Optimal. Leaf size=220 \[ \frac{16 (b c-a d)^{17/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{231 b^{9/4} d^3 \sqrt{a+b x}}-\frac{8 \sqrt{a+b x} \sqrt [4]{c+d x} (b c-a d)^3}{231 b^2 d^2}+\frac{4 (a+b x)^{3/2} \sqrt [4]{c+d x} (b c-a d)^2}{231 b^2 d}+\frac{4 (a+b x)^{5/2} \sqrt [4]{c+d x} (b c-a d)}{33 b^2}+\frac{4 (a+b x)^{5/2} (c+d x)^{5/4}}{15 b} \]
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Rubi [A] time = 0.168037, antiderivative size = 220, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {50, 63, 224, 221} \[ -\frac{8 \sqrt{a+b x} \sqrt [4]{c+d x} (b c-a d)^3}{231 b^2 d^2}+\frac{16 (b c-a d)^{17/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{231 b^{9/4} d^3 \sqrt{a+b x}}+\frac{4 (a+b x)^{3/2} \sqrt [4]{c+d x} (b c-a d)^2}{231 b^2 d}+\frac{4 (a+b x)^{5/2} \sqrt [4]{c+d x} (b c-a d)}{33 b^2}+\frac{4 (a+b x)^{5/2} (c+d x)^{5/4}}{15 b} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 224
Rule 221
Rubi steps
\begin{align*} \int (a+b x)^{3/2} (c+d x)^{5/4} \, dx &=\frac{4 (a+b x)^{5/2} (c+d x)^{5/4}}{15 b}+\frac{(b c-a d) \int (a+b x)^{3/2} \sqrt [4]{c+d x} \, dx}{3 b}\\ &=\frac{4 (b c-a d) (a+b x)^{5/2} \sqrt [4]{c+d x}}{33 b^2}+\frac{4 (a+b x)^{5/2} (c+d x)^{5/4}}{15 b}+\frac{(b c-a d)^2 \int \frac{(a+b x)^{3/2}}{(c+d x)^{3/4}} \, dx}{33 b^2}\\ &=\frac{4 (b c-a d)^2 (a+b x)^{3/2} \sqrt [4]{c+d x}}{231 b^2 d}+\frac{4 (b c-a d) (a+b x)^{5/2} \sqrt [4]{c+d x}}{33 b^2}+\frac{4 (a+b x)^{5/2} (c+d x)^{5/4}}{15 b}-\frac{\left (2 (b c-a d)^3\right ) \int \frac{\sqrt{a+b x}}{(c+d x)^{3/4}} \, dx}{77 b^2 d}\\ &=-\frac{8 (b c-a d)^3 \sqrt{a+b x} \sqrt [4]{c+d x}}{231 b^2 d^2}+\frac{4 (b c-a d)^2 (a+b x)^{3/2} \sqrt [4]{c+d x}}{231 b^2 d}+\frac{4 (b c-a d) (a+b x)^{5/2} \sqrt [4]{c+d x}}{33 b^2}+\frac{4 (a+b x)^{5/2} (c+d x)^{5/4}}{15 b}+\frac{\left (4 (b c-a d)^4\right ) \int \frac{1}{\sqrt{a+b x} (c+d x)^{3/4}} \, dx}{231 b^2 d^2}\\ &=-\frac{8 (b c-a d)^3 \sqrt{a+b x} \sqrt [4]{c+d x}}{231 b^2 d^2}+\frac{4 (b c-a d)^2 (a+b x)^{3/2} \sqrt [4]{c+d x}}{231 b^2 d}+\frac{4 (b c-a d) (a+b x)^{5/2} \sqrt [4]{c+d x}}{33 b^2}+\frac{4 (a+b x)^{5/2} (c+d x)^{5/4}}{15 b}+\frac{\left (16 (b c-a d)^4\right ) \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 )}{231 b^2 d^3}\\ &=-\frac{8 (b c-a d)^3 \sqrt{a+b x} \sqrt [4]{c+d x}}{231 b^2 d^2}+\frac{4 (b c-a d)^2 (a+b x)^{3/2} \sqrt [4]{c+d x}}{231 b^2 d}+\frac{4 (b c-a d) (a+b x)^{5/2} \sqrt [4]{c+d x}}{33 b^2}+\frac{4 (a+b x)^{5/2} (c+d x)^{5/4}}{15 b}+\frac{\left (16 (b c-a d)^4 \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 )}{231 b^2 d^3 \sqrt{a+b x}}\\ &=-\frac{8 (b c-a d)^3 \sqrt{a+b x} \sqrt [4]{c+d x}}{231 b^2 d^2}+\frac{4 (b c-a d)^2 (a+b x)^{3/2} \sqrt [4]{c+d x}}{231 b^2 d}+\frac{4 (b c-a d) (a+b x)^{5/2} \sqrt [4]{c+d x}}{33 b^2}+\frac{4 (a+b x)^{5/2} (c+d x)^{5/4}}{15 b}+\frac{16 (b c-a d)^{17/4} \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 )}{231 b^{9/4} d^3 \sqrt{a+b x}}\\ \end{align*}
Mathematica [C] time = 0.0680344, size = 73, normalized size = 0.33 \[ \frac{2 (a+b x)^{5/2} (c+d x)^{5/4} \, _2F_1\left (-\frac{5}{4},\frac{5}{2};\frac{7}{2};\frac{d (a+b x)}{a d-b c}\right )}{5 b \left (\frac{b (c+d x)}{b c-a d}\right )^{5/4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.022, size = 0, normalized size = 0. \begin{align*} \int \left ( bx+a \right ) ^{{\frac{3}{2}}} \left ( dx+c \right ) ^{{\frac{5}{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{\left (b x + a\right )}^{\frac{3}{2}}{\left (d x + c\right )}^{\frac{5}{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 ({\left (b d x^{2} + a c +{\left (b c + a d\right )} x\right )} \sqrt{b x + a}{\left (d x + c\right )}^{\frac{1}{4}}, 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 \left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{5}{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{\left (b x + a\right )}^{\frac{3}{2}}{\left (d x + c\right )}^{\frac{5}{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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