3.15.83 \(\int \frac {x^4 (4 b+a x^5)}{(-b+a x^5)^2 \sqrt [4]{-b+c x^4+a x^5}} \, dx\) [1483]

Optimal. Leaf size=104 \[ \frac {x \left (-b+c x^4+a x^5\right )^{3/4}}{c \left (b-a x^5\right )}+\frac {\text {ArcTan}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{-b+c x^4+a x^5}}\right )}{2 c^{5/4}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{-b+c x^4+a x^5}}\right )}{2 c^{5/4}} \]

[Out]

x*(a*x^5+c*x^4-b)^(3/4)/c/(-a*x^5+b)+1/2*arctan(c^(1/4)*x/(a*x^5+c*x^4-b)^(1/4))/c^(5/4)+1/2*arctanh(c^(1/4)*x
/(a*x^5+c*x^4-b)^(1/4))/c^(5/4)

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Rubi [F]
time = 1.77, 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 = {} \begin {gather*} \int \frac {x^4 \left (4 b+a x^5\right )}{\left (-b+a x^5\right )^2 \sqrt [4]{-b+c x^4+a x^5}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(x^4*(4*b + a*x^5))/((-b + a*x^5)^2*(-b + c*x^4 + a*x^5)^(1/4)),x]

[Out]

-1/5*Defer[Int][1/((b^(1/5) - a^(1/5)*x)*(-b + c*x^4 + a*x^5)^(1/4)), x]/a^(4/5) - Defer[Int][1/((-((-1)^(1/5)
*b^(1/5)) - a^(1/5)*x)*(-b + c*x^4 + a*x^5)^(1/4)), x]/(5*a^(4/5)) - Defer[Int][1/(((-1)^(2/5)*b^(1/5) - a^(1/
5)*x)*(-b + c*x^4 + a*x^5)^(1/4)), x]/(5*a^(4/5)) - Defer[Int][1/((-((-1)^(3/5)*b^(1/5)) - a^(1/5)*x)*(-b + c*
x^4 + a*x^5)^(1/4)), x]/(5*a^(4/5)) - Defer[Int][1/(((-1)^(4/5)*b^(1/5) - a^(1/5)*x)*(-b + c*x^4 + a*x^5)^(1/4
)), x]/(5*a^(4/5)) + 5*b*Defer[Int][x^4/((b - a*x^5)^2*(-b + c*x^4 + a*x^5)^(1/4)), x]

Rubi steps

\begin {align*} \int \frac {x^4 \left (4 b+a x^5\right )}{\left (-b+a x^5\right )^2 \sqrt [4]{-b+c x^4+a x^5}} \, dx &=\int \left (\frac {5 b x^4}{\left (b-a x^5\right )^2 \sqrt [4]{-b+c x^4+a x^5}}+\frac {x^4}{\left (-b+a x^5\right ) \sqrt [4]{-b+c x^4+a x^5}}\right ) \, dx\\ &=(5 b) \int \frac {x^4}{\left (b-a x^5\right )^2 \sqrt [4]{-b+c x^4+a x^5}} \, dx+\int \frac {x^4}{\left (-b+a x^5\right ) \sqrt [4]{-b+c x^4+a x^5}} \, dx\\ &=(5 b) \int \frac {x^4}{\left (b-a x^5\right )^2 \sqrt [4]{-b+c x^4+a x^5}} \, dx+\int \left (-\frac {1}{5 a^{4/5} \left (\sqrt [5]{b}-\sqrt [5]{a} x\right ) \sqrt [4]{-b+c x^4+a x^5}}-\frac {1}{5 a^{4/5} \left (-\sqrt [5]{-1} \sqrt [5]{b}-\sqrt [5]{a} x\right ) \sqrt [4]{-b+c x^4+a x^5}}-\frac {1}{5 a^{4/5} \left ((-1)^{2/5} \sqrt [5]{b}-\sqrt [5]{a} x\right ) \sqrt [4]{-b+c x^4+a x^5}}-\frac {1}{5 a^{4/5} \left (-(-1)^{3/5} \sqrt [5]{b}-\sqrt [5]{a} x\right ) \sqrt [4]{-b+c x^4+a x^5}}-\frac {1}{5 a^{4/5} \left ((-1)^{4/5} \sqrt [5]{b}-\sqrt [5]{a} x\right ) \sqrt [4]{-b+c x^4+a x^5}}\right ) \, dx\\ &=-\frac {\int \frac {1}{\left (\sqrt [5]{b}-\sqrt [5]{a} x\right ) \sqrt [4]{-b+c x^4+a x^5}} \, dx}{5 a^{4/5}}-\frac {\int \frac {1}{\left (-\sqrt [5]{-1} \sqrt [5]{b}-\sqrt [5]{a} x\right ) \sqrt [4]{-b+c x^4+a x^5}} \, dx}{5 a^{4/5}}-\frac {\int \frac {1}{\left ((-1)^{2/5} \sqrt [5]{b}-\sqrt [5]{a} x\right ) \sqrt [4]{-b+c x^4+a x^5}} \, dx}{5 a^{4/5}}-\frac {\int \frac {1}{\left (-(-1)^{3/5} \sqrt [5]{b}-\sqrt [5]{a} x\right ) \sqrt [4]{-b+c x^4+a x^5}} \, dx}{5 a^{4/5}}-\frac {\int \frac {1}{\left ((-1)^{4/5} \sqrt [5]{b}-\sqrt [5]{a} x\right ) \sqrt [4]{-b+c x^4+a x^5}} \, dx}{5 a^{4/5}}+(5 b) \int \frac {x^4}{\left (b-a x^5\right )^2 \sqrt [4]{-b+c x^4+a x^5}} \, dx\\ \end {align*}

