} \int{x}^{m}\sqrt{{a}^{2}c{x}^{2}+c}\arctan \left ( ax \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a^{2} c x^{2} + c} x^{m} \arctan \left (a x\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{a^{2} c x^{2} + c} x^{m} \arctan \left (a x\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m} \sqrt{c \left (a^{2} x^{2} + 1\right )} \operatorname{atan}{\left (a x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**m*sqrt(c*(a**2*x**2 + 1))*atan(a*x), x)

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError