Optimal. Leaf size=111 \[ \frac{8 b^{3/4} \sqrt [4]{b c-a d} \sqrt{-\frac{d (a+b x)}{b c-a d}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{3 d^2 \sqrt{a+b x}}-\frac{4 \sqrt{a+b x}}{3 d (c+d x)^{3/4}} \]
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Rubi [A] time = 0.0637834, 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 = {47, 63, 224, 221} \[ \frac{8 b^{3/4} \sqrt [4]{b c-a d} \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{3 d^2 \sqrt{a+b x}}-\frac{4 \sqrt{a+b x}}{3 d (c+d x)^{3/4}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 224
Rule 221
Rubi steps
\begin{align*} \int \frac{\sqrt{a+b x}}{(c+d x)^{7/4}} \, dx &=-\frac{4 \sqrt{a+b x}}{3 d (c+d x)^{3/4}}+\frac{(2 b) \int \frac{1}{\sqrt{a+b x} (c+d x)^{3/4}} \, dx}{3 d}\\ &=-\frac{4 \sqrt{a+b x}}{3 d (c+d x)^{3/4}}+\frac{(8 b) \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 d^2}\\ &=-\frac{4 \sqrt{a+b x}}{3 d (c+d x)^{3/4}}+\frac{\left (8 b \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 d^2 \sqrt{a+b x}}\\ &=-\frac{4 \sqrt{a+b x}}{3 d (c+d x)^{3/4}}+\frac{8 b^{3/4} \sqrt [4]{b c-a d} \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 d^2 \sqrt{a+b x}}\\ \end{align*}
Mathematica [C] time = 0.044321, size = 73, normalized size = 0.66 \[ \frac{2 (a+b x)^{3/2} \left (\frac{b (c+d x)}{b c-a d}\right )^{7/4} \, _2F_1\left (\frac{3}{2},\frac{7}{4};\frac{5}{2};\frac{d (a+b x)}{a d-b c}\right )}{3 b (c+d x)^{7/4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.037, size = 0, normalized size = 0. \begin{align*} \int{\sqrt{bx+a} \left ( dx+c \right ) ^{-{\frac{7}{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 \frac{\sqrt{b x + a}}{{\left (d x + c\right )}^{\frac{7}{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{1}{4}}}{d^{2} x^{2} + 2 \, c d x + c^{2}}, 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{a + b x}}{\left (c + d x\right )^{\frac{7}{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 \frac{\sqrt{b x + a}}{{\left (d x + c\right )}^{\frac{7}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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