Optimal. Leaf size=191 \[ -\frac{2 \sqrt{-\frac{d (a+b x)}{b c-a d}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{b^{3/4} \sqrt{a+b x} \sqrt [4]{b c-a d}}+\frac{2 \sqrt{-\frac{d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{b^{3/4} \sqrt{a+b x} \sqrt [4]{b c-a d}}-\frac{2 (c+d x)^{3/4}}{\sqrt{a+b x} (b c-a d)} \]
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Rubi [A] time = 0.224496, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421, Rules used = {51, 63, 307, 224, 221, 1200, 1199, 424} \[ -\frac{2 \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{b^{3/4} \sqrt{a+b x} \sqrt [4]{b c-a d}}+\frac{2 \sqrt{-\frac{d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{b^{3/4} \sqrt{a+b x} \sqrt [4]{b c-a d}}-\frac{2 (c+d x)^{3/4}}{\sqrt{a+b x} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 307
Rule 224
Rule 221
Rule 1200
Rule 1199
Rule 424
Rubi steps
\begin{align*} \int \frac{1}{(a+b x)^{3/2} \sqrt [4]{c+d x}} \, dx &=-\frac{2 (c+d x)^{3/4}}{(b c-a d) \sqrt{a+b x}}+\frac{d \int \frac{1}{\sqrt{a+b x} \sqrt [4]{c+d x}} \, dx}{2 (b c-a d)}\\ &=-\frac{2 (c+d x)^{3/4}}{(b c-a d) \sqrt{a+b x}}+\frac{2 \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a-\frac{b c}{d}+\frac{b x^4}{d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{b c-a d}\\ &=-\frac{2 (c+d x)^{3/4}}{(b c-a d) \sqrt{a+b x}}-\frac{2 \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 )}{\sqrt{b} \sqrt{b c-a d}}+\frac{2 \operatorname{Subst}\left (\int \frac{1+\frac{\sqrt{b} x^2}{\sqrt{b c-a d}}}{\sqrt{a-\frac{b c}{d}+\frac{b x^4}{d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{\sqrt{b} \sqrt{b c-a d}}\\ &=-\frac{2 (c+d x)^{3/4}}{(b c-a d) \sqrt{a+b x}}-\frac{\left (2 \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 )}{\sqrt{b} \sqrt{b c-a d} \sqrt{a+b x}}+\frac{\left (2 \sqrt{\frac{d (a+b x)}{-b c+a d}}\right ) \operatorname{Subst}\left (\int \frac{1+\frac{\sqrt{b} x^2}{\sqrt{b c-a d}}}{\sqrt{1+\frac{b x^4}{\left (a-\frac{b c}{d}\right ) d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{\sqrt{b} \sqrt{b c-a d} \sqrt{a+b x}}\\ &=-\frac{2 (c+d x)^{3/4}}{(b c-a d) \sqrt{a+b x}}-\frac{2 \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 )}{b^{3/4} \sqrt [4]{b c-a d} \sqrt{a+b x}}+\frac{\left (2 \sqrt{\frac{d (a+b x)}{-b c+a d}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{\sqrt{b} x^2}{\sqrt{b c-a d}}}}{\sqrt{1-\frac{\sqrt{b} x^2}{\sqrt{b c-a d}}}} \, dx,x,\sqrt [4]{c+d x}\right )}{\sqrt{b} \sqrt{b c-a d} \sqrt{a+b x}}\\ &=-\frac{2 (c+d x)^{3/4}}{(b c-a d) \sqrt{a+b x}}+\frac{2 \sqrt{-\frac{d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{b^{3/4} \sqrt [4]{b c-a d} \sqrt{a+b x}}-\frac{2 \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 )}{b^{3/4} \sqrt [4]{b c-a d} \sqrt{a+b x}}\\ \end{align*}
Mathematica [C] time = 0.0216798, size = 71, normalized size = 0.37 \[ -\frac{2 \sqrt [4]{\frac{b (c+d x)}{b c-a d}} \, _2F_1\left (-\frac{1}{2},\frac{1}{4};\frac{1}{2};\frac{d (a+b x)}{a d-b c}\right )}{b \sqrt{a+b x} \sqrt [4]{c+d x}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.04, size = 0, normalized size = 0. \begin{align*} \int{ \left ( bx+a \right ) ^{-{\frac{3}{2}}}{\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{1}{{\left (b x + a\right )}^{\frac{3}{2}}{\left (d x + c\right )}^{\frac{1}{4}}}\,{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{\sqrt{b x + a}{\left (d x + c\right )}^{\frac{3}{4}}}{b^{2} d x^{3} + a^{2} c +{\left (b^{2} c + 2 \, a b d\right )} x^{2} +{\left (2 \, a b c + a^{2} d\right )} x}, 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{1}{\left (a + b x\right )^{\frac{3}{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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x + a\right )}^{\frac{3}{2}}{\left (d x + c\right )}^{\frac{1}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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