Optimal. Leaf size=66 \[ \frac{4 d (c+d x)^{3/2}}{15 (a+b x)^{3/2} (b c-a d)^2}-\frac{2 (c+d x)^{3/2}}{5 (a+b x)^{5/2} (b c-a d)} \]
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Rubi [A] time = 0.0086535, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {45, 37} \[ \frac{4 d (c+d x)^{3/2}}{15 (a+b x)^{3/2} (b c-a d)^2}-\frac{2 (c+d x)^{3/2}}{5 (a+b x)^{5/2} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{\sqrt{c+d x}}{(a+b x)^{7/2}} \, dx &=-\frac{2 (c+d x)^{3/2}}{5 (b c-a d) (a+b x)^{5/2}}-\frac{(2 d) \int \frac{\sqrt{c+d x}}{(a+b x)^{5/2}} \, dx}{5 (b c-a d)}\\ &=-\frac{2 (c+d x)^{3/2}}{5 (b c-a d) (a+b x)^{5/2}}+\frac{4 d (c+d x)^{3/2}}{15 (b c-a d)^2 (a+b x)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0201384, size = 46, normalized size = 0.7 \[ \frac{2 (c+d x)^{3/2} (5 a d-3 b c+2 b d x)}{15 (a+b x)^{5/2} (b c-a d)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 54, normalized size = 0.8 \begin{align*}{\frac{4\,bdx+10\,ad-6\,bc}{15\,{a}^{2}{d}^{2}-30\,abcd+15\,{b}^{2}{c}^{2}} \left ( dx+c \right ) ^{{\frac{3}{2}}} \left ( bx+a \right ) ^{-{\frac{5}{2}}}} \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 = 4.31271, size = 365, normalized size = 5.53 \begin{align*} \frac{2 \,{\left (2 \, b d^{2} x^{2} - 3 \, b c^{2} + 5 \, a c d -{\left (b c d - 5 \, a d^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{15 \,{\left (a^{3} b^{2} c^{2} - 2 \, a^{4} b c d + a^{5} d^{2} +{\left (b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )} x^{3} + 3 \,{\left (a b^{4} c^{2} - 2 \, a^{2} b^{3} c d + a^{3} b^{2} d^{2}\right )} x^{2} + 3 \,{\left (a^{2} b^{3} c^{2} - 2 \, a^{3} b^{2} c d + a^{4} b d^{2}\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [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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Giac [B] time = 1.22328, size = 603, normalized size = 9.14 \begin{align*} \frac{8 \,{\left (\sqrt{b d} b^{7} c^{3} d^{2} - 3 \, \sqrt{b d} a b^{6} c^{2} d^{3} + 3 \, \sqrt{b d} a^{2} b^{5} c d^{4} - \sqrt{b d} a^{3} b^{4} d^{5} - 5 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2} b^{5} c^{2} d^{2} + 10 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2} a b^{4} c d^{3} - 5 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2} a^{2} b^{3} d^{4} - 5 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{4} b^{3} c d^{2} + 5 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{4} a b^{2} d^{3} - 15 \, \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{6} b d^{2}\right )}{\left | b \right |}}{15 \,{\left (b^{2} c - a b d -{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}^{5} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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