Optimal. Leaf size=158 \[ \frac {\left (8 a^2 d^2-12 a b c d+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{8 b^3 \sqrt {d}}-\frac {\sqrt {a} (b c-a d)^{3/2} \tan ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{b^3}+\frac {x \sqrt {c+d x^2} (5 b c-4 a d)}{8 b^2}+\frac {d x^3 \sqrt {c+d x^2}}{4 b} \]
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Rubi [A] time = 0.25, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {477, 582, 523, 217, 206, 377, 205} \begin {gather*} \frac {\left (8 a^2 d^2-12 a b c d+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{8 b^3 \sqrt {d}}+\frac {x \sqrt {c+d x^2} (5 b c-4 a d)}{8 b^2}-\frac {\sqrt {a} (b c-a d)^{3/2} \tan ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{b^3}+\frac {d x^3 \sqrt {c+d x^2}}{4 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 206
Rule 217
Rule 377
Rule 477
Rule 523
Rule 582
Rubi steps
\begin {align*} \int \frac {x^2 \left (c+d x^2\right )^{3/2}}{a+b x^2} \, dx &=\frac {d x^3 \sqrt {c+d x^2}}{4 b}+\frac {\int \frac {x^2 \left (c (4 b c-3 a d)+d (5 b c-4 a d) x^2\right )}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{4 b}\\ &=\frac {(5 b c-4 a d) x \sqrt {c+d x^2}}{8 b^2}+\frac {d x^3 \sqrt {c+d x^2}}{4 b}-\frac {\int \frac {a c d (5 b c-4 a d)-d \left (3 b^2 c^2-12 a b c d+8 a^2 d^2\right ) x^2}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{8 b^2 d}\\ &=\frac {(5 b c-4 a d) x \sqrt {c+d x^2}}{8 b^2}+\frac {d x^3 \sqrt {c+d x^2}}{4 b}-\frac {\left (a (b c-a d)^2\right ) \int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{b^3}+\frac {\left (3 b^2 c^2-12 a b c d+8 a^2 d^2\right ) \int \frac {1}{\sqrt {c+d x^2}} \, dx}{8 b^3}\\ &=\frac {(5 b c-4 a d) x \sqrt {c+d x^2}}{8 b^2}+\frac {d x^3 \sqrt {c+d x^2}}{4 b}-\frac {\left (a (b c-a d)^2\right ) \operatorname {Subst}\left (\int \frac {1}{a-(-b c+a d) x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{b^3}+\frac {\left (3 b^2 c^2-12 a b c d+8 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-d x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{8 b^3}\\ &=\frac {(5 b c-4 a d) x \sqrt {c+d x^2}}{8 b^2}+\frac {d x^3 \sqrt {c+d x^2}}{4 b}-\frac {\sqrt {a} (b c-a d)^{3/2} \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{b^3}+\frac {\left (3 b^2 c^2-12 a b c d+8 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{8 b^3 \sqrt {d}}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 139, normalized size = 0.88 \begin {gather*} \frac {\frac {\left (8 a^2 d^2-12 a b c d+3 b^2 c^2\right ) \log \left (\sqrt {d} \sqrt {c+d x^2}+d x\right )}{\sqrt {d}}+b x \sqrt {c+d x^2} \left (-4 a d+5 b c+2 b d x^2\right )-8 \sqrt {a} (b c-a d)^{3/2} \tan ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{8 b^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.34, size = 178, normalized size = 1.13 \begin {gather*} \frac {\sqrt {b c-a d} \left (\sqrt {a} b c-a^{3/2} d\right ) \tan ^{-1}\left (\frac {a \sqrt {d}-b x \sqrt {c+d x^2}+b \sqrt {d} x^2}{\sqrt {a} \sqrt {b c-a d}}\right )}{b^3}+\frac {\left (-8 a^2 d^2+12 a b c d-3 b^2 c^2\right ) \log \left (\sqrt {c+d x^2}-\sqrt {d} x\right )}{8 b^3 \sqrt {d}}+\frac {\sqrt {c+d x^2} \left (-4 a d x+5 b c x+2 b d x^3\right )}{8 b^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 2.99, size = 894, normalized size = 5.66 \begin {gather*} \left [\frac {{\left (3 \, b^{2} c^{2} - 12 \, a b c d + 8 \, a^{2} d^{2}\right )} \sqrt {d} \log \left (-2 \, d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) - 4 \, \sqrt {-a b c + a^{2} d} {\left (b c d - a d^{2}\right )} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} + 4 \, {\left ({\left (b c - 2 \, a d\right )} x^{3} - a c x\right )} \sqrt {-a b c + a^{2} d} \sqrt {d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 2 \, {\left (2 \, b^{2} d^{2} x^{3} + {\left (5 \, b^{2} c d - 4 \, a b d^{2}\right )} x\right )} \sqrt {d x^{2} + c}}{16 \, b^{3} d}, -\frac {{\left (3 \, b^{2} c^{2} - 12 \, a b c d + 8 \, a^{2} d^{2}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) + 2 \, \sqrt {-a b c + a^{2} d} {\left (b c d - a d^{2}\right )} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} + 4 \, {\left ({\left (b c - 2 \, a d\right )} x^{3} - a c x\right )} \sqrt {-a b c + a^{2} d} \sqrt {d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - {\left (2 \, b^{2} d^{2} x^{3} + {\left (5 \, b^{2} c d - 4 \, a b d^{2}\right )} x\right )} \sqrt {d x^{2} + c}}{8 \, b^{3} d}, -\frac {8 \, \sqrt {a b c - a^{2} d} {\left (b c d - a d^{2}\right )} \arctan \left (\frac {\sqrt {a b c - a^{2} d} {\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c}}{2 \, {\left ({\left (a b c d - a^{2} d^{2}\right )} x^{3} + {\left (a b c^{2} - a^{2} c d\right )} x\right )}}\right ) - {\left (3 \, b^{2} c^{2} - 12 \, a b c d + 8 \, a^{2} d^{2}\right )} \sqrt {d} \log \left (-2 \, d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) - 2 \, {\left (2 \, b^{2} d^{2} x^{3} + {\left (5 \, b^{2} c d - 4 \, a b d^{2}\right )} x\right )} \sqrt {d x^{2} + c}}{16 \, b^{3} d}, -\frac {4 \, \sqrt {a b c - a^{2} d} {\left (b c d - a d^{2}\right )} \arctan \left (\frac {\sqrt {a b c - a^{2} d} {\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c}}{2 \, {\left ({\left (a b c d - a^{2} d^{2}\right )} x^{3} + {\left (a b c^{2} - a^{2} c d\right )} x\right )}}\right ) + {\left (3 \, b^{2} c^{2} - 12 \, a b c d + 8 \, a^{2} d^{2}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) - {\left (2 \, b^{2} d^{2} x^{3} + {\left (5 \, b^{2} c d - 4 \, a b d^{2}\right )} x\right )} \sqrt {d x^{2} + c}}{8 \, b^{3} d}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 1973, normalized size = 12.49
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} x^{2}}{b x^{2} + a}\,{d x} \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 {x^2\,{\left (d\,x^2+c\right )}^{3/2}}{b\,x^2+a} \,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 {x^{2} \left (c + d x^{2}\right )^{\frac {3}{2}}}{a + b x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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