Optimal. Leaf size=152 \[ -\frac {\sqrt {d x-c} \left (2 a d^2+3 b c^2\right )}{2 d^5 \sqrt {c+d x}}-\frac {c \left (2 a d^2+3 b c^2\right )}{2 d^5 \sqrt {d x-c} \sqrt {c+d x}}+\frac {\left (2 a d^2+3 b c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d x-c}}{\sqrt {c+d x}}\right )}{d^5}+\frac {b x^3}{2 d^2 \sqrt {d x-c} \sqrt {c+d x}} \]
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Rubi [A] time = 0.11, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {460, 89, 12, 78, 63, 217, 206} \[ -\frac {\sqrt {d x-c} \left (2 a d^2+3 b c^2\right )}{2 d^5 \sqrt {c+d x}}-\frac {c \left (2 a d^2+3 b c^2\right )}{2 d^5 \sqrt {d x-c} \sqrt {c+d x}}+\frac {\left (2 a d^2+3 b c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d x-c}}{\sqrt {c+d x}}\right )}{d^5}+\frac {b x^3}{2 d^2 \sqrt {d x-c} \sqrt {c+d x}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 63
Rule 78
Rule 89
Rule 206
Rule 217
Rule 460
Rubi steps
\begin {align*} \int \frac {x^2 \left (a+b x^2\right )}{(-c+d x)^{3/2} (c+d x)^{3/2}} \, dx &=\frac {b x^3}{2 d^2 \sqrt {-c+d x} \sqrt {c+d x}}-\frac {1}{2} \left (-2 a-\frac {3 b c^2}{d^2}\right ) \int \frac {x^2}{(-c+d x)^{3/2} (c+d x)^{3/2}} \, dx\\ &=-\frac {c \left (3 b c^2+2 a d^2\right )}{2 d^5 \sqrt {-c+d x} \sqrt {c+d x}}+\frac {b x^3}{2 d^2 \sqrt {-c+d x} \sqrt {c+d x}}-\frac {\left (-2 a-\frac {3 b c^2}{d^2}\right ) \int \frac {c d^2 x}{\sqrt {-c+d x} (c+d x)^{3/2}} \, dx}{2 c d^3}\\ &=-\frac {c \left (3 b c^2+2 a d^2\right )}{2 d^5 \sqrt {-c+d x} \sqrt {c+d x}}+\frac {b x^3}{2 d^2 \sqrt {-c+d x} \sqrt {c+d x}}+\frac {\left (3 b c^2+2 a d^2\right ) \int \frac {x}{\sqrt {-c+d x} (c+d x)^{3/2}} \, dx}{2 d^3}\\ &=-\frac {c \left (3 b c^2+2 a d^2\right )}{2 d^5 \sqrt {-c+d x} \sqrt {c+d x}}+\frac {b x^3}{2 d^2 \sqrt {-c+d x} \sqrt {c+d x}}-\frac {\left (3 b c^2+2 a d^2\right ) \sqrt {-c+d x}}{2 d^5 \sqrt {c+d x}}+\frac {\left (3 b c^2+2 a d^2\right ) \int \frac {1}{\sqrt {-c+d x} \sqrt {c+d x}} \, dx}{2 d^4}\\ &=-\frac {c \left (3 b c^2+2 a d^2\right )}{2 d^5 \sqrt {-c+d x} \sqrt {c+d x}}+\frac {b x^3}{2 d^2 \sqrt {-c+d x} \sqrt {c+d x}}-\frac {\left (3 b c^2+2 a d^2\right ) \sqrt {-c+d x}}{2 d^5 \sqrt {c+d x}}+\frac {\left (3 b c^2+2 a d^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2 c+x^2}} \, dx,x,\sqrt {-c+d x}\right )}{d^5}\\ &=-\frac {c \left (3 b c^2+2 a d^2\right )}{2 d^5 \sqrt {-c+d x} \sqrt {c+d x}}+\frac {b x^3}{2 d^2 \sqrt {-c+d x} \sqrt {c+d x}}-\frac {\left (3 b c^2+2 a d^2\right ) \sqrt {-c+d x}}{2 d^5 \sqrt {c+d x}}+\frac {\left (3 b c^2+2 a d^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt {-c+d x}}{\sqrt {c+d x}}\right )}{d^5}\\ &=-\frac {c \left (3 b c^2+2 a d^2\right )}{2 d^5 \sqrt {-c+d x} \sqrt {c+d x}}+\frac {b x^3}{2 d^2 \sqrt {-c+d x} \sqrt {c+d x}}-\frac {\left (3 b c^2+2 a d^2\right ) \sqrt {-c+d x}}{2 d^5 \sqrt {c+d x}}+\frac {\left (3 b c^2+2 a d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {-c+d x}}{\sqrt {c+d x}}\right )}{d^5}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 