Optimal. Leaf size=325 \[ \frac{5 d^{3/4} (b c-a d)^{3/2} ((a+b x) (c+d x))^{3/4} \sqrt{(a d+b c+2 b d x)^2} \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}+1\right ) \sqrt{\frac{(a d+b (c+2 d x))^2}{(b c-a d)^2 \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}+1\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt [4]{d} \sqrt [4]{(a+b x) (c+d x)}}{\sqrt{b c-a d}}\right ),\frac{1}{2}\right )}{3 \sqrt{2} b^{9/4} (a+b x)^{3/4} (c+d x)^{3/4} (a d+b c+2 b d x) \sqrt{(a d+b (c+2 d x))^2}}+\frac{10 d \sqrt [4]{a+b x} \sqrt [4]{c+d x}}{3 b^2}-\frac{4 (c+d x)^{5/4}}{3 b (a+b x)^{3/4}} \]
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Rubi [A] time = 0.27846, antiderivative size = 325, 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, 62, 623, 220} \[ \frac{5 d^{3/4} (b c-a d)^{3/2} ((a+b x) (c+d x))^{3/4} \sqrt{(a d+b c+2 b d x)^2} \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}+1\right ) \sqrt{\frac{(a d+b (c+2 d x))^2}{(b c-a d)^2 \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}+1\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt [4]{d} \sqrt [4]{(a+b x) (c+d x)}}{\sqrt{b c-a d}}\right )|\frac{1}{2}\right )}{3 \sqrt{2} b^{9/4} (a+b x)^{3/4} (c+d x)^{3/4} (a d+b c+2 b d x) \sqrt{(a d+b (c+2 d x))^2}}+\frac{10 d \sqrt [4]{a+b x} \sqrt [4]{c+d x}}{3 b^2}-\frac{4 (c+d x)^{5/4}}{3 b (a+b x)^{3/4}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 62
Rule 623
Rule 220
Rubi steps
\begin{align*} \int \frac{(c+d x)^{5/4}}{(a+b x)^{7/4}} \, dx &=-\frac{4 (c+d x)^{5/4}}{3 b (a+b x)^{3/4}}+\frac{(5 d) \int \frac{\sqrt [4]{c+d x}}{(a+b x)^{3/4}} \, dx}{3 b}\\ &=\frac{10 d \sqrt [4]{a+b x} \sqrt [4]{c+d x}}{3 b^2}-\frac{4 (c+d x)^{5/4}}{3 b (a+b x)^{3/4}}+\frac{(5 d (b c-a d)) \int \frac{1}{(a+b x)^{3/4} (c+d x)^{3/4}} \, dx}{6 b^2}\\ &=\frac{10 d \sqrt [4]{a+b x} \sqrt [4]{c+d x}}{3 b^2}-\frac{4 (c+d x)^{5/4}}{3 b (a+b x)^{3/4}}+\frac{\left (5 d (b c-a d) ((a+b x) (c+d x))^{3/4}\right ) \int \frac{1}{\left (a c+(b c+a d) x+b d x^2\right )^{3/4}} \, dx}{6 b^2 (a+b x)^{3/4} (c+d x)^{3/4}}\\ &=\frac{10 d \sqrt [4]{a+b x} \sqrt [4]{c+d x}}{3 b^2}-\frac{4 (c+d x)^{5/4}}{3 b (a+b x)^{3/4}}+\frac{\left (10 d (b c-a d) ((a+b x) (c+d x))^{3/4} \sqrt{(b c+a d+2 b d x)^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-4 a b c d+(b c+a d)^2+4 b d x^4}} \, dx,x,\sqrt [4]{(a+b x) (c+d x)}\right )}{3 b^2 (a+b x)^{3/4} (c+d x)^{3/4} (b c+a d+2 b d x)}\\ &=\frac{10 d \sqrt [4]{a+b x} \sqrt [4]{c+d x}}{3 b^2}-\frac{4 (c+d x)^{5/4}}{3 b (a+b x)^{3/4}}+\frac{5 d^{3/4} (b c-a d)^{3/2} ((a+b x) (c+d x))^{3/4} \sqrt{(b c+a d+2 b d x)^2} \left (1+\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}\right ) \sqrt{\frac{(a d+b (c+2 d x))^2}{(b c-a d)^2 \left (1+\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt [4]{d} \sqrt [4]{(a+b x) (c+d x)}}{\sqrt{b c-a d}}\right )|\frac{1}{2}\right )}{3 \sqrt{2} b^{9/4} (a+b x)^{3/4} (c+d x)^{3/4} (b c+a d+2 b d x) \sqrt{(a d+b (c+2 d x))^2}}\\ \end{align*}
Mathematica [C] time = 0.040524, size = 73, normalized size = 0.22 \[ -\frac{4 (c+d x)^{5/4} \, _2F_1\left (-\frac{5}{4},-\frac{3}{4};\frac{1}{4};\frac{d (a+b x)}{a d-b c}\right )}{3 b (a+b x)^{3/4} \left (\frac{b (c+d x)}{b c-a d}\right )^{5/4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.046, size = 0, normalized size = 0. \begin{align*} \int{ \left ( dx+c \right ) ^{{\frac{5}{4}}} \left ( bx+a \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 (d x + c\right )}^{\frac{5}{4}}}{{\left (b x + a\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{1}{4}}{\left (d x + c\right )}^{\frac{5}{4}}}{b^{2} x^{2} + 2 \, a b x + a^{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 (c + d x\right )^{\frac{5}{4}}}{\left (a + b x\right )^{\frac{7}{4}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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