Optimal. Leaf size=136 \[ \frac{32 d^3 \sqrt{c+d x}}{35 \sqrt{a+b x} (b c-a d)^4}-\frac{16 d^2 \sqrt{c+d x}}{35 (a+b x)^{3/2} (b c-a d)^3}+\frac{12 d \sqrt{c+d x}}{35 (a+b x)^{5/2} (b c-a d)^2}-\frac{2 \sqrt{c+d x}}{7 (a+b x)^{7/2} (b c-a d)} \]
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Rubi [A] time = 0.0280244, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {45, 37} \[ \frac{32 d^3 \sqrt{c+d x}}{35 \sqrt{a+b x} (b c-a d)^4}-\frac{16 d^2 \sqrt{c+d x}}{35 (a+b x)^{3/2} (b c-a d)^3}+\frac{12 d \sqrt{c+d x}}{35 (a+b x)^{5/2} (b c-a d)^2}-\frac{2 \sqrt{c+d x}}{7 (a+b x)^{7/2} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{1}{(a+b x)^{9/2} \sqrt{c+d x}} \, dx &=-\frac{2 \sqrt{c+d x}}{7 (b c-a d) (a+b x)^{7/2}}-\frac{(6 d) \int \frac{1}{(a+b x)^{7/2} \sqrt{c+d x}} \, dx}{7 (b c-a d)}\\ &=-\frac{2 \sqrt{c+d x}}{7 (b c-a d) (a+b x)^{7/2}}+\frac{12 d \sqrt{c+d x}}{35 (b c-a d)^2 (a+b x)^{5/2}}+\frac{\left (24 d^2\right ) \int \frac{1}{(a+b x)^{5/2} \sqrt{c+d x}} \, dx}{35 (b c-a d)^2}\\ &=-\frac{2 \sqrt{c+d x}}{7 (b c-a d) (a+b x)^{7/2}}+\frac{12 d \sqrt{c+d x}}{35 (b c-a d)^2 (a+b x)^{5/2}}-\frac{16 d^2 \sqrt{c+d x}}{35 (b c-a d)^3 (a+b x)^{3/2}}-\frac{\left (16 d^3\right ) \int \frac{1}{(a+b x)^{3/2} \sqrt{c+d x}} \, dx}{35 (b c-a d)^3}\\ &=-\frac{2 \sqrt{c+d x}}{7 (b c-a d) (a+b x)^{7/2}}+\frac{12 d \sqrt{c+d x}}{35 (b c-a d)^2 (a+b x)^{5/2}}-\frac{16 d^2 \sqrt{c+d x}}{35 (b c-a d)^3 (a+b x)^{3/2}}+\frac{32 d^3 \sqrt{c+d x}}{35 (b c-a d)^4 \sqrt{a+b x}}\\ \end{align*}
Mathematica [A] time = 0.0456081, size = 116, normalized size = 0.85 \[ \frac{2 \sqrt{c+d x} \left (-35 a^2 b d^2 (c-2 d x)+35 a^3 d^3+7 a b^2 d \left (3 c^2-4 c d x+8 d^2 x^2\right )+b^3 \left (6 c^2 d x-5 c^3-8 c d^2 x^2+16 d^3 x^3\right )\right )}{35 (a+b x)^{7/2} (b c-a d)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 171, normalized size = 1.3 \begin{align*}{\frac{32\,{b}^{3}{d}^{3}{x}^{3}+112\,a{b}^{2}{d}^{3}{x}^{2}-16\,{b}^{3}c{d}^{2}{x}^{2}+140\,{a}^{2}b{d}^{3}x-56\,a{b}^{2}c{d}^{2}x+12\,{b}^{3}{c}^{2}dx+70\,{a}^{3}{d}^{3}-70\,{a}^{2}bc{d}^{2}+42\,a{b}^{2}{c}^{2}d-10\,{b}^{3}{c}^{3}}{35\,{d}^{4}{a}^{4}-140\,b{d}^{3}c{a}^{3}+210\,{b}^{2}{d}^{2}{c}^{2}{a}^{2}-140\,{b}^{3}d{c}^{3}a+35\,{b}^{4}{c}^{4}}\sqrt{dx+c} \left ( bx+a \right ) ^{-{\frac{7}{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 = 12.3486, size = 848, normalized size = 6.24 \begin{align*} \frac{2 \,{\left (16 \, b^{3} d^{3} x^{3} - 5 \, b^{3} c^{3} + 21 \, a b^{2} c^{2} d - 35 \, a^{2} b c d^{2} + 35 \, a^{3} d^{3} - 8 \,{\left (b^{3} c d^{2} - 7 \, a b^{2} d^{3}\right )} x^{2} + 2 \,{\left (3 \, b^{3} c^{2} d - 14 \, a b^{2} c d^{2} + 35 \, a^{2} b d^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{35 \,{\left (a^{4} b^{4} c^{4} - 4 \, a^{5} b^{3} c^{3} d + 6 \, a^{6} b^{2} c^{2} d^{2} - 4 \, a^{7} b c d^{3} + a^{8} d^{4} +{\left (b^{8} c^{4} - 4 \, a b^{7} c^{3} d + 6 \, a^{2} b^{6} c^{2} d^{2} - 4 \, a^{3} b^{5} c d^{3} + a^{4} b^{4} d^{4}\right )} x^{4} + 4 \,{\left (a b^{7} c^{4} - 4 \, a^{2} b^{6} c^{3} d + 6 \, a^{3} b^{5} c^{2} d^{2} - 4 \, a^{4} b^{4} c d^{3} + a^{5} b^{3} d^{4}\right )} x^{3} + 6 \,{\left (a^{2} b^{6} c^{4} - 4 \, a^{3} b^{5} c^{3} d + 6 \, a^{4} b^{4} c^{2} d^{2} - 4 \, a^{5} b^{3} c d^{3} + a^{6} b^{2} d^{4}\right )} x^{2} + 4 \,{\left (a^{3} b^{5} c^{4} - 4 \, a^{4} b^{4} c^{3} d + 6 \, a^{5} b^{3} c^{2} d^{2} - 4 \, a^{6} b^{2} c d^{3} + a^{7} b d^{4}\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 = 1.18414, size = 521, normalized size = 3.83 \begin{align*} \frac{64 \,{\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3} - 7 \,{\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^{4} c^{2} + 14 \,{\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^{3} c d - 7 \,{\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^{2} d^{2} + 21 \,{\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^{2} c - 21 \,{\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 d - 35 \,{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{6}\right )} \sqrt{b d} b^{4} d^{3}}{35 \,{\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 )}^{7}{\left | b \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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