Optimal. Leaf size=151 \[ \frac {2 \sqrt {a+b x} \left (-3 a^2 d^2-6 a b c d+b^2 c^2\right )}{3 b d \sqrt {c+d x} (b c-a d)^3}-\frac {2 \sqrt {a+b x} \left (3 a^2 d^2+b^2 c^2\right )}{3 b^2 d (c+d x)^{3/2} (b c-a d)^2}-\frac {2 a^2}{b^2 \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)} \]
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Rubi [A] time = 0.13, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {89, 78, 37} \[ \frac {2 \sqrt {a+b x} \left (-3 a^2 d^2-6 a b c d+b^2 c^2\right )}{3 b d \sqrt {c+d x} (b c-a d)^3}-\frac {2 \sqrt {a+b x} \left (3 a^2 d^2+b^2 c^2\right )}{3 b^2 d (c+d x)^{3/2} (b c-a d)^2}-\frac {2 a^2}{b^2 \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 37
Rule 78
Rule 89
Rubi steps
\begin {align*} \int \frac {x^2}{(a+b x)^{3/2} (c+d x)^{5/2}} \, dx &=-\frac {2 a^2}{b^2 (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}+\frac {2 \int \frac {-\frac {1}{2} a (b c+3 a d)+\frac {1}{2} b (b c-a d) x}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx}{b^2 (b c-a d)}\\ &=-\frac {2 a^2}{b^2 (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}-\frac {2 \left (b^2 c^2+3 a^2 d^2\right ) \sqrt {a+b x}}{3 b^2 d (b c-a d)^2 (c+d x)^{3/2}}+\frac {\left (b^2 c^2-6 a b c d-3 a^2 d^2\right ) \int \frac {1}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx}{3 b d (b c-a d)^2}\\ &=-\frac {2 a^2}{b^2 (b c-a d) \sqrt {a+b x} (c+d x)^{3/2}}-\frac {2 \left (b^2 c^2+3 a^2 d^2\right ) \sqrt {a+b x}}{3 b^2 d (b c-a d)^2 (c+d x)^{3/2}}+\frac {2 \left (b^2 c^2-6 a b c d-3 a^2 d^2\right ) \sqrt {a+b x}}{3 b d (b c-a d)^3 \sqrt {c+d x}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 82, normalized size = 0.54 \[ \frac {-2 a^2 \left (8 c^2+12 c d x+3 d^2 x^2\right )-4 a b c x (2 c+3 d x)+2 b^2 c^2 x^2}{3 \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 2.25, size = 275, normalized size = 1.82 \[ -\frac {2 \, {\left (8 \, a^{2} c^{2} - {\left (b^{2} c^{2} - 6 \, a b c d - 3 \, a^{2} d^{2}\right )} x^{2} + 4 \, {\left (a b c^{2} + 3 \, a^{2} c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3 \, {\left (a b^{3} c^{5} - 3 \, a^{2} b^{2} c^{4} d + 3 \, a^{3} b c^{3} d^{2} - a^{4} c^{2} d^{3} + {\left (b^{4} c^{3} d^{2} - 3 \, a b^{3} c^{2} d^{3} + 3 \, a^{2} b^{2} c d^{4} - a^{3} b d^{5}\right )} x^{3} + {\left (2 \, b^{4} c^{4} d - 5 \, a b^{3} c^{3} d^{2} + 3 \, a^{2} b^{2} c^{2} d^{3} + a^{3} b c d^{4} - a^{4} d^{5}\right )} x^{2} + {\left (b^{4} c^{5} - a b^{3} c^{4} d - 3 \, a^{2} b^{2} c^{3} d^{2} + 5 \, a^{3} b c^{2} d^{3} - 2 \, a^{4} c d^{4}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.36, size = 395, normalized size = 2.62 \[ -\frac {4 \, \sqrt {b d} a^{2} b}{{\left (b^{2} c^{2} {\left | b \right |} - 2 \, a b c d {\left | b \right |} + a^{2} d^{2} {\left | b \right |}\right )} {\left (b^{2} c - a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}} + \frac {2 \, \sqrt {b x + a} {\left (\frac {{\left (b^{6} c^{4} d {\left | b \right |} - 8 \, a b^{5} c^{3} d^{2} {\left | b \right |} + 13 \, a^{2} b^{4} c^{2} d^{3} {\left | b \right |} - 6 \, a^{3} b^{3} c d^{4} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{7} c^{5} d - 5 \, a b^{6} c^{4} d^{2} + 10 \, a^{2} b^{5} c^{3} d^{3} - 10 \, a^{3} b^{4} c^{2} d^{4} + 5 \, a^{4} b^{3} c d^{5} - a^{5} b^{2} d^{6}} - \frac {6 \, {\left (a b^{6} c^{4} d {\left | b \right |} - 3 \, a^{2} b^{5} c^{3} d^{2} {\left | b \right |} + 3 \, a^{3} b^{4} c^{2} d^{3} {\left | b \right |} - a^{4} b^{3} c d^{4} {\left | b \right |}\right )}}{b^{7} c^{5} d - 5 \, a b^{6} c^{4} d^{2} + 10 \, a^{2} b^{5} c^{3} d^{3} - 10 \, a^{3} b^{4} c^{2} d^{4} + 5 \, a^{4} b^{3} c d^{5} - a^{5} b^{2} d^{6}}\right )}}{3 \, {\left (b^{2} c + {\left (b x + a\right )} b d - a b d\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 111, normalized size = 0.74 \[ \frac {2 a^{2} d^{2} x^{2}+4 a b c d \,x^{2}-\frac {2}{3} b^{2} c^{2} x^{2}+8 a^{2} c d x +\frac {8}{3} a b \,c^{2} x +\frac {16}{3} a^{2} c^{2}}{\sqrt {b x +a}\, \left (d x +c \right )^{\frac {3}{2}} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )} \]
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 = 1.63, size = 139, normalized size = 0.92 \[ \frac {\sqrt {c+d\,x}\,\left (\frac {x^2\,\left (6\,a^2\,d^2+12\,a\,b\,c\,d-2\,b^2\,c^2\right )}{3\,d^2\,{\left (a\,d-b\,c\right )}^3}+\frac {16\,a^2\,c^2}{3\,d^2\,{\left (a\,d-b\,c\right )}^3}+\frac {8\,a\,c\,x\,\left (3\,a\,d+b\,c\right )}{3\,d^2\,{\left (a\,d-b\,c\right )}^3}\right )}{x^2\,\sqrt {a+b\,x}+\frac {c^2\,\sqrt {a+b\,x}}{d^2}+\frac {2\,c\,x\,\sqrt {a+b\,x}}{d}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\left (a + b x\right )^{\frac {3}{2}} \left (c + d x\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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