Optimal. Leaf size=105 \[ \frac {x (b c-a d) (3 a d+2 b c)}{3 a^2 b^2 \sqrt {a+b x^2}}+\frac {d^2 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{b^{5/2}}+\frac {x \left (c+d x^2\right ) (b c-a d)}{3 a b \left (a+b x^2\right )^{3/2}} \]
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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} \[ \frac {x (b c-a d) (3 a d+2 b c)}{3 a^2 b^2 \sqrt {a+b x^2}}+\frac {d^2 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{b^{5/2}}+\frac {x \left (c+d x^2\right ) (b c-a d)}{3 a b \left (a+b x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 385
Rule 413
Rubi steps
\begin {align*} \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{5/2}} \, dx &=\frac {(b c-a d) x \left (c+d x^2\right )}{3 a b \left (a+b x^2\right )^{3/2}}+\frac {\int \frac {c (2 b c+a d)+3 a d^2 x^2}{\left (a+b x^2\right )^{3/2}} \, dx}{3 a b}\\ &=\frac {(b c-a d) (2 b c+3 a d) x}{3 a^2 b^2 \sqrt {a+b x^2}}+\frac {(b c-a d) x \left (c+d x^2\right )}{3 a b \left (a+b x^2\right )^{3/2}}+\frac {d^2 \int \frac {1}{\sqrt {a+b x^2}} \, dx}{b^2}\\ &=\frac {(b c-a d) (2 b c+3 a d) x}{3 a^2 b^2 \sqrt {a+b x^2}}+\frac {(b c-a d) x \left (c+d x^2\right )}{3 a b \left (a+b x^2\right )^{3/2}}+\frac {d^2 \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{b^2}\\ &=\frac {(b c-a d) (2 b c+3 a d) x}{3 a^2 b^2 \sqrt {a+b x^2}}+\frac {(b c-a d) x \left (c+d x^2\right )}{3 a b \left (a+b x^2\right )^{3/2}}+\frac {d^2 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{b^{5/2}}\\ \end {align*}
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Mathematica [C] time = 4.15, size = 214, normalized size = 2.04 \[ \frac {\sqrt {\frac {b x^2}{a}+1} \left (-16 b^3 x^6 \left (c+d x^2\right )^2 \, _3F_2\left (\frac {3}{2},2,\frac {7}{2};1,\frac {9}{2};-\frac {b x^2}{a}\right )+\frac {7 a^2 \left (15 c^2+10 c d x^2+3 d^2 x^4\right ) \left (\sqrt {-\frac {b x^2 \left (a+b x^2\right )}{a^2}} \left (2 b x^2-3 a\right )+3 a \sin ^{-1}\left (\sqrt {-\frac {b x^2}{a}}\right )\right )}{\sqrt {-\frac {b x^2}{a}}}-32 b^3 x^6 \left (2 c^2+3 c d x^2+d^2 x^4\right ) \, _2F_1\left (\frac {3}{2},\frac {7}{2};\frac {9}{2};-\frac {b x^2}{a}\right )\right )}{168 a^3 b^2 x^3 \sqrt {a+b x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.47, size = 318, normalized size = 3.03 \[ \left [\frac {3 \, {\left (a^{2} b^{2} d^{2} x^{4} + 2 \, a^{3} b d^{2} x^{2} + a^{4} d^{2}\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (2 \, {\left (b^{4} c^{2} + a b^{3} c d - 2 \, a^{2} b^{2} d^{2}\right )} x^{3} + 3 \, {\left (a b^{3} c^{2} - a^{3} b d^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{6 \, {\left (a^{2} b^{5} x^{4} + 2 \, a^{3} b^{4} x^{2} + a^{4} b^{3}\right )}}, -\frac {3 \, {\left (a^{2} b^{2} d^{2} x^{4} + 2 \, a^{3} b d^{2} x^{2} + a^{4} d^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (2 \, {\left (b^{4} c^{2} + a b^{3} c d - 2 \, a^{2} b^{2} d^{2}\right )} x^{3} + 3 \, {\left (a b^{3} c^{2} - a^{3} b d^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{3 \, {\left (a^{2} b^{5} x^{4} + 2 \, a^{3} b^{4} x^{2} + a^{4} b^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.63, size = 103, normalized size = 0.98 \[ \frac {x {\left (\frac {2 \, {\left (b^{4} c^{2} + a b^{3} c d - 2 \, a^{2} b^{2} d^{2}\right )} x^{2}}{a^{2} b^{3}} + \frac {3 \, {\left (a b^{3} c^{2} - a^{3} b d^{2}\right )}}{a^{2} b^{3}}\right )}}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}}} - \frac {d^{2} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{b^{\frac {5}{2}}} \]
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 \[ -\frac {d^{2} x^{3}}{3 \left (b \,x^{2}+a \right )^{\frac {3}{2}} b}+\frac {c^{2} x}{3 \left (b \,x^{2}+a \right )^{\frac {3}{2}} a}-\frac {2 c d x}{3 \left (b \,x^{2}+a \right )^{\frac {3}{2}} b}+\frac {2 c d x}{3 \sqrt {b \,x^{2}+a}\, a b}+\frac {2 c^{2} x}{3 \sqrt {b \,x^{2}+a}\, a^{2}}-\frac {d^{2} x}{\sqrt {b \,x^{2}+a}\, b^{2}}+\frac {d^{2} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.39, size = 147, normalized size = 1.40 \[ -\frac {1}{3} \, d^{2} x {\left (\frac {3 \, x^{2}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {2 \, a}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}}\right )} + \frac {2 \, c^{2} x}{3 \, \sqrt {b x^{2} + a} a^{2}} + \frac {c^{2} x}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a} - \frac {2 \, c d x}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {2 \, c d x}{3 \, \sqrt {b x^{2} + a} a b} - \frac {d^{2} x}{3 \, \sqrt {b x^{2} + a} b^{2}} + \frac {d^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{b^{\frac {5}{2}}} \]
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 \[ \int \frac {{\left (d\,x^2+c\right )}^2}{{\left (b\,x^2+a\right )}^{5/2}} \,d x \]
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 \[ \int \frac {\left (c + d x^{2}\right )^{2}}{\left (a + b x^{2}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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