Optimal. Leaf size=146 \[ -\frac{10 b^{3/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{3 \sqrt{a+b x} (b c-a d)^{7/4}}-\frac{10 d \sqrt{a+b x}}{3 (c+d x)^{3/4} (b c-a d)^2}-\frac{2}{\sqrt{a+b x} (c+d x)^{3/4} (b c-a d)} \]
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Rubi [A] time = 0.0784827, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {51, 63, 224, 221} \[ -\frac{10 b^{3/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{3 \sqrt{a+b x} (b c-a d)^{7/4}}-\frac{10 d \sqrt{a+b x}}{3 (c+d x)^{3/4} (b c-a d)^2}-\frac{2}{\sqrt{a+b x} (c+d x)^{3/4} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 224
Rule 221
Rubi steps
\begin{align*} \int \frac{1}{(a+b x)^{3/2} (c+d x)^{7/4}} \, dx &=-\frac{2}{(b c-a d) \sqrt{a+b x} (c+d x)^{3/4}}-\frac{(5 d) \int \frac{1}{\sqrt{a+b x} (c+d x)^{7/4}} \, dx}{2 (b c-a d)}\\ &=-\frac{2}{(b c-a d) \sqrt{a+b x} (c+d x)^{3/4}}-\frac{10 d \sqrt{a+b x}}{3 (b c-a d)^2 (c+d x)^{3/4}}-\frac{(5 b d) \int \frac{1}{\sqrt{a+b x} (c+d x)^{3/4}} \, dx}{6 (b c-a d)^2}\\ &=-\frac{2}{(b c-a d) \sqrt{a+b x} (c+d x)^{3/4}}-\frac{10 d \sqrt{a+b x}}{3 (b c-a d)^2 (c+d x)^{3/4}}-\frac{(10 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 (b c-a d)^2}\\ &=-\frac{2}{(b c-a d) \sqrt{a+b x} (c+d x)^{3/4}}-\frac{10 d \sqrt{a+b x}}{3 (b c-a d)^2 (c+d x)^{3/4}}-\frac{\left (10 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 (b c-a d)^2 \sqrt{a+b x}}\\ &=-\frac{2}{(b c-a d) \sqrt{a+b x} (c+d x)^{3/4}}-\frac{10 d \sqrt{a+b x}}{3 (b c-a d)^2 (c+d x)^{3/4}}-\frac{10 b^{3/4} \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 c-a d)^{7/4} \sqrt{a+b x}}\\ \end{align*}
Mathematica [C] time = 0.0355567, size = 71, normalized size = 0.49 \[ -\frac{2 \left (\frac{b (c+d x)}{b c-a d}\right )^{7/4} \, _2F_1\left (-\frac{1}{2},\frac{7}{4};\frac{1}{2};\frac{d (a+b x)}{a d-b c}\right )}{b \sqrt{a+b x} (c+d x)^{7/4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.042, 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{1}{{\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{\sqrt{b x + a}{\left (d x + c\right )}^{\frac{1}{4}}}{b^{2} d^{2} x^{4} + a^{2} c^{2} + 2 \,{\left (b^{2} c d + a b d^{2}\right )} x^{3} +{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{2} + 2 \,{\left (a b c^{2} + a^{2} c 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}} \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{1}{{\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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