Optimal. Leaf size=173 \[ \frac{b^{3/4} \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 d}+\frac{b^{3/4} \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 d}-\frac{(b c-a d)^{3/4} \tan ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} d}-\frac{(b c-a d)^{3/4} \tanh ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.103124, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381, Rules used = {408, 240, 212, 206, 203, 377, 208, 205} \[ \frac{b^{3/4} \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 d}+\frac{b^{3/4} \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 d}-\frac{(b c-a d)^{3/4} \tan ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} d}-\frac{(b c-a d)^{3/4} \tanh ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 408
Rule 240
Rule 212
Rule 206
Rule 203
Rule 377
Rule 208
Rule 205
Rubi steps
\begin{align*} \int \frac{\left (a+b x^4\right )^{3/4}}{c+d x^4} \, dx &=\frac{b \int \frac{1}{\sqrt [4]{a+b x^4}} \, dx}{d}-\frac{(b c-a d) \int \frac{1}{\sqrt [4]{a+b x^4} \left (c+d x^4\right )} \, dx}{d}\\ &=\frac{b \operatorname{Subst}\left (\int \frac{1}{1-b x^4} \, dx,x,\frac{x}{\sqrt [4]{a+b x^4}}\right )}{d}-\frac{(b c-a d) \operatorname{Subst}\left (\int \frac{1}{c-(b c-a d) x^4} \, dx,x,\frac{x}{\sqrt [4]{a+b x^4}}\right )}{d}\\ &=\frac{b \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{b} x^2} \, dx,x,\frac{x}{\sqrt [4]{a+b x^4}}\right )}{2 d}+\frac{b \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{b} x^2} \, dx,x,\frac{x}{\sqrt [4]{a+b x^4}}\right )}{2 d}-\frac{(b c-a d) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c}-\sqrt{b c-a d} x^2} \, dx,x,\frac{x}{\sqrt [4]{a+b x^4}}\right )}{2 \sqrt{c} d}-\frac{(b c-a d) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c}+\sqrt{b c-a d} x^2} \, dx,x,\frac{x}{\sqrt [4]{a+b x^4}}\right )}{2 \sqrt{c} d}\\ &=\frac{b^{3/4} \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 d}-\frac{(b c-a d)^{3/4} \tan ^{-1}\left (\frac{\sqrt [4]{b c-a d} x}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} d}+\frac{b^{3/4} \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 d}-\frac{(b c-a d)^{3/4} \tanh ^{-1}\left (\frac{\sqrt [4]{b c-a d} x}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} d}\\ \end{align*}
Mathematica [C] time = 0.158452, size = 161, normalized size = 0.93 \[ \frac{5 a c x \left (a+b x^4\right )^{3/4} F_1\left (\frac{1}{4};-\frac{3}{4},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{\left (c+d x^4\right ) \left (x^4 \left (3 b c F_1\left (\frac{5}{4};\frac{1}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )-4 a d F_1\left (\frac{5}{4};-\frac{3}{4},2;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )+5 a c F_1\left (\frac{1}{4};-\frac{3}{4},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.415, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{d{x}^{4}+c} \left ( b{x}^{4}+a \right ) ^{{\frac{3}{4}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{d x^{4} + c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.1963, size = 1794, normalized size = 10.37 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b x^{4}\right )^{\frac{3}{4}}}{c + d x^{4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{d x^{4} + c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]