Optimal. Leaf size=174 \[ \frac{7 b (a+b x)^{5/2} \sqrt{c+d x}}{3 d^2}-\frac{35 b (a+b x)^{3/2} \sqrt{c+d x} (b c-a d)}{12 d^3}+\frac{35 b \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^2}{8 d^4}-\frac{35 \sqrt{b} (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 d^{9/2}}-\frac{2 (a+b x)^{7/2}}{d \sqrt{c+d x}} \]
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Rubi [A] time = 0.0891534, antiderivative size = 174, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {47, 50, 63, 217, 206} \[ \frac{7 b (a+b x)^{5/2} \sqrt{c+d x}}{3 d^2}-\frac{35 b (a+b x)^{3/2} \sqrt{c+d x} (b c-a d)}{12 d^3}+\frac{35 b \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^2}{8 d^4}-\frac{35 \sqrt{b} (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 d^{9/2}}-\frac{2 (a+b x)^{7/2}}{d \sqrt{c+d x}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{(a+b x)^{7/2}}{(c+d x)^{3/2}} \, dx &=-\frac{2 (a+b x)^{7/2}}{d \sqrt{c+d x}}+\frac{(7 b) \int \frac{(a+b x)^{5/2}}{\sqrt{c+d x}} \, dx}{d}\\ &=-\frac{2 (a+b x)^{7/2}}{d \sqrt{c+d x}}+\frac{7 b (a+b x)^{5/2} \sqrt{c+d x}}{3 d^2}-\frac{(35 b (b c-a d)) \int \frac{(a+b x)^{3/2}}{\sqrt{c+d x}} \, dx}{6 d^2}\\ &=-\frac{2 (a+b x)^{7/2}}{d \sqrt{c+d x}}-\frac{35 b (b c-a d) (a+b x)^{3/2} \sqrt{c+d x}}{12 d^3}+\frac{7 b (a+b x)^{5/2} \sqrt{c+d x}}{3 d^2}+\frac{\left (35 b (b c-a d)^2\right ) \int \frac{\sqrt{a+b x}}{\sqrt{c+d x}} \, dx}{8 d^3}\\ &=-\frac{2 (a+b x)^{7/2}}{d \sqrt{c+d x}}+\frac{35 b (b c-a d)^2 \sqrt{a+b x} \sqrt{c+d x}}{8 d^4}-\frac{35 b (b c-a d) (a+b x)^{3/2} \sqrt{c+d x}}{12 d^3}+\frac{7 b (a+b x)^{5/2} \sqrt{c+d x}}{3 d^2}-\frac{\left (35 b (b c-a d)^3\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx}{16 d^4}\\ &=-\frac{2 (a+b x)^{7/2}}{d \sqrt{c+d x}}+\frac{35 b (b c-a d)^2 \sqrt{a+b x} \sqrt{c+d x}}{8 d^4}-\frac{35 b (b c-a d) (a+b x)^{3/2} \sqrt{c+d x}}{12 d^3}+\frac{7 b (a+b x)^{5/2} \sqrt{c+d x}}{3 d^2}-\frac{\left (35 (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 d^4}\\ &=-\frac{2 (a+b x)^{7/2}}{d \sqrt{c+d x}}+\frac{35 b (b c-a d)^2 \sqrt{a+b x} \sqrt{c+d x}}{8 d^4}-\frac{35 b (b c-a d) (a+b x)^{3/2} \sqrt{c+d x}}{12 d^3}+\frac{7 b (a+b x)^{5/2} \sqrt{c+d x}}{3 d^2}-\frac{\left (35 (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 d^4}\\ &=-\frac{2 (a+b x)^{7/2}}{d \sqrt{c+d x}}+\frac{35 b (b c-a d)^2 \sqrt{a+b x} \sqrt{c+d x}}{8 d^4}-\frac{35 b (b c-a d) (a+b x)^{3/2} \sqrt{c+d x}}{12 d^3}+\frac{7 b (a+b x)^{5/2} \sqrt{c+d x}}{3 d^2}-\frac{35 \sqrt{b} (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 d^{9/2}}\\ \end{align*}
