Optimal. Leaf size=148 \[ \frac{5 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^2}{8 d^3}-\frac{5 (a+b x)^{3/2} \sqrt{c+d x} (b c-a d)}{12 d^2}-\frac{5 (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 \sqrt{b} d^{7/2}}+\frac{(a+b x)^{5/2} \sqrt{c+d x}}{3 d} \]
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Rubi [A] time = 0.0723475, antiderivative size = 148, 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, 217, 206} \[ \frac{5 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^2}{8 d^3}-\frac{5 (a+b x)^{3/2} \sqrt{c+d x} (b c-a d)}{12 d^2}-\frac{5 (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 \sqrt{b} d^{7/2}}+\frac{(a+b x)^{5/2} \sqrt{c+d x}}{3 d} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{(a+b x)^{5/2}}{\sqrt{c+d x}} \, dx &=\frac{(a+b x)^{5/2} \sqrt{c+d x}}{3 d}-\frac{(5 (b c-a d)) \int \frac{(a+b x)^{3/2}}{\sqrt{c+d x}} \, dx}{6 d}\\ &=-\frac{5 (b c-a d) (a+b x)^{3/2} \sqrt{c+d x}}{12 d^2}+\frac{(a+b x)^{5/2} \sqrt{c+d x}}{3 d}+\frac{\left (5 (b c-a d)^2\right ) \int \frac{\sqrt{a+b x}}{\sqrt{c+d x}} \, dx}{8 d^2}\\ &=\frac{5 (b c-a d)^2 \sqrt{a+b x} \sqrt{c+d x}}{8 d^3}-\frac{5 (b c-a d) (a+b x)^{3/2} \sqrt{c+d x}}{12 d^2}+\frac{(a+b x)^{5/2} \sqrt{c+d x}}{3 d}-\frac{\left (5 (b c-a d)^3\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx}{16 d^3}\\ &=\frac{5 (b c-a d)^2 \sqrt{a+b x} \sqrt{c+d x}}{8 d^3}-\frac{5 (b c-a d) (a+b x)^{3/2} \sqrt{c+d x}}{12 d^2}+\frac{(a+b x)^{5/2} \sqrt{c+d x}}{3 d}-\frac{\left (5 (b c-a d)^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c-\frac{a d}{b}+\frac{d x^2}{b}}} \, dx,x,\sqrt{a+b x}\right )}{8 b d^3}\\ &=\frac{5 (b c-a d)^2 \sqrt{a+b x} \sqrt{c+d x}}{8 d^3}-\frac{5 (b c-a d) (a+b x)^{3/2} \sqrt{c+d x}}{12 d^2}+\frac{(a+b x)^{5/2} \sqrt{c+d x}}{3 d}-\frac{\left (5 (b c-a d)^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{d x^2}{b}} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{c+d x}}\right )}{8 b d^3}\\ &=\frac{5 (b c-a d)^2 \sqrt{a+b x} \sqrt{c+d x}}{8 d^3}-\frac{5 (b c-a d) (a+b x)^{3/2} \sqrt{c+d x}}{12 d^2}+\frac{(a+b x)^{5/2} \sqrt{c+d x}}{3 d}-\frac{5 (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 \sqrt{b} d^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.497898, size = 150, normalized size = 1.01 \[ \frac{\sqrt{d} \sqrt{a+b x} (c+d x) \left (33 a^2 d^2+2 a b d (13 d x-20 c)+b^2 \left (15 c^2-10 c d x+8 d^2 x^2\right )\right )-\frac{15 (b c-a d)^{7/2} \sqrt{\frac{b (c+d x)}{b c-a d}} \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right )}{b}}{24 d^{7/2} \sqrt{c+d x}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.007, size = 465, normalized size = 3.1 \begin{align*}{\frac{1}{3\,d} \left ( bx+a \right ) ^{{\frac{5}{2}}}\sqrt{dx+c}}+{\frac{5\,a}{12\,d} \left ( bx+a \right ) ^{{\frac{3}{2}}}\sqrt{dx+c}}-{\frac{5\,bc}{12\,{d}^{2}} \left ( bx+a \right ) ^{{\frac{3}{2}}}\sqrt{dx+c}}+{\frac{5\,{a}^{2}}{8\,d}\sqrt{bx+a}\sqrt{dx+c}}-{\frac{5\,abc}{4\,{d}^{2}}\sqrt{bx+a}\sqrt{dx+c}}+{\frac{5\,{b}^{2}{c}^{2}}{8\,{d}^{3}}\sqrt{bx+a}\sqrt{dx+c}}+{\frac{5\,{a}^{3}}{16}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\ln \left ({ \left ({\frac{ad}{2}}+{\frac{bc}{2}}+bdx \right ){\frac{1}{\sqrt{bd}}}}+\sqrt{d{x}^{2}b+ \left ( ad+bc \right ) x+ac} \right ){\frac{1}{\sqrt{bx+a}}}{\frac{1}{\sqrt{dx+c}}}{\frac{1}{\sqrt{bd}}}}-{\frac{15\,{a}^{2}bc}{16\,d}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\ln \left ({ \left ({\frac{ad}{2}}+{\frac{bc}{2}}+bdx \right ){\frac{1}{\sqrt{bd}}}}+\sqrt{d{x}^{2}b+ \left ( ad+bc \right ) x+ac} \right ){\frac{1}{\sqrt{bx+a}}}{\frac{1}{\sqrt{dx+c}}}{\frac{1}{\sqrt{bd}}}}+{\frac{15\,a{b}^{2}{c}^{2}}{16\,{d}^{2}}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\ln \left ({ \left ({\frac{ad}{2}}+{\frac{bc}{2}}+bdx \right ){\frac{1}{\sqrt{bd}}}}+\sqrt{d{x}^{2}b+ \left ( ad+bc \right ) x+ac} \right ){\frac{1}{\sqrt{bx+a}}}{\frac{1}{\sqrt{dx+c}}}{\frac{1}{\sqrt{bd}}}}-{\frac{5\,{b}^{3}{c}^{3}}{16\,{d}^{3}}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\ln \left ({ \left ({\frac{ad}{2}}+{\frac{bc}{2}}+bdx \right ){\frac{1}{\sqrt{bd}}}}+\sqrt{d{x}^{2}b+ \left ( ad+bc \right ) x+ac} \right ){\frac{1}{\sqrt{bx+a}}}{\frac{1}{\sqrt{dx+c}}}{\frac{1}{\sqrt{bd}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.52327, size = 932, normalized size = 6.3 \begin{align*} \left [-\frac{15 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt{b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \,{\left (2 \, b d x + b c + a d\right )} \sqrt{b d} \sqrt{b x + a} \sqrt{d x + c} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \,{\left (8 \, b^{3} d^{3} x^{2} + 15 \, b^{3} c^{2} d - 40 \, a b^{2} c d^{2} + 33 \, a^{2} b d^{3} - 2 \,{\left (5 \, b^{3} c d^{2} - 13 \, a b^{2} d^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{96 \, b d^{4}}, \frac{15 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt{-b d} \arctan \left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{-b d} \sqrt{b x + a} \sqrt{d x + c}}{2 \,{\left (b^{2} d^{2} x^{2} + a b c d +{\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) + 2 \,{\left (8 \, b^{3} d^{3} x^{2} + 15 \, b^{3} c^{2} d - 40 \, a b^{2} c d^{2} + 33 \, a^{2} b d^{3} - 2 \,{\left (5 \, b^{3} c d^{2} - 13 \, a b^{2} d^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{48 \, b d^{4}}\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 (a + b x\right )^{\frac{5}{2}}}{\sqrt{c + d x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13627, size = 267, normalized size = 1.8 \begin{align*} \frac{{\left (\sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \sqrt{b x + a}{\left (2 \,{\left (b x + a\right )}{\left (\frac{4 \,{\left (b x + a\right )}}{b d} - \frac{5 \,{\left (b c d^{3} - a d^{4}\right )}}{b d^{5}}\right )} + \frac{15 \,{\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )}}{b d^{5}}\right )} + \frac{15 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt{b d} d^{3}}\right )} b}{24 \,{\left | b \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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