Optimal. Leaf size=115 \[ \frac {2 \sqrt {d x-c} \sqrt {c+d x} \left (3 a d^2+4 b c^2\right )}{3 d^6}-\frac {x^2 \left (3 a d^2+4 b c^2\right )}{3 d^4 \sqrt {d x-c} \sqrt {c+d x}}+\frac {b x^4}{3 d^2 \sqrt {d x-c} \sqrt {c+d x}} \]
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Rubi [A] time = 0.09, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {460, 98, 21, 74} \[ -\frac {x^2 \left (3 a d^2+4 b c^2\right )}{3 d^4 \sqrt {d x-c} \sqrt {c+d x}}+\frac {2 \sqrt {d x-c} \sqrt {c+d x} \left (3 a d^2+4 b c^2\right )}{3 d^6}+\frac {b x^4}{3 d^2 \sqrt {d x-c} \sqrt {c+d x}} \]
Antiderivative was successfully verified.
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Rule 21
Rule 74
Rule 98
Rule 460
Rubi steps
\begin {align*} \int \frac {x^3 \left (a+b x^2\right )}{(-c+d x)^{3/2} (c+d x)^{3/2}} \, dx &=\frac {b x^4}{3 d^2 \sqrt {-c+d x} \sqrt {c+d x}}-\frac {1}{3} \left (-3 a-\frac {4 b c^2}{d^2}\right ) \int \frac {x^3}{(-c+d x)^{3/2} (c+d x)^{3/2}} \, dx\\ &=-\frac {\left (4 b c^2+3 a d^2\right ) x^2}{3 d^4 \sqrt {-c+d x} \sqrt {c+d x}}+\frac {b x^4}{3 d^2 \sqrt {-c+d x} \sqrt {c+d x}}-\frac {\left (3 a+\frac {4 b c^2}{d^2}\right ) \int \frac {x \left (-2 c^2-2 c d x\right )}{\sqrt {-c+d x} (c+d x)^{3/2}} \, dx}{3 c d^2}\\ &=-\frac {\left (4 b c^2+3 a d^2\right ) x^2}{3 d^4 \sqrt {-c+d x} \sqrt {c+d x}}+\frac {b x^4}{3 d^2 \sqrt {-c+d x} \sqrt {c+d x}}+\frac {\left (2 \left (3 a+\frac {4 b c^2}{d^2}\right )\right ) \int \frac {x}{\sqrt {-c+d x} \sqrt {c+d x}} \, dx}{3 d^2}\\ &=-\frac {\left (4 b c^2+3 a d^2\right ) x^2}{3 d^4 \sqrt {-c+d x} \sqrt {c+d x}}+\frac {b x^4}{3 d^2 \sqrt {-c+d x} \sqrt {c+d x}}+\frac {2 \left (4 b c^2+3 a d^2\right ) \sqrt {-c+d x} \sqrt {c+d x}}{3 d^6}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 72, normalized size = 0.63 \[ \frac {-6 a c^2 d^2+3 a d^4 x^2-8 b c^4+4 b c^2 d^2 x^2+b d^4 x^4}{3 d^6 \sqrt {d x-c} \sqrt {c+d x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.18, size = 80, normalized size = 0.70 \[ \frac {{\left (b d^{4} x^{4} - 8 \, b c^{4} - 6 \, a c^{2} d^{2} + {\left (4 \, b c^{2} d^{2} + 3 \, a d^{4}\right )} x^{2}\right )} \sqrt {d x + c} \sqrt {d x - c}}{3 \, {\left (d^{8} x^{2} - c^{2} d^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.43, size = 200, normalized size = 1.74 \[ \frac {{\left (2 \, {\left (d x + c\right )} {\left ({\left (d x + c\right )} {\left (\frac {{\left (d x + c\right )} b}{d^{6}} - \frac {4 \, b c}{d^{6}}\right )} + \frac {10 \, b c^{2} d^{24} + 3 \, a d^{26}}{d^{30}}\right )} - \frac {3 \, {\left (9 \, b c^{3} d^{24} + 5 \, a c d^{26}\right )}}{d^{30}}\right )} \sqrt {d x + c}}{6 \, \sqrt {d x - c}} + \frac {2 \, {\left (b^{2} c^{8} + 2 \, a b c^{6} d^{2} + a^{2} c^{4} d^{4}\right )}}{{\left (b c^{4} {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{2} + a c^{2} d^{2} {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{2} + 2 \, b c^{5} + 2 \, a c^{3} d^{2}\right )} d^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 68, normalized size = 0.59 \[ -\frac {-b \,d^{4} x^{4}-3 a \,d^{4} x^{2}-4 b \,c^{2} d^{2} x^{2}+6 a \,c^{2} d^{2}+8 b \,c^{4}}{3 \sqrt {d x +c}\, \sqrt {d x -c}\, d^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 123, normalized size = 1.07 \[ \frac {b x^{4}}{3 \, \sqrt {d^{2} x^{2} - c^{2}} d^{2}} + \frac {4 \, b c^{2} x^{2}}{3 \, \sqrt {d^{2} x^{2} - c^{2}} d^{4}} + \frac {a x^{2}}{\sqrt {d^{2} x^{2} - c^{2}} d^{2}} - \frac {8 \, b c^{4}}{3 \, \sqrt {d^{2} x^{2} - c^{2}} d^{6}} - \frac {2 \, a c^{2}}{\sqrt {d^{2} x^{2} - c^{2}} d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.80, size = 90, normalized size = 0.78 \[ \frac {\sqrt {d\,x-c}\,\left (\frac {x^2\,\left (4\,b\,c^2\,d^2+3\,a\,d^4\right )}{3\,d^7}-\frac {8\,b\,c^4+6\,a\,c^2\,d^2}{3\,d^7}+\frac {b\,x^4}{3\,d^3}\right )}{x\,\sqrt {c+d\,x}-\frac {c\,\sqrt {c+d\,x}}{d}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 177.26, size = 226, normalized size = 1.97 \[ a \left (\frac {c {G_{6, 6}^{6, 2}\left (\begin {matrix} - \frac {3}{4}, - \frac {1}{4} & -1, 0, \frac {1}{2}, 1 \\- \frac {3}{4}, - \frac {1}{2}, - \frac {1}{4}, 0, \frac {1}{2}, 0 & \end {matrix} \middle | {\frac {c^{2}}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac {3}{2}} d^{4}} - \frac {i c {G_{6, 6}^{2, 6}\left (\begin {matrix} -2, - \frac {3}{2}, - \frac {5}{4}, -1, - \frac {3}{4}, 1 & \\- \frac {5}{4}, - \frac {3}{4} & -2, - \frac {3}{2}, - \frac {1}{2}, 0 \end {matrix} \middle | {\frac {c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac {3}{2}} d^{4}}\right ) + b \left (\frac {c^{3} {G_{6, 6}^{6, 2}\left (\begin {matrix} - \frac {7}{4}, - \frac {5}{4} & -2, -1, - \frac {1}{2}, 1 \\- \frac {7}{4}, - \frac {3}{2}, - \frac {5}{4}, -1, - \frac {1}{2}, 0 & \end {matrix} \middle | {\frac {c^{2}}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac {3}{2}} d^{6}} - \frac {i c^{3} {G_{6, 6}^{2, 6}\left (\begin {matrix} -3, - \frac {5}{2}, - \frac {9}{4}, -2, - \frac {7}{4}, 1 & \\- \frac {9}{4}, - \frac {7}{4} & -3, - \frac {5}{2}, - \frac {3}{2}, 0 \end {matrix} \middle | {\frac {c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac {3}{2}} d^{6}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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