Mathematica [C] time = 0.0671263, size = 73, normalized size = 0.42 \[ \frac{2 (a+b x)^{9/2} \left (\frac{b (c+d x)}{b c-a d}\right )^{3/2} \, _2F_1\left (\frac{3}{2},\frac{9}{2};\frac{11}{2};\frac{d (a+b x)}{a d-b c}\right )}{9 b (c+d x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.033, size = 0, normalized size = 0. \begin{align*} \int{ \left ( bx+a \right ) ^{{\frac{7}{2}}} \left ( dx+c \right ) ^{-{\frac{3}{2}}}}\, dx \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 [B] time = 7.12697, size = 1312, normalized size = 7.54 \begin{align*} \left [-\frac{105 \,{\left (b^{3} c^{4} - 3 \, a b^{2} c^{3} d + 3 \, a^{2} b c^{2} d^{2} - a^{3} c d^{3} +{\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - a^{3} d^{4}\right )} x\right )} \sqrt{\frac{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^{2} x + b c d + a d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} \sqrt{\frac{b}{d}} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \,{\left (8 \, b^{3} d^{3} x^{3} + 105 \, b^{3} c^{3} - 280 \, a b^{2} c^{2} d + 231 \, a^{2} b c d^{2} - 48 \, a^{3} d^{3} - 2 \,{\left (7 \, b^{3} c d^{2} - 19 \, a b^{2} d^{3}\right )} x^{2} +{\left (35 \, b^{3} c^{2} d - 98 \, a b^{2} c d^{2} + 87 \, a^{2} b d^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{96 \,{\left (d^{5} x + c d^{4}\right )}}, \frac{105 \,{\left (b^{3} c^{4} - 3 \, a b^{2} c^{3} d + 3 \, a^{2} b c^{2} d^{2} - a^{3} c d^{3} +{\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - a^{3} d^{4}\right )} x\right )} \sqrt{-\frac{b}{d}} \arctan \left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{b x + a} \sqrt{d x + c} \sqrt{-\frac{b}{d}}}{2 \,{\left (b^{2} d x^{2} + a b c +{\left (b^{2} c + a b d\right )} x\right )}}\right ) + 2 \,{\left (8 \, b^{3} d^{3} x^{3} + 105 \, b^{3} c^{3} - 280 \, a b^{2} c^{2} d + 231 \, a^{2} b c d^{2} - 48 \, a^{3} d^{3} - 2 \,{\left (7 \, b^{3} c d^{2} - 19 \, a b^{2} d^{3}\right )} x^{2} +{\left (35 \, b^{3} c^{2} d - 98 \, a b^{2} c d^{2} + 87 \, a^{2} b d^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{48 \,{\left (d^{5} x + c d^{4}\right )}}\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{7}{2}}}{\left (c + d x\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.16745, size = 423, normalized size = 2.43 \begin{align*} \frac{{\left ({\left (2 \,{\left (\frac{4 \,{\left (b x + a\right )} b^{2} d^{6}}{b^{10} c d^{8} - a b^{9} d^{9}} - \frac{7 \,{\left (b^{3} c d^{5} - a b^{2} d^{6}\right )}}{b^{10} c d^{8} - a b^{9} d^{9}}\right )}{\left (b x + a\right )} + \frac{35 \,{\left (b^{4} c^{2} d^{4} - 2 \, a b^{3} c d^{5} + a^{2} b^{2} d^{6}\right )}}{b^{10} c d^{8} - a b^{9} d^{9}}\right )}{\left (b x + a\right )} + \frac{105 \,{\left (b^{5} c^{3} d^{3} - 3 \, a b^{4} c^{2} d^{4} + 3 \, a^{2} b^{3} c d^{5} - a^{3} b^{2} d^{6}\right )}}{b^{10} c d^{8} - a b^{9} d^{9}}\right )} \sqrt{b x + a}}{184320 \, \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}} + \frac{7 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\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 )}{12288 \, \sqrt{b d} b^{7} d^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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