Optimal. Leaf size=232 \[ \frac{8 (b c-a d)^{11/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{15 b^{7/4} d^2 \sqrt{a+b x}}-\frac{8 (b c-a d)^{11/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{15 b^{7/4} d^2 \sqrt{a+b x}}+\frac{4 \sqrt{a+b x} (c+d x)^{3/4} (b c-a d)}{15 b d}+\frac{4 (a+b x)^{3/2} (c+d x)^{3/4}}{9 b} \]
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Rubi [A] time = 0.245906, antiderivative size = 232, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421, Rules used = {50, 63, 307, 224, 221, 1200, 1199, 424} \[ \frac{8 (b c-a d)^{11/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{15 b^{7/4} d^2 \sqrt{a+b x}}-\frac{8 (b c-a d)^{11/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{15 b^{7/4} d^2 \sqrt{a+b x}}+\frac{4 \sqrt{a+b x} (c+d x)^{3/4} (b c-a d)}{15 b d}+\frac{4 (a+b x)^{3/2} (c+d x)^{3/4}}{9 b} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 307
Rule 224
Rule 221
Rule 1200
Rule 1199
Rule 424
Rubi steps
\begin{align*} \int \sqrt{a+b x} (c+d x)^{3/4} \, dx &=\frac{4 (a+b x)^{3/2} (c+d x)^{3/4}}{9 b}+\frac{(b c-a d) \int \frac{\sqrt{a+b x}}{\sqrt [4]{c+d x}} \, dx}{3 b}\\ &=\frac{4 (b c-a d) \sqrt{a+b x} (c+d x)^{3/4}}{15 b d}+\frac{4 (a+b x)^{3/2} (c+d x)^{3/4}}{9 b}-\frac{\left (2 (b c-a d)^2\right ) \int \frac{1}{\sqrt{a+b x} \sqrt [4]{c+d x}} \, dx}{15 b d}\\ &=\frac{4 (b c-a d) \sqrt{a+b x} (c+d x)^{3/4}}{15 b d}+\frac{4 (a+b x)^{3/2} (c+d x)^{3/4}}{9 b}-\frac{\left (8 (b c-a d)^2\right ) \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 )}{15 b d^2}\\ &=\frac{4 (b c-a d) \sqrt{a+b x} (c+d x)^{3/4}}{15 b d}+\frac{4 (a+b x)^{3/2} (c+d x)^{3/4}}{9 b}+\frac{\left (8 (b c-a d)^{5/2}\right ) \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 )}{15 b^{3/2} d^2}-\frac{\left (8 (b c-a d)^{5/2}\right ) \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 )}{15 b^{3/2} d^2}\\ &=\frac{4 (b c-a d) \sqrt{a+b x} (c+d x)^{3/4}}{15 b d}+\frac{4 (a+b x)^{3/2} (c+d x)^{3/4}}{9 b}+\frac{\left (8 (b c-a d)^{5/2} \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 )}{15 b^{3/2} d^2 \sqrt{a+b x}}-\frac{\left (8 (b c-a d)^{5/2} \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 )}{15 b^{3/2} d^2 \sqrt{a+b x}}\\ &=\frac{4 (b c-a d) \sqrt{a+b x} (c+d x)^{3/4}}{15 b d}+\frac{4 (a+b x)^{3/2} (c+d x)^{3/4}}{9 b}+\frac{8 (b c-a d)^{11/4} \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 )}{15 b^{7/4} d^2 \sqrt{a+b x}}-\frac{\left (8 (b c-a d)^{5/2} \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 )}{15 b^{3/2} d^2 \sqrt{a+b x}}\\ &=\frac{4 (b c-a d) \sqrt{a+b x} (c+d x)^{3/4}}{15 b d}+\frac{4 (a+b x)^{3/2} (c+d x)^{3/4}}{9 b}-\frac{8 (b c-a d)^{11/4} \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 )}{15 b^{7/4} d^2 \sqrt{a+b x}}+\frac{8 (b c-a d)^{11/4} \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 )}{15 b^{7/4} d^2 \sqrt{a+b x}}\\ \end{align*}
Mathematica [C] time = 0.0265635, size = 73, normalized size = 0.31 \[ \frac{2 (a+b x)^{3/2} (c+d x)^{3/4} \, _2F_1\left (-\frac{3}{4},\frac{3}{2};\frac{5}{2};\frac{d (a+b x)}{a d-b c}\right )}{3 b \left (\frac{b (c+d x)}{b c-a d}\right )^{3/4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.016, size = 0, normalized size = 0. \begin{align*} \int \sqrt{bx+a} \left ( dx+c \right ) ^{{\frac{3}{4}}}\, 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 \sqrt{b x + a}{\left (d x + c\right )}^{\frac{3}{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 (\sqrt{b x + a}{\left (d x + c\right )}^{\frac{3}{4}}, 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 \sqrt{a + b x} \left (c + d x\right )^{\frac{3}{4}}\, 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 \sqrt{b x + a}{\left (d x + c\right )}^{\frac{3}{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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