Optimal. Leaf size=111 \[ \frac{4 (b c-a d)^{5/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{3 b^{5/4} d \sqrt{a+b x}}+\frac{4 \sqrt{a+b x} \sqrt [4]{c+d x}}{3 b} \]
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Rubi [A] time = 0.0673849, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {50, 63, 224, 221} \[ \frac{4 (b c-a d)^{5/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{3 b^{5/4} d \sqrt{a+b x}}+\frac{4 \sqrt{a+b x} \sqrt [4]{c+d x}}{3 b} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 224
Rule 221
Rubi steps
\begin{align*} \int \frac{\sqrt [4]{c+d x}}{\sqrt{a+b x}} \, dx &=\frac{4 \sqrt{a+b x} \sqrt [4]{c+d x}}{3 b}+\frac{(b c-a d) \int \frac{1}{\sqrt{a+b x} (c+d x)^{3/4}} \, dx}{3 b}\\ &=\frac{4 \sqrt{a+b x} \sqrt [4]{c+d x}}{3 b}+\frac{(4 (b c-a d)) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a-\frac{b c}{d}+\frac{b x^4}{d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{3 b d}\\ &=\frac{4 \sqrt{a+b x} \sqrt [4]{c+d x}}{3 b}+\frac{\left (4 (b c-a d) \sqrt{\frac{d (a+b x)}{-b c+a d}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{b x^4}{\left (a-\frac{b c}{d}\right ) d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{3 b d \sqrt{a+b x}}\\ &=\frac{4 \sqrt{a+b x} \sqrt [4]{c+d x}}{3 b}+\frac{4 (b c-a d)^{5/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{3 b^{5/4} d \sqrt{a+b x}}\\ \end{align*}
Mathematica [C] time = 0.0228263, size = 71, normalized size = 0.64 \[ \frac{2 \sqrt{a+b x} \sqrt [4]{c+d x} \, _2F_1\left (-\frac{1}{4},\frac{1}{2};\frac{3}{2};\frac{d (a+b x)}{a d-b c}\right )}{b \sqrt [4]{\frac{b (c+d x)}{b c-a d}}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.016, size = 0, normalized size = 0. \begin{align*} \int{\sqrt [4]{dx+c}{\frac{1}{\sqrt{bx+a}}}}\, 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 (d x + c\right )}^{\frac{1}{4}}}{\sqrt{b x + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (d x + c\right )}^{\frac{1}{4}}}{\sqrt{b x + a}}, x\right ) \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]{c + d x}}{\sqrt{a + b x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x + c\right )}^{\frac{1}{4}}}{\sqrt{b x + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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