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Mathematica [A]
time = 1.79, size = 95, normalized size = 0.91 \begin {gather*} \frac {\frac {2 \sqrt [4]{c} x \left (-b+x^4 (c+a x)\right )^{3/4}}{b-a x^5}+\text {ArcTan}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{-b+x^4 (c+a x)}}\right )+\tanh ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{-b+x^4 (c+a x)}}\right )}{2 c^{5/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^4*(4*b + a*x^5))/((-b + a*x^5)^2*(-b + c*x^4 + a*x^5)^(1/4)),x]

[Out]

((2*c^(1/4)*x*(-b + x^4*(c + a*x))^(3/4))/(b - a*x^5) + ArcTan[(c^(1/4)*x)/(-b + x^4*(c + a*x))^(1/4)] + ArcTa
nh[(c^(1/4)*x)/(-b + x^4*(c + a*x))^(1/4)])/(2*c^(5/4))

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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {x^{4} \left (a \,x^{5}+4 b \right )}{\left (a \,x^{5}-b \right )^{2} \left (a \,x^{5}+c \,x^{4}-b \right )^{\frac {1}{4}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(a*x^5+4*b)/(a*x^5-b)^2/(a*x^5+c*x^4-b)^(1/4),x)

[Out]

int(x^4*(a*x^5+4*b)/(a*x^5-b)^2/(a*x^5+c*x^4-b)^(1/4),x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a*x^5+4*b)/(a*x^5-b)^2/(a*x^5+c*x^4-b)^(1/4),x, algorithm="maxima")

[Out]

integrate((a*x^5 + 4*b)*x^4/((a*x^5 + c*x^4 - b)^(1/4)*(a*x^5 - b)^2), x)

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a*x^5+4*b)/(a*x^5-b)^2/(a*x^5+c*x^4-b)^(1/4),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(a*x**5+4*b)/(a*x**5-b)**2/(a*x**5+c*x**4-b)**(1/4),x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a*x^5+4*b)/(a*x^5-b)^2/(a*x^5+c*x^4-b)^(1/4),x, algorithm="giac")

[Out]

integrate((a*x^5 + 4*b)*x^4/((a*x^5 + c*x^4 - b)^(1/4)*(a*x^5 - b)^2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^4\,\left (a\,x^5+4\,b\right )}{{\left (b-a\,x^5\right )}^2\,{\left (a\,x^5+c\,x^4-b\right )}^{1/4}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4*(4*b + a*x^5))/((b - a*x^5)^2*(a*x^5 - b + c*x^4)^(1/4)),x)

[Out]

int((x^4*(4*b + a*x^5))/((b - a*x^5)^2*(a*x^5 - b + c*x^4)^(1/4)), x)

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