3.66 \(\int \frac {\tan ^{-1}(a+b x)}{\sqrt [3]{(1+a^2) c+2 a b c x+b^2 c x^2}} \, dx\)

Optimal. Leaf size=25 \[ \text {Int}\left (\frac {\tan ^{-1}(a+b x)}{\sqrt [3]{c (a+b x)^2+c}},x\right ) \]

[Out]

Unintegrable(arctan(b*x+a)/(c+c*(b*x+a)^2)^(1/3),x)

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Rubi [A]  time = 0.05, 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 = {} \[ \int \frac {\tan ^{-1}(a+b x)}{\sqrt [3]{\left (1+a^2\right ) c+2 a b c x+b^2 c x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[ArcTan[a + b*x]/((1 + a^2)*c + 2*a*b*c*x + b^2*c*x^2)^(1/3),x]

[Out]

Defer[Subst][Defer[Int][ArcTan[x]/(c + c*x^2)^(1/3), x], x, a + b*x]/b

Rubi steps

\begin {align*} \int \frac {\tan ^{-1}(a+b x)}{\sqrt [3]{\left (1+a^2\right ) c+2 a b c x+b^2 c x^2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\tan ^{-1}(x)}{\sqrt [3]{c+c x^2}} \, dx,x,a+b x\right )}{b}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 165, normalized size = 6.60 \[ \frac {\frac {5 \sqrt [3]{2} \sqrt {\pi } \Gamma \left (\frac {5}{3}\right ) \, _3F_2\left (1,\frac {4}{3},\frac {4}{3};\frac {11}{6},\frac {7}{3};\frac {1}{(a+b x)^2+1}\right )}{(a+b x)^2+1}+6 \Gamma \left (\frac {11}{6}\right ) \Gamma \left (\frac {7}{3}\right ) \left (\frac {4 (a+b x) \, _2F_1\left (1,\frac {4}{3};\frac {11}{6};\frac {1}{(a+b x)^2+1}\right ) \tan ^{-1}(a+b x)}{(a+b x)^2+1}+10 (a+b x) \tan ^{-1}(a+b x)+15\right )}{20 b \Gamma \left (\frac {11}{6}\right ) \Gamma \left (\frac {7}{3}\right ) \sqrt [3]{c \left (a^2+2 a b x+b^2 x^2+1\right )}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcTan[a + b*x]/((1 + a^2)*c + 2*a*b*c*x + b^2*c*x^2)^(1/3),x]

[Out]

(6*Gamma[11/6]*Gamma[7/3]*(15 + 10*(a + b*x)*ArcTan[a + b*x] + (4*(a + b*x)*ArcTan[a + b*x]*Hypergeometric2F1[
1, 4/3, 11/6, (1 + (a + b*x)^2)^(-1)])/(1 + (a + b*x)^2)) + (5*2^(1/3)*Sqrt[Pi]*Gamma[5/3]*HypergeometricPFQ[{
1, 4/3, 4/3}, {11/6, 7/3}, (1 + (a + b*x)^2)^(-1)])/(1 + (a + b*x)^2))/(20*b*(c*(1 + a^2 + 2*a*b*x + b^2*x^2))
^(1/3)*Gamma[11/6]*Gamma[7/3])

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fricas [A]  time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\arctan \left (b x + a\right )}{{\left (b^{2} c x^{2} + 2 \, a b c x + {\left (a^{2} + 1\right )} c\right )}^{\frac {1}{3}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(b*x+a)/((a^2+1)*c+2*a*b*c*x+c*x^2*b^2)^(1/3),x, algorithm="fricas")

[Out]

integral(arctan(b*x + a)/(b^2*c*x^2 + 2*a*b*c*x + (a^2 + 1)*c)^(1/3), x)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(b*x+a)/((a^2+1)*c+2*a*b*c*x+c*x^2*b^2)^(1/3),x, algorithm="giac")

[Out]

sage0*x

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maple [A]  time = 1.38, size = 0, normalized size = 0.00 \[ \int \frac {\arctan \left (b x +a \right )}{\left (\left (a^{2}+1\right ) c +2 a b c x +b^{2} c \,x^{2}\right )^{\frac {1}{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctan(b*x+a)/((a^2+1)*c+2*a*b*c*x+b^2*c*x^2)^(1/3),x)

[Out]

int(arctan(b*x+a)/((a^2+1)*c+2*a*b*c*x+b^2*c*x^2)^(1/3),x)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\arctan \left (b x + a\right )}{{\left (b^{2} c x^{2} + 2 \, a b c x + {\left (a^{2} + 1\right )} c\right )}^{\frac {1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(b*x+a)/((a^2+1)*c+2*a*b*c*x+c*x^2*b^2)^(1/3),x, algorithm="maxima")

[Out]

integrate(arctan(b*x + a)/(b^2*c*x^2 + 2*a*b*c*x + (a^2 + 1)*c)^(1/3), x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {\mathrm {atan}\left (a+b\,x\right )}{{\left (c\,b^2\,x^2+2\,a\,c\,b\,x+c\,\left (a^2+1\right )\right )}^{1/3}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(atan(a + b*x)/(c*(a^2 + 1) + b^2*c*x^2 + 2*a*b*c*x)^(1/3),x)

[Out]

int(atan(a + b*x)/(c*(a^2 + 1) + b^2*c*x^2 + 2*a*b*c*x)^(1/3), x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {atan}{\left (a + b x \right )}}{\sqrt [3]{c \left (a^{2} + 2 a b x + b^{2} x^{2} + 1\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(atan(b*x+a)/((a**2+1)*c+2*a*b*c*x+c*x**2*b**2)**(1/3),x)

[Out]

Integral(atan(a + b*x)/(c*(a**2 + 2*a*b*x + b**2*x**2 + 1))**(1/3), x)

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