Optimal. Leaf size=316 \[ \frac{\sqrt{a} b^{3/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} (6 b c-11 a d) \text{EllipticF}\left (\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right ),2\right )}{12 d^2 \left (a+b x^4\right )^{3/4}}-\frac{b x \sqrt [4]{a+b x^4} (6 b c-11 a d)}{12 d^2}+\frac{\sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} (b c-a d)^2 \Pi \left (-\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c d^2}+\frac{\sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} (b c-a d)^2 \Pi \left (\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c d^2}+\frac{b x \left (a+b x^4\right )^{5/4}}{6 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.338013, antiderivative size = 316, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.476, Rules used = {416, 528, 529, 237, 335, 275, 231, 407, 409, 1218} \[ \frac{\sqrt{a} b^{3/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} (6 b c-11 a d) F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{12 d^2 \left (a+b x^4\right )^{3/4}}-\frac{b x \sqrt [4]{a+b x^4} (6 b c-11 a d)}{12 d^2}+\frac{\sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} (b c-a d)^2 \Pi \left (-\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c d^2}+\frac{\sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} (b c-a d)^2 \Pi \left (\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c d^2}+\frac{b x \left (a+b x^4\right )^{5/4}}{6 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 416
Rule 528
Rule 529
Rule 237
Rule 335
Rule 275
Rule 231
Rule 407
Rule 409
Rule 1218
Rubi steps
\begin{align*} \int \frac{\left (a+b x^4\right )^{9/4}}{c+d x^4} \, dx &=\frac{b x \left (a+b x^4\right )^{5/4}}{6 d}+\frac{\int \frac{\sqrt [4]{a+b x^4} \left (-a (b c-6 a d)-b (6 b c-11 a d) x^4\right )}{c+d x^4} \, dx}{6 d}\\ &=-\frac{b (6 b c-11 a d) x \sqrt [4]{a+b x^4}}{12 d^2}+\frac{b x \left (a+b x^4\right )^{5/4}}{6 d}+\frac{\int \frac{a \left (6 b^2 c^2-13 a b c d+12 a^2 d^2\right )+b \left (12 b^2 c^2-30 a b c d+23 a^2 d^2\right ) x^4}{\left (a+b x^4\right )^{3/4} \left (c+d x^4\right )} \, dx}{12 d^2}\\ &=-\frac{b (6 b c-11 a d) x \sqrt [4]{a+b x^4}}{12 d^2}+\frac{b x \left (a+b x^4\right )^{5/4}}{6 d}-\frac{(a b (6 b c-11 a d)) \int \frac{1}{\left (a+b x^4\right )^{3/4}} \, dx}{12 d^2}+\frac{(b c-a d)^2 \int \frac{\sqrt [4]{a+b x^4}}{c+d x^4} \, dx}{d^2}\\ &=-\frac{b (6 b c-11 a d) x \sqrt [4]{a+b x^4}}{12 d^2}+\frac{b x \left (a+b x^4\right )^{5/4}}{6 d}-\frac{\left (a b (6 b c-11 a d) \left (1+\frac{a}{b x^4}\right )^{3/4} x^3\right ) \int \frac{1}{\left (1+\frac{a}{b x^4}\right )^{3/4} x^3} \, dx}{12 d^2 \left (a+b x^4\right )^{3/4}}+\frac{\left ((b c-a d)^2 \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-b x^4} \left (c-(b c-a d) x^4\right )} \, dx,x,\frac{x}{\sqrt [4]{a+b x^4}}\right )}{d^2}\\ &=-\frac{b (6 b c-11 a d) x \sqrt [4]{a+b x^4}}{12 d^2}+\frac{b x \left (a+b x^4\right )^{5/4}}{6 d}+\frac{\left (a b (6 b c-11 a d) \left (1+\frac{a}{b x^4}\right )^{3/4} x^3\right ) \operatorname{Subst}\left (\int \frac{x}{\left (1+\frac{a x^4}{b}\right )^{3/4}} \, dx,x,\frac{1}{x}\right )}{12 d^2 \left (a+b x^4\right )^{3/4}}+\frac{\left ((b c-a d)^2 \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-\frac{\sqrt{b c-a d} x^2}{\sqrt{c}}\right ) \sqrt{1-b x^4}} \, dx,x,\frac{x}{\sqrt [4]{a+b x^4}}\right )}{2 c d^2}+\frac{\left ((b c-a d)^2 \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+\frac{\sqrt{b c-a d} x^2}{\sqrt{c}}\right ) \sqrt{1-b x^4}} \, dx,x,\frac{x}{\sqrt [4]{a+b x^4}}\right )}{2 c d^2}\\ &=-\frac{b (6 b c-11 a d) x \sqrt [4]{a+b x^4}}{12 d^2}+\frac{b x \left (a+b x^4\right )^{5/4}}{6 d}+\frac{(b c-a d)^2 \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} \Pi \left (-\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )\right |-1\right )}{2 \sqrt [4]{b} c d^2}+\frac{(b c-a d)^2 \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} \Pi \left (\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )\right |-1\right )}{2 \sqrt [4]{b} c d^2}+\frac{\left (a b (6 b c-11 a d) \left (1+\frac{a}{b x^4}\right )^{3/4} x^3\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+\frac{a x^2}{b}\right )^{3/4}} \, dx,x,\frac{1}{x^2}\right )}{24 d^2 \left (a+b x^4\right )^{3/4}}\\ &=-\frac{b (6 b c-11 a d) x \sqrt [4]{a+b x^4}}{12 d^2}+\frac{b x \left (a+b x^4\right )^{5/4}}{6 d}+\frac{\sqrt{a} b^{3/2} (6 b c-11 a d) \left (1+\frac{a}{b x^4}\right )^{3/4} x^3 F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{12 d^2 \left (a+b x^4\right )^{3/4}}+\frac{(b c-a d)^2 \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} \Pi \left (-\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )\right |-1\right )}{2 \sqrt [4]{b} c d^2}+\frac{(b c-a d)^2 \sqrt{\frac{a}{a+b x^4}} \sqrt{a+b x^4} \Pi \left (\frac{\sqrt{b c-a d}}{\sqrt{b} \sqrt{c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )\right |-1\right )}{2 \sqrt [4]{b} c d^2}\\ \end{align*}
Mathematica [C] time = 0.659961, size = 294, normalized size = 0.93 \[ \frac{x \left (\frac{b x^4 \left (\frac{b x^4}{a}+1\right )^{3/4} \left (23 a^2 d^2-30 a b c d+12 b^2 c^2\right ) F_1\left (\frac{5}{4};\frac{3}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{c}-\frac{25 a^2 c \left (12 a^2 d^2-13 a b c d+6 b^2 c^2\right ) F_1\left (\frac{1}{4};\frac{3}{4},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{\left (c+d x^4\right ) \left (x^4 \left (4 a d F_1\left (\frac{5}{4};\frac{3}{4},2;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )+3 b c F_1\left (\frac{5}{4};\frac{7}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )-5 a c F_1\left (\frac{1}{4};\frac{3}{4},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )}+5 b \left (a+b x^4\right ) \left (13 a d-6 b c+2 b d x^4\right )\right )}{60 d^2 \left (a+b x^4\right )^{3/4}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.431, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{d{x}^{4}+c} \left ( b{x}^{4}+a \right ) ^{{\frac{9}{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{9}{4}}}{d x^{4} + c}\,{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{{\left (b x^{4} + a\right )}^{\frac{9}{4}}}{d x^{4} + c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]