Optimal. Leaf size=90 \[ \frac {d (4 b c-3 a d) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{5/2}}+\frac {x (b c-a d)^2}{a b^2 \sqrt {a+b x^2}}+\frac {d^2 x \sqrt {a+b x^2}}{2 b^2} \]
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Rubi [A] time = 0.06, antiderivative size = 105, normalized size of antiderivative = 1.17, 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, 388, 217, 206} \begin {gather*} -\frac {d x \sqrt {a+b x^2} (2 b c-3 a d)}{2 a b^2}+\frac {d (4 b c-3 a d) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{5/2}}+\frac {x \left (c+d x^2\right ) (b c-a d)}{a b \sqrt {a+b x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 388
Rule 413
Rubi steps
\begin {align*} \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{3/2}} \, dx &=\frac {(b c-a d) x \left (c+d x^2\right )}{a b \sqrt {a+b x^2}}+\frac {\int \frac {a c d-d (2 b c-3 a d) x^2}{\sqrt {a+b x^2}} \, dx}{a b}\\ &=-\frac {d (2 b c-3 a d) x \sqrt {a+b x^2}}{2 a b^2}+\frac {(b c-a d) x \left (c+d x^2\right )}{a b \sqrt {a+b x^2}}+\frac {(d (4 b c-3 a d)) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{2 b^2}\\ &=-\frac {d (2 b c-3 a d) x \sqrt {a+b x^2}}{2 a b^2}+\frac {(b c-a d) x \left (c+d x^2\right )}{a b \sqrt {a+b x^2}}+\frac {(d (4 b c-3 a d)) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{2 b^2}\\ &=-\frac {d (2 b c-3 a d) x \sqrt {a+b x^2}}{2 a b^2}+\frac {(b c-a d) x \left (c+d x^2\right )}{a b \sqrt {a+b x^2}}+\frac {d (4 b c-3 a d) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{5/2}}\\ \end {align*}
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Mathematica [C] time = 2.53, size = 160, normalized size = 1.78 \begin {gather*} \frac {x \sqrt {\frac {b x^2}{a}+1} \left (-6 b x^2 \left (c+d x^2\right )^2 \, _3F_2\left (\frac {3}{2},2,\frac {5}{2};1,\frac {9}{2};-\frac {b x^2}{a}\right )-12 b x^2 \left (2 c^2+3 c d x^2+d^2 x^4\right ) \, _2F_1\left (\frac {3}{2},\frac {5}{2};\frac {9}{2};-\frac {b x^2}{a}\right )+7 a \left (15 c^2+10 c d x^2+3 d^2 x^4\right ) \, _2F_1\left (\frac {1}{2},\frac {3}{2};\frac {7}{2};-\frac {b x^2}{a}\right )\right )}{105 a^2 \sqrt {a+b x^2}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.17, size = 99, normalized size = 1.10 \begin {gather*} \frac {3 a^2 d^2 x-4 a b c d x+a b d^2 x^3+2 b^2 c^2 x}{2 a b^2 \sqrt {a+b x^2}}+\frac {\left (3 a d^2-4 b c d\right ) \log \left (\sqrt {a+b x^2}-\sqrt {b} x\right )}{2 b^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.94, size = 276, normalized size = 3.07 \begin {gather*} \left [-\frac {{\left (4 \, a^{2} b c d - 3 \, a^{3} d^{2} + {\left (4 \, a b^{2} c d - 3 \, a^{2} b d^{2}\right )} x^{2}\right )} \sqrt {b} \log \left (-2 \, b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, {\left (a b^{2} d^{2} x^{3} + {\left (2 \, b^{3} c^{2} - 4 \, a b^{2} c d + 3 \, a^{2} b d^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{4 \, {\left (a b^{4} x^{2} + a^{2} b^{3}\right )}}, -\frac {{\left (4 \, a^{2} b c d - 3 \, a^{3} d^{2} + {\left (4 \, a b^{2} c d - 3 \, a^{2} b d^{2}\right )} x^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (a b^{2} d^{2} x^{3} + {\left (2 \, b^{3} c^{2} - 4 \, a b^{2} c d + 3 \, a^{2} b d^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{2 \, {\left (a b^{4} x^{2} + a^{2} b^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.63, size = 92, normalized size = 1.02 \begin {gather*} \frac {{\left (\frac {d^{2} x^{2}}{b} + \frac {2 \, b^{3} c^{2} - 4 \, a b^{2} c d + 3 \, a^{2} b d^{2}}{a b^{3}}\right )} x}{2 \, \sqrt {b x^{2} + a}} - \frac {{\left (4 \, b c d - 3 \, a d^{2}\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{2 \, b^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 123, normalized size = 1.37 \begin {gather*} \frac {d^{2} x^{3}}{2 \sqrt {b \,x^{2}+a}\, b}+\frac {3 a \,d^{2} x}{2 \sqrt {b \,x^{2}+a}\, b^{2}}+\frac {c^{2} x}{\sqrt {b \,x^{2}+a}\, a}-\frac {2 c d x}{\sqrt {b \,x^{2}+a}\, b}-\frac {3 a \,d^{2} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {5}{2}}}+\frac {2 c d \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.38, size = 108, normalized size = 1.20 \begin {gather*} \frac {d^{2} x^{3}}{2 \, \sqrt {b x^{2} + a} b} + \frac {c^{2} x}{\sqrt {b x^{2} + a} a} - \frac {2 \, c d x}{\sqrt {b x^{2} + a} b} + \frac {3 \, a d^{2} x}{2 \, \sqrt {b x^{2} + a} b^{2}} + \frac {2 \, c d \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{b^{\frac {3}{2}}} - \frac {3 \, a d^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, b^{\frac {5}{2}}} \end {gather*}
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 \begin {gather*} \int \frac {{\left (d\,x^2+c\right )}^2}{{\left (b\,x^2+a\right )}^{3/2}} \,d x \end {gather*}
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 \begin {gather*} \int \frac {\left (c + d x^{2}\right )^{2}}{\left (a + b x^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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