Optimal. Leaf size=169 \[ \frac {3 d \left (5 a^2 d^2-12 a b c d+8 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 b^{7/2}}-\frac {d x \sqrt {a+b x^2} (2 b c-5 a d) (4 b c-3 a d)}{8 a b^3}-\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right ) (4 b c-5 a d)}{4 a b^2}+\frac {x \left (c+d x^2\right )^2 (b c-a d)}{a b \sqrt {a+b x^2}} \]
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Rubi [A] time = 0.20, antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {413, 528, 388, 217, 206} \begin {gather*} \frac {3 d \left (5 a^2 d^2-12 a b c d+8 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 b^{7/2}}-\frac {d x \sqrt {a+b x^2} \left (c+d x^2\right ) (4 b c-5 a d)}{4 a b^2}-\frac {d x \sqrt {a+b x^2} (2 b c-5 a d) (4 b c-3 a d)}{8 a b^3}+\frac {x \left (c+d x^2\right )^2 (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
Rule 528
Rubi steps
\begin {align*} \int \frac {\left (c+d x^2\right )^3}{\left (a+b x^2\right )^{3/2}} \, dx &=\frac {(b c-a d) x \left (c+d x^2\right )^2}{a b \sqrt {a+b x^2}}+\frac {\int \frac {\left (c+d x^2\right ) \left (a c d-d (4 b c-5 a d) x^2\right )}{\sqrt {a+b x^2}} \, dx}{a b}\\ &=-\frac {d (4 b c-5 a d) x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 a b^2}+\frac {(b c-a d) x \left (c+d x^2\right )^2}{a b \sqrt {a+b x^2}}+\frac {\int \frac {a c d (8 b c-5 a d)-d (2 b c-5 a d) (4 b c-3 a d) x^2}{\sqrt {a+b x^2}} \, dx}{4 a b^2}\\ &=-\frac {d (2 b c-5 a d) (4 b c-3 a d) x \sqrt {a+b x^2}}{8 a b^3}-\frac {d (4 b c-5 a d) x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 a b^2}+\frac {(b c-a d) x \left (c+d x^2\right )^2}{a b \sqrt {a+b x^2}}+\frac {\left (3 d \left (8 b^2 c^2-12 a b c d+5 a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{8 b^3}\\ &=-\frac {d (2 b c-5 a d) (4 b c-3 a d) x \sqrt {a+b x^2}}{8 a b^3}-\frac {d (4 b c-5 a d) x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 a b^2}+\frac {(b c-a d) x \left (c+d x^2\right )^2}{a b \sqrt {a+b x^2}}+\frac {\left (3 d \left (8 b^2 c^2-12 a b c d+5 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{8 b^3}\\ &=-\frac {d (2 b c-5 a d) (4 b c-3 a d) x \sqrt {a+b x^2}}{8 a b^3}-\frac {d (4 b c-5 a d) x \sqrt {a+b x^2} \left (c+d x^2\right )}{4 a b^2}+\frac {(b c-a d) x \left (c+d x^2\right )^2}{a b \sqrt {a+b x^2}}+\frac {3 d \left (8 b^2 c^2-12 a b c d+5 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 b^{7/2}}\\ \end {align*}
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Mathematica [A] time = 5.10, size = 122, normalized size = 0.72 \begin {gather*} \frac {3 d \left (5 a^2 d^2-12 a b c d+8 b^2 c^2\right ) \log \left (\sqrt {b} \sqrt {a+b x^2}+b x\right )}{8 b^{7/2}}+\frac {x \sqrt {a+b x^2} \left (d^2 (12 b c-7 a d)+\frac {8 (b c-a d)^3}{a \left (a+b x^2\right )}+2 b d^3 x^2\right )}{8 b^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.28, size = 156, normalized size = 0.92 \begin {gather*} \frac {-15 a^3 d^3 x+36 a^2 b c d^2 x-5 a^2 b d^3 x^3-24 a b^2 c^2 d x+12 a b^2 c d^2 x^3+2 a b^2 d^3 x^5+8 b^3 c^3 x}{8 a b^3 \sqrt {a+b x^2}}-\frac {3 \left (5 a^2 d^3-12 a b c d^2+8 b^2 c^2 d\right ) \log \left (\sqrt {a+b x^2}-\sqrt {b} x\right )}{8 b^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 416, normalized size = 2.46 \begin {gather*} \left [\frac {3 \, {\left (8 \, a^{2} b^{2} c^{2} d - 12 \, a^{3} b c d^{2} + 5 \, a^{4} d^{3} + {\left (8 \, a b^{3} c^{2} d - 12 \, a^{2} b^{2} c d^{2} + 5 \, a^{3} b d^{3}\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 (2 \, a b^{3} d^{3} x^{5} + {\left (12 \, a b^{3} c d^{2} - 5 \, a^{2} b^{2} d^{3}\right )} x^{3} + {\left (8 \, b^{4} c^{3} - 24 \, a b^{3} c^{2} d + 36 \, a^{2} b^{2} c d^{2} - 15 \, a^{3} b d^{3}\right )} x\right )} \sqrt {b x^{2} + a}}{16 \, {\left (a b^{5} x^{2} + a^{2} b^{4}\right )}}, -\frac {3 \, {\left (8 \, a^{2} b^{2} c^{2} d - 12 \, a^{3} b c d^{2} + 5 \, a^{4} d^{3} + {\left (8 \, a b^{3} c^{2} d - 12 \, a^{2} b^{2} c d^{2} + 5 \, a^{3} b d^{3}\right )} x^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (2 \, a b^{3} d^{3} x^{5} + {\left (12 \, a b^{3} c d^{2} - 5 \, a^{2} b^{2} d^{3}\right )} x^{3} + {\left (8 \, b^{4} c^{3} - 24 \, a b^{3} c^{2} d + 36 \, a^{2} b^{2} c d^{2} - 15 \, a^{3} b d^{3}\right )} x\right )} \sqrt {b x^{2} + a}}{8 \, {\left (a b^{5} x^{2} + a^{2} b^{4}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.64, size = 157, normalized size = 0.93 \begin {gather*} \frac {{\left ({\left (\frac {2 \, d^{3} x^{2}}{b} + \frac {12 \, a b^{4} c d^{2} - 5 \, a^{2} b^{3} d^{3}}{a b^{5}}\right )} x^{2} + \frac {8 \, b^{5} c^{3} - 24 \, a b^{4} c^{2} d + 36 \, a^{2} b^{3} c d^{2} - 15 \, a^{3} b^{2} d^{3}}{a b^{5}}\right )} x}{8 \, \sqrt {b x^{2} + a}} - \frac {3 \, {\left (8 \, b^{2} c^{2} d - 12 \, a b c d^{2} + 5 \, a^{2} d^{3}\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{8 \, b^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 219, normalized size = 1.30 \begin {gather*} \frac {d^{3} x^{5}}{4 \sqrt {b \,x^{2}+a}\, b}-\frac {5 a \,d^{3} x^{3}}{8 \sqrt {b \,x^{2}+a}\, b^{2}}+\frac {3 c \,d^{2} x^{3}}{2 \sqrt {b \,x^{2}+a}\, b}-\frac {15 a^{2} d^{3} x}{8 \sqrt {b \,x^{2}+a}\, b^{3}}+\frac {9 a c \,d^{2} x}{2 \sqrt {b \,x^{2}+a}\, b^{2}}+\frac {c^{3} x}{\sqrt {b \,x^{2}+a}\, a}-\frac {3 c^{2} d x}{\sqrt {b \,x^{2}+a}\, b}+\frac {15 a^{2} d^{3} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{8 b^{\frac {7}{2}}}-\frac {9 a c \,d^{2} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {5}{2}}}+\frac {3 c^{2} 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 = 1.40, size = 197, normalized size = 1.17 \begin {gather*} \frac {d^{3} x^{5}}{4 \, \sqrt {b x^{2} + a} b} + \frac {3 \, c d^{2} x^{3}}{2 \, \sqrt {b x^{2} + a} b} - \frac {5 \, a d^{3} x^{3}}{8 \, \sqrt {b x^{2} + a} b^{2}} + \frac {c^{3} x}{\sqrt {b x^{2} + a} a} - \frac {3 \, c^{2} d x}{\sqrt {b x^{2} + a} b} + \frac {9 \, a c d^{2} x}{2 \, \sqrt {b x^{2} + a} b^{2}} - \frac {15 \, a^{2} d^{3} x}{8 \, \sqrt {b x^{2} + a} b^{3}} + \frac {3 \, c^{2} d \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{b^{\frac {3}{2}}} - \frac {9 \, a c d^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, b^{\frac {5}{2}}} + \frac {15 \, a^{2} d^{3} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{8 \, b^{\frac {7}{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 )}^3}{{\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 )^{3}}{\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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