Optimal. Leaf size=32 \[ -\frac {2 (c+d x)^{7/2}}{7 (a+b x)^{7/2} (b c-a d)} \]
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Rubi [A] time = 0.00, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {37} \[ -\frac {2 (c+d x)^{7/2}}{7 (a+b x)^{7/2} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 37
Rubi steps
\begin {align*} \int \frac {(c+d x)^{5/2}}{(a+b x)^{9/2}} \, dx &=-\frac {2 (c+d x)^{7/2}}{7 (b c-a d) (a+b x)^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 32, normalized size = 1.00 \[ -\frac {2 (c+d x)^{7/2}}{7 (a+b x)^{7/2} (b c-a d)} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.44, size = 138, normalized size = 4.31 \[ -\frac {2 \, {\left (d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}\right )} \sqrt {b x + a} \sqrt {d x + c}}{7 \, {\left (a^{4} b c - a^{5} d + {\left (b^{5} c - a b^{4} d\right )} x^{4} + 4 \, {\left (a b^{4} c - a^{2} b^{3} d\right )} x^{3} + 6 \, {\left (a^{2} b^{3} c - a^{3} b^{2} d\right )} x^{2} + 4 \, {\left (a^{3} b^{2} c - a^{4} b d\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.54, size = 706, normalized size = 22.06 \[ -\frac {4 \, {\left (\sqrt {b d} b^{12} c^{6} d^{3} {\left | b \right |} - 6 \, \sqrt {b d} a b^{11} c^{5} d^{4} {\left | b \right |} + 15 \, \sqrt {b d} a^{2} b^{10} c^{4} d^{5} {\left | b \right |} - 20 \, \sqrt {b d} a^{3} b^{9} c^{3} d^{6} {\left | b \right |} + 15 \, \sqrt {b d} a^{4} b^{8} c^{2} d^{7} {\left | b \right |} - 6 \, \sqrt {b d} a^{5} b^{7} c d^{8} {\left | b \right |} + \sqrt {b d} a^{6} b^{6} d^{9} {\left | b \right |} + 21 \, \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^{8} c^{4} d^{3} {\left | b \right |} - 84 \, \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^{7} c^{3} d^{4} {\left | b \right |} + 126 \, \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^{2} b^{6} c^{2} d^{5} {\left | b \right |} - 84 \, \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^{3} b^{5} c d^{6} {\left | b \right |} + 21 \, \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^{4} b^{4} d^{7} {\left | b \right |} + 35 \, \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 )}^{8} b^{4} c^{2} d^{3} {\left | b \right |} - 70 \, \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 )}^{8} a b^{3} c d^{4} {\left | b \right |} + 35 \, \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 )}^{8} a^{2} b^{2} d^{5} {\left | b \right |} + 7 \, \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 )}^{12} d^{3} {\left | b \right |}\right )}}{7 \, {\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} b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 27, normalized size = 0.84 \[ \frac {2 \left (d x +c \right )^{\frac {7}{2}}}{7 \left (b x +a \right )^{\frac {7}{2}} \left (a d -b c \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.97, size = 27, normalized size = 0.84 \[ \frac {2\,{\left (c+d\,x\right )}^{7/2}}{\left (7\,a\,d-7\,b\,c\right )\,{\left (a+b\,x\right )}^{7/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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