Optimal. Leaf size=115 \[ -\frac {a (b c-a d) \sqrt {c+d x^2}}{b^3}-\frac {a \left (c+d x^2\right )^{3/2}}{3 b^2}+\frac {\left (c+d x^2\right )^{5/2}}{5 b d}+\frac {a (b c-a d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{b^{7/2}} \]
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Rubi [A]
time = 0.07, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {457, 81, 52, 65,
214} \begin {gather*} \frac {a (b c-a d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{b^{7/2}}-\frac {a \sqrt {c+d x^2} (b c-a d)}{b^3}-\frac {a \left (c+d x^2\right )^{3/2}}{3 b^2}+\frac {\left (c+d x^2\right )^{5/2}}{5 b d} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 81
Rule 214
Rule 457
Rubi steps
\begin {align*} \int \frac {x^3 \left (c+d x^2\right )^{3/2}}{a+b x^2} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x (c+d x)^{3/2}}{a+b x} \, dx,x,x^2\right )\\ &=\frac {\left (c+d x^2\right )^{5/2}}{5 b d}-\frac {a \text {Subst}\left (\int \frac {(c+d x)^{3/2}}{a+b x} \, dx,x,x^2\right )}{2 b}\\ &=-\frac {a \left (c+d x^2\right )^{3/2}}{3 b^2}+\frac {\left (c+d x^2\right )^{5/2}}{5 b d}-\frac {(a (b c-a d)) \text {Subst}\left (\int \frac {\sqrt {c+d x}}{a+b x} \, dx,x,x^2\right )}{2 b^2}\\ &=-\frac {a (b c-a d) \sqrt {c+d x^2}}{b^3}-\frac {a \left (c+d x^2\right )^{3/2}}{3 b^2}+\frac {\left (c+d x^2\right )^{5/2}}{5 b d}-\frac {\left (a (b c-a d)^2\right ) \text {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,x^2\right )}{2 b^3}\\ &=-\frac {a (b c-a d) \sqrt {c+d x^2}}{b^3}-\frac {a \left (c+d x^2\right )^{3/2}}{3 b^2}+\frac {\left (c+d x^2\right )^{5/2}}{5 b d}-\frac {\left (a (b c-a d)^2\right ) \text {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x^2}\right )}{b^3 d}\\ &=-\frac {a (b c-a d) \sqrt {c+d x^2}}{b^3}-\frac {a \left (c+d x^2\right )^{3/2}}{3 b^2}+\frac {\left (c+d x^2\right )^{5/2}}{5 b d}+\frac {a (b c-a d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{b^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.21, size = 109, normalized size = 0.95 \begin {gather*} \frac {\sqrt {c+d x^2} \left (15 a^2 d^2+3 b^2 \left (c+d x^2\right )^2-5 a b d \left (4 c+d x^2\right )\right )}{15 b^3 d}-\frac {a (-b c+a d)^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {-b c+a d}}\right )}{b^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1255\) vs.
\(2(95)=190\).
time = 0.10, size = 1256, normalized size = 10.92 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.39, size = 397, normalized size = 3.45 \begin {gather*} \left [-\frac {15 \, {\left (a b c d - a^{2} d^{2}\right )} \sqrt {\frac {b c - a d}{b}} \log \left (\frac {b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \, {\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2} - 4 \, {\left (b^{2} d x^{2} + 2 \, b^{2} c - a b d\right )} \sqrt {d x^{2} + c} \sqrt {\frac {b c - a d}{b}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - 4 \, {\left (3 \, b^{2} d^{2} x^{4} + 3 \, b^{2} c^{2} - 20 \, a b c d + 15 \, a^{2} d^{2} + {\left (6 \, b^{2} c d - 5 \, a b d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{60 \, b^{3} d}, \frac {15 \, {\left (a b c d - a^{2} d^{2}\right )} \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {b c - a d}{b}}}{2 \, {\left (b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x^{2}\right )}}\right ) + 2 \, {\left (3 \, b^{2} d^{2} x^{4} + 3 \, b^{2} c^{2} - 20 \, a b c d + 15 \, a^{2} d^{2} + {\left (6 \, b^{2} c d - 5 \, a b d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{30 \, b^{3} d}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 19.70, size = 104, normalized size = 0.90 \begin {gather*} - \frac {a \left (c + d x^{2}\right )^{\frac {3}{2}}}{3 b^{2}} - \frac {a \left (a d - b c\right )^{2} \operatorname {atan}{\left (\frac {\sqrt {c + d x^{2}}}{\sqrt {\frac {a d - b c}{b}}} \right )}}{b^{4} \sqrt {\frac {a d - b c}{b}}} + \frac {\left (c + d x^{2}\right )^{\frac {5}{2}}}{5 b d} + \frac {\sqrt {c + d x^{2}} \left (a^{2} d - a b c\right )}{b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.81, size = 151, normalized size = 1.31 \begin {gather*} -\frac {{\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} \arctan \left (\frac {\sqrt {d x^{2} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{\sqrt {-b^{2} c + a b d} b^{3}} + \frac {3 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{4} d^{4} - 5 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b^{3} d^{5} - 15 \, \sqrt {d x^{2} + c} a b^{3} c d^{5} + 15 \, \sqrt {d x^{2} + c} a^{2} b^{2} d^{6}}{15 \, b^{5} d^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.37, size = 179, normalized size = 1.56 \begin {gather*} \frac {{\left (d\,x^2+c\right )}^{5/2}}{5\,b\,d}-{\left (d\,x^2+c\right )}^{3/2}\,\left (\frac {c}{3\,b\,d}+\frac {a\,d^2-b\,c\,d}{3\,b^2\,d^2}\right )-\frac {a\,\mathrm {atan}\left (\frac {a\,\sqrt {b}\,\sqrt {d\,x^2+c}\,{\left (a\,d-b\,c\right )}^{3/2}}{a^3\,d^2-2\,a^2\,b\,c\,d+a\,b^2\,c^2}\right )\,{\left (a\,d-b\,c\right )}^{3/2}}{b^{7/2}}+\frac {\sqrt {d\,x^2+c}\,\left (a\,d^2-b\,c\,d\right )\,\left (\frac {c}{b\,d}+\frac {a\,d^2-b\,c\,d}{b^2\,d^2}\right )}{b\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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