Optimal. Leaf size=66 \[ \frac {4 d (c+d x)^{5/2}}{35 (a+b x)^{5/2} (b c-a d)^2}-\frac {2 (c+d x)^{5/2}}{7 (a+b x)^{7/2} (b c-a d)} \]
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Rubi [A] time = 0.01, antiderivative size = 66, normalized size of antiderivative = 1.00, 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} \[ \frac {4 d (c+d x)^{5/2}}{35 (a+b x)^{5/2} (b c-a d)^2}-\frac {2 (c+d x)^{5/2}}{7 (a+b x)^{7/2} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rubi steps
\begin {align*} \int \frac {(c+d x)^{3/2}}{(a+b x)^{9/2}} \, dx &=-\frac {2 (c+d x)^{5/2}}{7 (b c-a d) (a+b x)^{7/2}}-\frac {(2 d) \int \frac {(c+d x)^{3/2}}{(a+b x)^{7/2}} \, dx}{7 (b c-a d)}\\ &=-\frac {2 (c+d x)^{5/2}}{7 (b c-a d) (a+b x)^{7/2}}+\frac {4 d (c+d x)^{5/2}}{35 (b c-a d)^2 (a+b x)^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 46, normalized size = 0.70 \[ \frac {2 (c+d x)^{5/2} (7 a d-5 b c+2 b d x)}{35 (a+b x)^{7/2} (b c-a d)^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.35, size = 235, normalized size = 3.56 \[ \frac {2 \, {\left (2 \, b d^{3} x^{3} - 5 \, b c^{3} + 7 \, a c^{2} d - {\left (b c d^{2} - 7 \, a d^{3}\right )} x^{2} - 2 \, {\left (4 \, b c^{2} d - 7 \, a c d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{35 \, {\left (a^{4} b^{2} c^{2} - 2 \, a^{5} b c d + a^{6} d^{2} + {\left (b^{6} c^{2} - 2 \, a b^{5} c d + a^{2} b^{4} d^{2}\right )} x^{4} + 4 \, {\left (a b^{5} c^{2} - 2 \, a^{2} b^{4} c d + a^{3} b^{3} d^{2}\right )} x^{3} + 6 \, {\left (a^{2} b^{4} c^{2} - 2 \, a^{3} b^{3} c d + a^{4} b^{2} d^{2}\right )} x^{2} + 4 \, {\left (a^{3} b^{3} c^{2} - 2 \, a^{4} b^{2} c d + a^{5} b d^{2}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.12, size = 1024, normalized size = 15.52 \[ \frac {8 \, {\left (\sqrt {b d} b^{10} c^{5} d^{3} {\left | b \right |} - 5 \, \sqrt {b d} a b^{9} c^{4} d^{4} {\left | b \right |} + 10 \, \sqrt {b d} a^{2} b^{8} c^{3} d^{5} {\left | b \right |} - 10 \, \sqrt {b d} a^{3} b^{7} c^{2} d^{6} {\left | b \right |} + 5 \, \sqrt {b d} a^{4} b^{6} c d^{7} {\left | b \right |} - \sqrt {b d} a^{5} b^{5} d^{8} {\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 )}^{2} b^{8} c^{4} d^{3} {\left | b \right |} + 28 \, \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^{7} c^{3} d^{4} {\left | b \right |} - 42 \, \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^{6} c^{2} d^{5} {\left | b \right |} + 28 \, \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^{3} b^{5} c d^{6} {\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 )}^{2} a^{4} b^{4} d^{7} {\left | b \right |} - 14 \, \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^{6} c^{3} d^{3} {\left | b \right |} + 42 \, \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^{5} c^{2} d^{4} {\left | b \right |} - 42 \, \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^{4} c d^{5} {\left | b \right |} + 14 \, \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^{3} d^{6} {\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 )}^{6} b^{4} c^{2} d^{3} {\left | b \right |} + 140 \, \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} a b^{3} c d^{4} {\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 )}^{6} a^{2} b^{2} d^{5} {\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^{2} c d^{3} {\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 b 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 )}^{10} d^{3} {\left | b \right |}\right )}}{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} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 54, normalized size = 0.82 \[ \frac {2 \left (d x +c \right )^{\frac {5}{2}} \left (2 b d x +7 a d -5 b c \right )}{35 \left (b x +a \right )^{\frac {7}{2}} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )} \]
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 = 0.93, size = 178, normalized size = 2.70 \[ \frac {\sqrt {c+d\,x}\,\left (\frac {4\,d^3\,x^3}{35\,b^2\,{\left (a\,d-b\,c\right )}^2}-\frac {10\,b\,c^3-14\,a\,c^2\,d}{35\,b^3\,{\left (a\,d-b\,c\right )}^2}+\frac {x^2\,\left (14\,a\,d^3-2\,b\,c\,d^2\right )}{35\,b^3\,{\left (a\,d-b\,c\right )}^2}+\frac {4\,c\,d\,x\,\left (7\,a\,d-4\,b\,c\right )}{35\,b^3\,{\left (a\,d-b\,c\right )}^2}\right )}{x^3\,\sqrt {a+b\,x}+\frac {a^3\,\sqrt {a+b\,x}}{b^3}+\frac {3\,a\,x^2\,\sqrt {a+b\,x}}{b}+\frac {3\,a^2\,x\,\sqrt {a+b\,x}}{b^2}} \]
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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