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]
________________________________________________________________________________________
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]
[Out]
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*}
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________