Optimal. Leaf size=105 \[ -\frac {x (b c-a d) (2 a d+3 b c)}{3 c^2 d^2 \sqrt {c+d x^2}}-\frac {x \left (a+b x^2\right ) (b c-a d)}{3 c d \left (c+d x^2\right )^{3/2}}+\frac {b^2 \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{d^{5/2}} \]
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Rubi [A] time = 0.05, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {413, 385, 217, 206} \[ -\frac {x (b c-a d) (2 a d+3 b c)}{3 c^2 d^2 \sqrt {c+d x^2}}-\frac {x \left (a+b x^2\right ) (b c-a d)}{3 c d \left (c+d x^2\right )^{3/2}}+\frac {b^2 \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{d^{5/2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 385
Rule 413
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx &=-\frac {(b c-a d) x \left (a+b x^2\right )}{3 c d \left (c+d x^2\right )^{3/2}}+\frac {\int \frac {a (b c+2 a d)+3 b^2 c x^2}{\left (c+d x^2\right )^{3/2}} \, dx}{3 c d}\\ &=-\frac {(b c-a d) x \left (a+b x^2\right )}{3 c d \left (c+d x^2\right )^{3/2}}-\frac {(b c-a d) (3 b c+2 a d) x}{3 c^2 d^2 \sqrt {c+d x^2}}+\frac {b^2 \int \frac {1}{\sqrt {c+d x^2}} \, dx}{d^2}\\ &=-\frac {(b c-a d) x \left (a+b x^2\right )}{3 c d \left (c+d x^2\right )^{3/2}}-\frac {(b c-a d) (3 b c+2 a d) x}{3 c^2 d^2 \sqrt {c+d x^2}}+\frac {b^2 \operatorname {Subst}\left (\int \frac {1}{1-d x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{d^2}\\ &=-\frac {(b c-a d) x \left (a+b x^2\right )}{3 c d \left (c+d x^2\right )^{3/2}}-\frac {(b c-a d) (3 b c+2 a d) x}{3 c^2 d^2 \sqrt {c+d x^2}}+\frac {b^2 \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{d^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 101, normalized size = 0.96 \[ \frac {x \left (a^2 d^2 \left (3 c+2 d x^2\right )+2 a b c d^2 x^2-b^2 c^2 \left (3 c+4 d x^2\right )\right )}{3 c^2 d^2 \left (c+d x^2\right )^{3/2}}+\frac {b^2 \log \left (\sqrt {d} \sqrt {c+d x^2}+d x\right )}{d^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 321, normalized size = 3.06 \[ \left [\frac {3 \, {\left (b^{2} c^{2} d^{2} x^{4} + 2 \, b^{2} c^{3} d x^{2} + b^{2} c^{4}\right )} \sqrt {d} \log \left (-2 \, d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) - 2 \, {\left (2 \, {\left (2 \, b^{2} c^{2} d^{2} - a b c d^{3} - a^{2} d^{4}\right )} x^{3} + 3 \, {\left (b^{2} c^{3} d - a^{2} c d^{3}\right )} x\right )} \sqrt {d x^{2} + c}}{6 \, {\left (c^{2} d^{5} x^{4} + 2 \, c^{3} d^{4} x^{2} + c^{4} d^{3}\right )}}, -\frac {3 \, {\left (b^{2} c^{2} d^{2} x^{4} + 2 \, b^{2} c^{3} d x^{2} + b^{2} c^{4}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) + {\left (2 \, {\left (2 \, b^{2} c^{2} d^{2} - a b c d^{3} - a^{2} d^{4}\right )} x^{3} + 3 \, {\left (b^{2} c^{3} d - a^{2} c d^{3}\right )} x\right )} \sqrt {d x^{2} + c}}{3 \, {\left (c^{2} d^{5} x^{4} + 2 \, c^{3} d^{4} x^{2} + c^{4} d^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.52, size = 105, normalized size = 1.00 \[ -\frac {x {\left (\frac {2 \, {\left (2 \, b^{2} c^{2} d^{2} - a b c d^{3} - a^{2} d^{4}\right )} x^{2}}{c^{2} d^{3}} + \frac {3 \, {\left (b^{2} c^{3} d - a^{2} c d^{3}\right )}}{c^{2} d^{3}}\right )}}{3 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}}} - \frac {b^{2} \log \left ({\left | -\sqrt {d} x + \sqrt {d x^{2} + c} \right |}\right )}{d^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 136, normalized size = 1.30 \[ -\frac {b^{2} x^{3}}{3 \left (d \,x^{2}+c \right )^{\frac {3}{2}} d}+\frac {a^{2} x}{3 \left (d \,x^{2}+c \right )^{\frac {3}{2}} c}-\frac {2 a b x}{3 \left (d \,x^{2}+c \right )^{\frac {3}{2}} d}+\frac {2 a^{2} x}{3 \sqrt {d \,x^{2}+c}\, c^{2}}+\frac {2 a b x}{3 \sqrt {d \,x^{2}+c}\, c d}-\frac {b^{2} x}{\sqrt {d \,x^{2}+c}\, d^{2}}+\frac {b^{2} \ln \left (\sqrt {d}\, x +\sqrt {d \,x^{2}+c}\right )}{d^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.95, size = 147, normalized size = 1.40 \[ -\frac {1}{3} \, b^{2} x {\left (\frac {3 \, x^{2}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} d} + \frac {2 \, c}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} d^{2}}\right )} + \frac {2 \, a^{2} x}{3 \, \sqrt {d x^{2} + c} c^{2}} + \frac {a^{2} x}{3 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} c} - \frac {b^{2} x}{3 \, \sqrt {d x^{2} + c} d^{2}} - \frac {2 \, a b x}{3 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} d} + \frac {2 \, a b x}{3 \, \sqrt {d x^{2} + c} c d} + \frac {b^{2} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right )}{d^{\frac {5}{2}}} \]
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 \[ \int \frac {{\left (b\,x^2+a\right )}^2}{{\left (d\,x^2+c\right )}^{5/2}} \,d x \]
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 \[ \int \frac {\left (a + b x^{2}\right )^{2}}{\left (c + d x^{2}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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