90, normalized size = 0.59 \[ \frac {c \sqrt {1-\frac {d^2 x^2}{c^2}} \left (2 a d^2+3 b c^2\right ) \sin ^{-1}\left (\frac {d x}{c}\right )-2 a d^3 x-3 b c^2 d x+b d^3 x^3}{2 d^5 \sqrt {d x-c} \sqrt {c+d x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.98, size = 159, normalized size = 1.05 \[ \frac {2 \, b c^{4} + 2 \, a c^{2} d^{2} - 2 \, {\left (b c^{2} d^{2} + a d^{4}\right )} x^{2} + {\left (b d^{3} x^{3} - {\left (3 \, b c^{2} d + 2 \, a d^{3}\right )} x\right )} \sqrt {d x + c} \sqrt {d x - c} + {\left (3 \, b c^{4} + 2 \, a c^{2} d^{2} - {\left (3 \, b c^{2} d^{2} + 2 \, a d^{4}\right )} x^{2}\right )} \log \left (-d x + \sqrt {d x + c} \sqrt {d x - c}\right )}{2 \, {\left (d^{7} x^{2} - c^{2} d^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 147, normalized size = 0.97 \[ \frac {\sqrt {d x + c} {\left ({\left (d x + c\right )} {\left (\frac {{\left (d x + c\right )} b}{d^{5}} - \frac {3 \, b c}{d^{5}}\right )} + \frac {b c^{2} d^{15} - a d^{17}}{d^{20}}\right )}}{2 \, \sqrt {d x - c}} - \frac {{\left (3 \, b c^{2} + 2 \, a d^{2}\right )} \log \left ({\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{2}\right )}{2 \, d^{5}} - \frac {2 \, {\left (b c^{3} + a c d^{2}\right )}}{{\left ({\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{2} + 2 \, c\right )} d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.08, size = 254, normalized size = 1.67 \[ \frac {\left (2 a \,d^{4} x^{2} \ln \left (\left (d x +\sqrt {d^{2} x^{2}-c^{2}}\, \mathrm {csgn}\relax (d )\right ) \mathrm {csgn}\relax (d )\right )+3 b \,c^{2} d^{2} x^{2} \ln \left (\left (d x +\sqrt {d^{2} x^{2}-c^{2}}\, \mathrm {csgn}\relax (d )\right ) \mathrm {csgn}\relax (d )\right )+\sqrt {d^{2} x^{2}-c^{2}}\, b \,d^{3} x^{3} \mathrm {csgn}\relax (d )-2 a \,c^{2} d^{2} \ln \left (\left (d x +\sqrt {d^{2} x^{2}-c^{2}}\, \mathrm {csgn}\relax (d )\right ) \mathrm {csgn}\relax (d )\right )-2 \sqrt {d^{2} x^{2}-c^{2}}\, a \,d^{3} x \,\mathrm {csgn}\relax (d )-3 b \,c^{4} \ln \left (\left (d x +\sqrt {d^{2} x^{2}-c^{2}}\, \mathrm {csgn}\relax (d )\right ) \mathrm {csgn}\relax (d )\right )-3 \sqrt {d^{2} x^{2}-c^{2}}\, b \,c^{2} d x \,\mathrm {csgn}\relax (d )\right ) \mathrm {csgn}\relax (d )}{2 \sqrt {d^{2} x^{2}-c^{2}}\, \sqrt {d x +c}\, \sqrt {d x -c}\, d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 138, normalized size = 0.91 \[ \frac {b x^{3}}{2 \, \sqrt {d^{2} x^{2} - c^{2}} d^{2}} - \frac {3 \, b c^{2} x}{2 \, \sqrt {d^{2} x^{2} - c^{2}} d^{4}} - \frac {a x}{\sqrt {d^{2} x^{2} - c^{2}} d^{2}} + \frac {3 \, b c^{2} \log \left (2 \, d^{2} x + 2 \, \sqrt {d^{2} x^{2} - c^{2}} d\right )}{2 \, d^{5}} + \frac {a \log \left (2 \, d^{2} x + 2 \, \sqrt {d^{2} x^{2} - c^{2}} d\right )}{d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^2\,\left (b\,x^2+a\right )}{{\left (c+d\,x\right )}^{3/2}\,{\left (d\,x-c\right )}^{3/2}} \,d x \]
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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