Optimal. Leaf size=149 \[ \frac {c (3 b c-4 a d) \tan ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{8 a^{5/2} (b c-a d)^{3/2}}+\frac {x \sqrt {c+d x^2} (3 b c-4 a d)}{8 a^2 \left (a+b x^2\right ) (b c-a d)}+\frac {b x \left (c+d x^2\right )^{3/2}}{4 a \left (a+b x^2\right )^2 (b c-a d)} \]
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Rubi [A] time = 0.08, antiderivative size = 149, 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 = {382, 378, 377, 205} \[ \frac {x \sqrt {c+d x^2} (3 b c-4 a d)}{8 a^2 \left (a+b x^2\right ) (b c-a d)}+\frac {c (3 b c-4 a d) \tan ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{8 a^{5/2} (b c-a d)^{3/2}}+\frac {b x \left (c+d x^2\right )^{3/2}}{4 a \left (a+b x^2\right )^2 (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 205
Rule 377
Rule 378
Rule 382
Rubi steps
\begin {align*} \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^3} \, dx &=\frac {b x \left (c+d x^2\right )^{3/2}}{4 a (b c-a d) \left (a+b x^2\right )^2}+\frac {(3 b c-4 a d) \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^2} \, dx}{4 a (b c-a d)}\\ &=\frac {(3 b c-4 a d) x \sqrt {c+d x^2}}{8 a^2 (b c-a d) \left (a+b x^2\right )}+\frac {b x \left (c+d x^2\right )^{3/2}}{4 a (b c-a d) \left (a+b x^2\right )^2}+\frac {(c (3 b c-4 a d)) \int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{8 a^2 (b c-a d)}\\ &=\frac {(3 b c-4 a d) x \sqrt {c+d x^2}}{8 a^2 (b c-a d) \left (a+b x^2\right )}+\frac {b x \left (c+d x^2\right )^{3/2}}{4 a (b c-a d) \left (a+b x^2\right )^2}+\frac {(c (3 b c-4 a d)) \operatorname {Subst}\left (\int \frac {1}{a-(-b c+a d) x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{8 a^2 (b c-a d)}\\ &=\frac {(3 b c-4 a d) x \sqrt {c+d x^2}}{8 a^2 (b c-a d) \left (a+b x^2\right )}+\frac {b x \left (c+d x^2\right )^{3/2}}{4 a (b c-a d) \left (a+b x^2\right )^2}+\frac {c (3 b c-4 a d) \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{8 a^{5/2} (b c-a d)^{3/2}}\\ \end {align*}
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Mathematica [A] time = 5.17, size = 130, normalized size = 0.87 \[ \frac {\frac {\sqrt {a} x \sqrt {c+d x^2} \left (-4 a^2 d+a b \left (5 c-2 d x^2\right )+3 b^2 c x^2\right )}{\left (a+b x^2\right )^2 (b c-a d)}+\frac {c (3 b c-4 a d) \tan ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{(b c-a d)^{3/2}}}{8 a^{5/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.01, size = 698, normalized size = 4.68 \[ \left [-\frac {{\left (3 \, a^{2} b c^{2} - 4 \, a^{3} c d + {\left (3 \, b^{3} c^{2} - 4 \, a b^{2} c d\right )} x^{4} + 2 \, {\left (3 \, a b^{2} c^{2} - 4 \, a^{2} b c d\right )} x^{2}\right )} \sqrt {-a b c + a^{2} d} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} - 4 \, {\left ({\left (b c - 2 \, a d\right )} x^{3} - a c x\right )} \sqrt {-a b c + a^{2} d} \sqrt {d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - 4 \, {\left ({\left (3 \, a b^{3} c^{2} - 5 \, a^{2} b^{2} c d + 2 \, a^{3} b d^{2}\right )} x^{3} + {\left (5 \, a^{2} b^{2} c^{2} - 9 \, a^{3} b c d + 4 \, a^{4} d^{2}\right )} x\right )} \sqrt {d x^{2} + c}}{32 \, {\left (a^{5} b^{2} c^{2} - 2 \, a^{6} b c d + a^{7} d^{2} + {\left (a^{3} b^{4} c^{2} - 2 \, a^{4} b^{3} c d + a^{5} b^{2} d^{2}\right )} x^{4} + 2 \, {\left (a^{4} b^{3} c^{2} - 2 \, a^{5} b^{2} c d + a^{6} b d^{2}\right )} x^{2}\right )}}, \frac {{\left (3 \, a^{2} b c^{2} - 4 \, a^{3} c d + {\left (3 \, b^{3} c^{2} - 4 \, a b^{2} c d\right )} x^{4} + 2 \, {\left (3 \, a b^{2} c^{2} - 4 \, a^{2} b c d\right )} x^{2}\right )} \sqrt {a b c - a^{2} d} \arctan \left (\frac {\sqrt {a b c - a^{2} d} {\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c}}{2 \, {\left ({\left (a b c d - a^{2} d^{2}\right )} x^{3} + {\left (a b c^{2} - a^{2} c d\right )} x\right )}}\right ) + 2 \, {\left ({\left (3 \, a b^{3} c^{2} - 5 \, a^{2} b^{2} c d + 2 \, a^{3} b d^{2}\right )} x^{3} + {\left (5 \, a^{2} b^{2} c^{2} - 9 \, a^{3} b c d + 4 \, a^{4} d^{2}\right )} x\right )} \sqrt {d x^{2} + c}}{16 \, {\left (a^{5} b^{2} c^{2} - 2 \, a^{6} b c d + a^{7} d^{2} + {\left (a^{3} b^{4} c^{2} - 2 \, a^{4} b^{3} c d + a^{5} b^{2} d^{2}\right )} x^{4} + 2 \, {\left (a^{4} b^{3} c^{2} - 2 \, a^{5} b^{2} c d + a^{6} b d^{2}\right )} x^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 3.13, size = 487, normalized size = 3.27 \[ -\frac {{\left (3 \, b c^{2} \sqrt {d} - 4 \, a c d^{\frac {3}{2}}\right )} \arctan \left (\frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{8 \, {\left (a^{2} b c - a^{3} d\right )} \sqrt {a b c d - a^{2} d^{2}}} - \frac {3 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} b^{3} c^{2} \sqrt {d} - 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} a b^{2} c d^{\frac {3}{2}} - 9 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b^{3} c^{3} \sqrt {d} + 30 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a b^{2} c^{2} d^{\frac {3}{2}} - 40 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a^{2} b c d^{\frac {5}{2}} + 16 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a^{3} d^{\frac {7}{2}} + 9 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b^{3} c^{4} \sqrt {d} - 28 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a b^{2} c^{3} d^{\frac {3}{2}} + 16 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a^{2} b c^{2} d^{\frac {5}{2}} - 3 \, b^{3} c^{5} \sqrt {d} + 2 \, a b^{2} c^{4} d^{\frac {3}{2}}}{4 \, {\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c + 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a d + b c^{2}\right )}^{2} {\left (a^{2} b^{2} c - a^{3} b d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 5177, normalized size = 34.74 \[ \text {output too large to display} \]
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 \[ \int \frac {\sqrt {d x^{2} + c}}{{\left (b x^{2} + a\right )}^{3}}\,{d x} \]
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 \[ \int \frac {\sqrt {d\,x^2+c}}{{\left (b\,x^2+a\right )}^3} \,d x \]
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 \[ \int \frac {\sqrt {c + d x^{2}}}{\left (a + b x^{2}\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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