Optimal. Leaf size=131 \[ -\frac{(b c-a d) \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{2 a^{3/4} c^{5/4}}-\frac{(b c-a d) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{2 a^{3/4} c^{5/4}}-\frac{\sqrt [4]{a+b x} (c+d x)^{3/4}}{c x} \]
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Rubi [A] time = 0.0565925, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {94, 93, 212, 208, 205} \[ -\frac{(b c-a d) \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{2 a^{3/4} c^{5/4}}-\frac{(b c-a d) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{2 a^{3/4} c^{5/4}}-\frac{\sqrt [4]{a+b x} (c+d x)^{3/4}}{c x} \]
Antiderivative was successfully verified.
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Rule 94
Rule 93
Rule 212
Rule 208
Rule 205
Rubi steps
\begin{align*} \int \frac{\sqrt [4]{a+b x}}{x^2 \sqrt [4]{c+d x}} \, dx &=-\frac{\sqrt [4]{a+b x} (c+d x)^{3/4}}{c x}+\frac{(b c-a d) \int \frac{1}{x (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx}{4 c}\\ &=-\frac{\sqrt [4]{a+b x} (c+d x)^{3/4}}{c x}+\frac{(b c-a d) \operatorname{Subst}\left (\int \frac{1}{-a+c x^4} \, dx,x,\frac{\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}\right )}{c}\\ &=-\frac{\sqrt [4]{a+b x} (c+d x)^{3/4}}{c x}-\frac{(b c-a d) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a}-\sqrt{c} x^2} \, dx,x,\frac{\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}\right )}{2 \sqrt{a} c}-\frac{(b c-a d) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a}+\sqrt{c} x^2} \, dx,x,\frac{\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}\right )}{2 \sqrt{a} c}\\ &=-\frac{\sqrt [4]{a+b x} (c+d x)^{3/4}}{c x}-\frac{(b c-a d) \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{2 a^{3/4} c^{5/4}}-\frac{(b c-a d) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{2 a^{3/4} c^{5/4}}\\ \end{align*}
Mathematica [C] time = 0.0189081, size = 73, normalized size = 0.56 \[ \frac{\sqrt [4]{a+b x} \left ((a d x-b c x) \, _2F_1\left (\frac{1}{4},1;\frac{5}{4};\frac{c (a+b x)}{a (c+d x)}\right )-a (c+d x)\right )}{a c x \sqrt [4]{c+d x}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.025, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}}\sqrt [4]{bx+a}{\frac{1}{\sqrt [4]{dx+c}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{\frac{1}{4}}}{{\left (d x + c\right )}^{\frac{1}{4}} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.08567, size = 1758, normalized size = 13.42 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt [4]{a + b x}}{x^{2} \sqrt [4]{c + d x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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.
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