Optimal. Leaf size=137 \[ -\frac{16 b^{3/4} (b c-a d)^{5/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{3 d^3 \sqrt{a+b x}}+\frac{8 b \sqrt{a+b x} \sqrt [4]{c+d x}}{3 d^2}-\frac{4 (a+b x)^{3/2}}{3 d (c+d x)^{3/4}} \]
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Rubi [A] time = 0.0804734, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {47, 50, 63, 224, 221} \[ -\frac{16 b^{3/4} (b c-a d)^{5/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{3 d^3 \sqrt{a+b x}}+\frac{8 b \sqrt{a+b x} \sqrt [4]{c+d x}}{3 d^2}-\frac{4 (a+b x)^{3/2}}{3 d (c+d x)^{3/4}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 224
Rule 221
Rubi steps
\begin{align*} \int \frac{(a+b x)^{3/2}}{(c+d x)^{7/4}} \, dx &=-\frac{4 (a+b x)^{3/2}}{3 d (c+d x)^{3/4}}+\frac{(2 b) \int \frac{\sqrt{a+b x}}{(c+d x)^{3/4}} \, dx}{d}\\ &=-\frac{4 (a+b x)^{3/2}}{3 d (c+d x)^{3/4}}+\frac{8 b \sqrt{a+b x} \sqrt [4]{c+d x}}{3 d^2}-\frac{(4 b (b c-a d)) \int \frac{1}{\sqrt{a+b x} (c+d x)^{3/4}} \, dx}{3 d^2}\\ &=-\frac{4 (a+b x)^{3/2}}{3 d (c+d x)^{3/4}}+\frac{8 b \sqrt{a+b x} \sqrt [4]{c+d x}}{3 d^2}-\frac{(16 b (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 d^3}\\ &=-\frac{4 (a+b x)^{3/2}}{3 d (c+d x)^{3/4}}+\frac{8 b \sqrt{a+b x} \sqrt [4]{c+d x}}{3 d^2}-\frac{\left (16 b (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 d^3 \sqrt{a+b x}}\\ &=-\frac{4 (a+b x)^{3/2}}{3 d (c+d x)^{3/4}}+\frac{8 b \sqrt{a+b x} \sqrt [4]{c+d x}}{3 d^2}-\frac{16 b^{3/4} (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 d^3 \sqrt{a+b x}}\\ \end{align*}
Mathematica [C] time = 0.0488485, size = 73, normalized size = 0.53 \[ \frac{2 (a+b x)^{5/2} \left (\frac{b (c+d x)}{b c-a d}\right )^{7/4} \, _2F_1\left (\frac{7}{4},\frac{5}{2};\frac{7}{2};\frac{d (a+b x)}{a d-b c}\right )}{5 b (c+d x)^{7/4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.044, size = 0, normalized size = 0. \begin{align*} \int{ \left ( bx+a \right ) ^{{\frac{3}{2}}} \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{{\left (b x + a\right )}^{\frac{3}{2}}}{{\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{{\left (b x + a\right )}^{\frac{3}{2}}{\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{\left (a + b x\right )^{\frac{3}{2}}}{\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{{\left (b x + a\right )}^{\frac{3}{2}}}{{\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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