Optimal. Leaf size=233 \[ \frac{b x \left (113 a^2 d^2-100 a b c d+32 b^2 c^2\right )}{45 a^3 \sqrt [4]{a+b x^4} (b c-a d)^3}+\frac{b x (8 b c-17 a d)}{45 a^2 \left (a+b x^4\right )^{5/4} (b c-a d)^2}-\frac{d^3 \tan ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} (b c-a d)^{13/4}}-\frac{d^3 \tanh ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} (b c-a d)^{13/4}}+\frac{b x}{9 a \left (a+b x^4\right )^{9/4} (b c-a d)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.294959, antiderivative size = 233, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {414, 527, 12, 377, 212, 208, 205} \[ \frac{b x \left (113 a^2 d^2-100 a b c d+32 b^2 c^2\right )}{45 a^3 \sqrt [4]{a+b x^4} (b c-a d)^3}+\frac{b x (8 b c-17 a d)}{45 a^2 \left (a+b x^4\right )^{5/4} (b c-a d)^2}-\frac{d^3 \tan ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} (b c-a d)^{13/4}}-\frac{d^3 \tanh ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} (b c-a d)^{13/4}}+\frac{b x}{9 a \left (a+b x^4\right )^{9/4} (b c-a d)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 414
Rule 527
Rule 12
Rule 377
Rule 212
Rule 208
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b x^4\right )^{13/4} \left (c+d x^4\right )} \, dx &=\frac{b x}{9 a (b c-a d) \left (a+b x^4\right )^{9/4}}-\frac{\int \frac{-8 b c+9 a d-8 b d x^4}{\left (a+b x^4\right )^{9/4} \left (c+d x^4\right )} \, dx}{9 a (b c-a d)}\\ &=\frac{b x}{9 a (b c-a d) \left (a+b x^4\right )^{9/4}}+\frac{b (8 b c-17 a d) x}{45 a^2 (b c-a d)^2 \left (a+b x^4\right )^{5/4}}+\frac{\int \frac{32 b^2 c^2-68 a b c d+45 a^2 d^2+4 b d (8 b c-17 a d) x^4}{\left (a+b x^4\right )^{5/4} \left (c+d x^4\right )} \, dx}{45 a^2 (b c-a d)^2}\\ &=\frac{b x}{9 a (b c-a d) \left (a+b x^4\right )^{9/4}}+\frac{b (8 b c-17 a d) x}{45 a^2 (b c-a d)^2 \left (a+b x^4\right )^{5/4}}+\frac{b \left (32 b^2 c^2-100 a b c d+113 a^2 d^2\right ) x}{45 a^3 (b c-a d)^3 \sqrt [4]{a+b x^4}}-\frac{\int \frac{45 a^3 d^3}{\sqrt [4]{a+b x^4} \left (c+d x^4\right )} \, dx}{45 a^3 (b c-a d)^3}\\ &=\frac{b x}{9 a (b c-a d) \left (a+b x^4\right )^{9/4}}+\frac{b (8 b c-17 a d) x}{45 a^2 (b c-a d)^2 \left (a+b x^4\right )^{5/4}}+\frac{b \left (32 b^2 c^2-100 a b c d+113 a^2 d^2\right ) x}{45 a^3 (b c-a d)^3 \sqrt [4]{a+b x^4}}-\frac{d^3 \int \frac{1}{\sqrt [4]{a+b x^4} \left (c+d x^4\right )} \, dx}{(b c-a d)^3}\\ &=\frac{b x}{9 a (b c-a d) \left (a+b x^4\right )^{9/4}}+\frac{b (8 b c-17 a d) x}{45 a^2 (b c-a d)^2 \left (a+b x^4\right )^{5/4}}+\frac{b \left (32 b^2 c^2-100 a b c d+113 a^2 d^2\right ) x}{45 a^3 (b c-a d)^3 \sqrt [4]{a+b x^4}}-\frac{d^3 \operatorname{Subst}\left (\int \frac{1}{c-(b c-a d) x^4} \, dx,x,\frac{x}{\sqrt [4]{a+b x^4}}\right )}{(b c-a d)^3}\\ &=\frac{b x}{9 a (b c-a d) \left (a+b x^4\right )^{9/4}}+\frac{b (8 b c-17 a d) x}{45 a^2 (b c-a d)^2 \left (a+b x^4\right )^{5/4}}+\frac{b \left (32 b^2 c^2-100 a b c d+113 a^2 d^2\right ) x}{45 a^3 (b c-a d)^3 \sqrt [4]{a+b x^4}}-\frac{d^3 \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} (b c-a d)^3}-\frac{d^3 \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} (b c-a d)^3}\\ &=\frac{b x}{9 a (b c-a d) \left (a+b x^4\right )^{9/4}}+\frac{b (8 b c-17 a d) x}{45 a^2 (b c-a d)^2 \left (a+b x^4\right )^{5/4}}+\frac{b \left (32 b^2 c^2-100 a b c d+113 a^2 d^2\right ) x}{45 a^3 (b c-a d)^3 \sqrt [4]{a+b x^4}}-\frac{d^3 \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} (b c-a d)^{13/4}}-\frac{d^3 \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} (b c-a d)^{13/4}}\\ \end{align*}
Mathematica [C] time = 4.36266, size = 1172, normalized size = 5.03 \[ \text{result too large to display} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.425, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{d{x}^{4}+c} \left ( b{x}^{4}+a \right ) ^{-{\frac{13}{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{1}{{\left (b x^{4} + a\right )}^{\frac{13}{4}}{\left (d x^{4} + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{1}{{\left (b x^{4} + a\right )}^{\frac{13}{4}}{\left (d x^{4} + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]