Optimal. Leaf size=178 \[ \frac{5 b^{3/4} d \sqrt{-\frac{d (a+b x)}{b c-a d}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{\sqrt{a+b x} (b c-a d)^{11/4}}+\frac{5 d^2 \sqrt{a+b x}}{(c+d x)^{3/4} (b c-a d)^3}+\frac{3 d}{\sqrt{a+b x} (c+d x)^{3/4} (b c-a d)^2}-\frac{2}{3 (a+b x)^{3/2} (c+d x)^{3/4} (b c-a d)} \]
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Rubi [A] time = 0.103164, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {51, 63, 224, 221} \[ \frac{5 b^{3/4} d \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{\sqrt{a+b x} (b c-a d)^{11/4}}+\frac{5 d^2 \sqrt{a+b x}}{(c+d x)^{3/4} (b c-a d)^3}+\frac{3 d}{\sqrt{a+b x} (c+d x)^{3/4} (b c-a d)^2}-\frac{2}{3 (a+b x)^{3/2} (c+d x)^{3/4} (b c-a d)} \]
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)^{5/2} (c+d x)^{7/4}} \, dx &=-\frac{2}{3 (b c-a d) (a+b x)^{3/2} (c+d x)^{3/4}}-\frac{(3 d) \int \frac{1}{(a+b x)^{3/2} (c+d x)^{7/4}} \, dx}{2 (b c-a d)}\\ &=-\frac{2}{3 (b c-a d) (a+b x)^{3/2} (c+d x)^{3/4}}+\frac{3 d}{(b c-a d)^2 \sqrt{a+b x} (c+d x)^{3/4}}+\frac{\left (15 d^2\right ) \int \frac{1}{\sqrt{a+b x} (c+d x)^{7/4}} \, dx}{4 (b c-a d)^2}\\ &=-\frac{2}{3 (b c-a d) (a+b x)^{3/2} (c+d x)^{3/4}}+\frac{3 d}{(b c-a d)^2 \sqrt{a+b x} (c+d x)^{3/4}}+\frac{5 d^2 \sqrt{a+b x}}{(b c-a d)^3 (c+d x)^{3/4}}+\frac{\left (5 b d^2\right ) \int \frac{1}{\sqrt{a+b x} (c+d x)^{3/4}} \, dx}{4 (b c-a d)^3}\\ &=-\frac{2}{3 (b c-a d) (a+b x)^{3/2} (c+d x)^{3/4}}+\frac{3 d}{(b c-a d)^2 \sqrt{a+b x} (c+d x)^{3/4}}+\frac{5 d^2 \sqrt{a+b x}}{(b c-a d)^3 (c+d x)^{3/4}}+\frac{(5 b 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 )}{(b c-a d)^3}\\ &=-\frac{2}{3 (b c-a d) (a+b x)^{3/2} (c+d x)^{3/4}}+\frac{3 d}{(b c-a d)^2 \sqrt{a+b x} (c+d x)^{3/4}}+\frac{5 d^2 \sqrt{a+b x}}{(b c-a d)^3 (c+d x)^{3/4}}+\frac{\left (5 b 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 )}{(b c-a d)^3 \sqrt{a+b x}}\\ &=-\frac{2}{3 (b c-a d) (a+b x)^{3/2} (c+d x)^{3/4}}+\frac{3 d}{(b c-a d)^2 \sqrt{a+b x} (c+d x)^{3/4}}+\frac{5 d^2 \sqrt{a+b x}}{(b c-a d)^3 (c+d x)^{3/4}}+\frac{5 b^{3/4} 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 )}{(b c-a d)^{11/4} \sqrt{a+b x}}\\ \end{align*}
Mathematica [C] time = 0.0382015, size = 73, normalized size = 0.41 \[ -\frac{2 \left (\frac{b (c+d x)}{b c-a d}\right )^{7/4} \, _2F_1\left (-\frac{3}{2},\frac{7}{4};-\frac{1}{2};\frac{d (a+b x)}{a d-b c}\right )}{3 b (a+b x)^{3/2} (c+d x)^{7/4}} \]
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{5}{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{5}{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^{3} d^{2} x^{5} + a^{3} c^{2} +{\left (2 \, b^{3} c d + 3 \, a b^{2} d^{2}\right )} x^{4} +{\left (b^{3} c^{2} + 6 \, a b^{2} c d + 3 \, a^{2} b d^{2}\right )} x^{3} +{\left (3 \, a b^{2} c^{2} + 6 \, a^{2} b c d + a^{3} d^{2}\right )} x^{2} +{\left (3 \, a^{2} b c^{2} + 2 \, a^{3} 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{5}{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{5}{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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