Optimal. Leaf size=182 \[ -\frac{40 (b c-a d)^{13/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{231 b^{9/4} d^2 \sqrt{a+b x}}+\frac{20 \sqrt{a+b x} \sqrt [4]{c+d x} (b c-a d)^2}{231 b^2 d}+\frac{20 (a+b x)^{3/2} \sqrt [4]{c+d x} (b c-a d)}{77 b^2}+\frac{4 (a+b x)^{3/2} (c+d x)^{5/4}}{11 b} \]
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Rubi [A] time = 0.104016, antiderivative size = 182, 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 = {50, 63, 224, 221} \[ -\frac{40 (b c-a d)^{13/4} \sqrt{-\frac{d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{231 b^{9/4} d^2 \sqrt{a+b x}}+\frac{20 \sqrt{a+b x} \sqrt [4]{c+d x} (b c-a d)^2}{231 b^2 d}+\frac{20 (a+b x)^{3/2} \sqrt [4]{c+d x} (b c-a d)}{77 b^2}+\frac{4 (a+b x)^{3/2} (c+d x)^{5/4}}{11 b} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 224
Rule 221
Rubi steps
\begin{align*} \int \sqrt{a+b x} (c+d x)^{5/4} \, dx &=\frac{4 (a+b x)^{3/2} (c+d x)^{5/4}}{11 b}+\frac{(5 (b c-a d)) \int \sqrt{a+b x} \sqrt [4]{c+d x} \, dx}{11 b}\\ &=\frac{20 (b c-a d) (a+b x)^{3/2} \sqrt [4]{c+d x}}{77 b^2}+\frac{4 (a+b x)^{3/2} (c+d x)^{5/4}}{11 b}+\frac{\left (5 (b c-a d)^2\right ) \int \frac{\sqrt{a+b x}}{(c+d x)^{3/4}} \, dx}{77 b^2}\\ &=\frac{20 (b c-a d)^2 \sqrt{a+b x} \sqrt [4]{c+d x}}{231 b^2 d}+\frac{20 (b c-a d) (a+b x)^{3/2} \sqrt [4]{c+d x}}{77 b^2}+\frac{4 (a+b x)^{3/2} (c+d x)^{5/4}}{11 b}-\frac{\left (10 (b c-a d)^3\right ) \int \frac{1}{\sqrt{a+b x} (c+d x)^{3/4}} \, dx}{231 b^2 d}\\ &=\frac{20 (b c-a d)^2 \sqrt{a+b x} \sqrt [4]{c+d x}}{231 b^2 d}+\frac{20 (b c-a d) (a+b x)^{3/2} \sqrt [4]{c+d x}}{77 b^2}+\frac{4 (a+b x)^{3/2} (c+d x)^{5/4}}{11 b}-\frac{\left (40 (b c-a d)^3\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^2}\\ &=\frac{20 (b c-a d)^2 \sqrt{a+b x} \sqrt [4]{c+d x}}{231 b^2 d}+\frac{20 (b c-a d) (a+b x)^{3/2} \sqrt [4]{c+d x}}{77 b^2}+\frac{4 (a+b x)^{3/2} (c+d x)^{5/4}}{11 b}-\frac{\left (40 (b c-a d)^3 \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^2 \sqrt{a+b x}}\\ &=\frac{20 (b c-a d)^2 \sqrt{a+b x} \sqrt [4]{c+d x}}{231 b^2 d}+\frac{20 (b c-a d) (a+b x)^{3/2} \sqrt [4]{c+d x}}{77 b^2}+\frac{4 (a+b x)^{3/2} (c+d x)^{5/4}}{11 b}-\frac{40 (b c-a d)^{13/4} \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^2 \sqrt{a+b x}}\\ \end{align*}
Mathematica [C] time = 0.046882, size = 73, normalized size = 0.4 \[ \frac{2 (a+b x)^{3/2} (c+d x)^{5/4} \, _2F_1\left (-\frac{5}{4},\frac{3}{2};\frac{5}{2};\frac{d (a+b x)}{a d-b c}\right )}{3 b \left (\frac{b (c+d x)}{b c-a d}\right )^{5/4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.017, size = 0, normalized size = 0. \begin{align*} \int \sqrt{bx+a} \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 \sqrt{b x + a}{\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 (\sqrt{b x + a}{\left (d x + c\right )}^{\frac{5}{4}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 70.5203, size = 218, normalized size = 1.2 \begin{align*} - \frac{2 a d \left (a + b x\right )^{\frac{3}{2}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{\frac{a d e^{i \pi }}{b \operatorname{polar\_lift}{\left (- \frac{a d}{b} + c \right )}} + \frac{d x e^{i \pi }}{\operatorname{polar\_lift}{\left (- \frac{a d}{b} + c \right )}}} \right )} \sqrt [4]{\operatorname{polar\_lift}{\left (- \frac{a d}{b} + c \right )}}}{3 b^{2}} + \frac{2 c \left (a + b x\right )^{\frac{3}{2}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{\frac{a d e^{i \pi }}{b \operatorname{polar\_lift}{\left (- \frac{a d}{b} + c \right )}} + \frac{d x e^{i \pi }}{\operatorname{polar\_lift}{\left (- \frac{a d}{b} + c \right )}}} \right )} \sqrt [4]{\operatorname{polar\_lift}{\left (- \frac{a d}{b} + c \right )}}}{3 b} + \frac{2 d \left (a + b x\right )^{\frac{5}{2}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{\frac{a d e^{i \pi }}{b \operatorname{polar\_lift}{\left (- \frac{a d}{b} + c \right )}} + \frac{d x e^{i \pi }}{\operatorname{polar\_lift}{\left (- \frac{a d}{b} + c \right )}}} \right )} \sqrt [4]{\operatorname{polar\_lift}{\left (- \frac{a d}{b} + c \right )}}}{5 b^{2}} \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 \sqrt{b x + a}{\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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