Optimal. Leaf size=115 \[ \frac {x (a d+2 b c)}{3 c \sqrt {c+d x^2} (b c-a d)^2}+\frac {x}{3 \left (c+d x^2\right )^{3/2} (b c-a d)}-\frac {\sqrt {a} b \tan ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{(b c-a d)^{5/2}} \]
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Rubi [A] time = 0.09, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {471, 527, 12, 377, 205} \begin {gather*} \frac {x (a d+2 b c)}{3 c \sqrt {c+d x^2} (b c-a d)^2}+\frac {x}{3 \left (c+d x^2\right )^{3/2} (b c-a d)}-\frac {\sqrt {a} b \tan ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{(b c-a d)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 205
Rule 377
Rule 471
Rule 527
Rubi steps
\begin {align*} \int \frac {x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}} \, dx &=\frac {x}{3 (b c-a d) \left (c+d x^2\right )^{3/2}}-\frac {\int \frac {a-2 b x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx}{3 (b c-a d)}\\ &=\frac {x}{3 (b c-a d) \left (c+d x^2\right )^{3/2}}+\frac {(2 b c+a d) x}{3 c (b c-a d)^2 \sqrt {c+d x^2}}-\frac {\int \frac {3 a b c}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{3 c (b c-a d)^2}\\ &=\frac {x}{3 (b c-a d) \left (c+d x^2\right )^{3/2}}+\frac {(2 b c+a d) x}{3 c (b c-a d)^2 \sqrt {c+d x^2}}-\frac {(a b) \int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{(b c-a d)^2}\\ &=\frac {x}{3 (b c-a d) \left (c+d x^2\right )^{3/2}}+\frac {(2 b c+a d) x}{3 c (b c-a d)^2 \sqrt {c+d x^2}}-\frac {(a b) \operatorname {Subst}\left (\int \frac {1}{a-(-b c+a d) x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{(b c-a d)^2}\\ &=\frac {x}{3 (b c-a d) \left (c+d x^2\right )^{3/2}}+\frac {(2 b c+a d) x}{3 c (b c-a d)^2 \sqrt {c+d x^2}}-\frac {\sqrt {a} b \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{(b c-a d)^{5/2}}\\ \end {align*}
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Mathematica [C] time = 3.08, size = 257, normalized size = 2.23 \begin {gather*} \frac {12 x^6 \left (c+d x^2\right ) (b c-a d)^3 \, _2F_1\left (2,2;\frac {9}{2};\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right )-\frac {35 c \left (a+b x^2\right ) \left (5 c+2 d x^2\right ) \left (-3 a^2 \left (c+d x^2\right )^2 \sin ^{-1}\left (\sqrt {\frac {x^2 (b c-a d)}{c \left (a+b x^2\right )}}\right )-c \left (a+b x^2\right ) \sqrt {\frac {a x^2 \left (c+d x^2\right ) (b c-a d)}{c^2 \left (a+b x^2\right )^2}} \left (-3 a c-4 a d x^2+b c x^2\right )\right )}{\sqrt {\frac {a x^2 \left (c+d x^2\right ) (b c-a d)}{c^2 \left (a+b x^2\right )^2}}}}{315 c^3 x \left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2} (b c-a d)^2} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.41, size = 156, normalized size = 1.36 \begin {gather*} \frac {a d^2 x^3+3 b c^2 x+2 b c d x^3}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)^2}+\frac {\sqrt {a} b \tan ^{-1}\left (\frac {b \sqrt {d} x^2}{\sqrt {a} \sqrt {b c-a d}}-\frac {b x \sqrt {c+d x^2}}{\sqrt {a} \sqrt {b c-a d}}+\frac {\sqrt {a} \sqrt {d}}{\sqrt {b c-a d}}\right )}{(b c-a d)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.79, size = 550, normalized size = 4.78 \begin {gather*} \left [\frac {3 \, {\left (b c d^{2} x^{4} + 2 \, b c^{2} d x^{2} + b c^{3}\right )} \sqrt {-\frac {a}{b c - a 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^{2} c^{2} - 3 \, a b c d + 2 \, a^{2} d^{2}\right )} x^{3} - {\left (a b c^{2} - a^{2} c d\right )} x\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {a}{b c - a d}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 4 \, {\left (3 \, b c^{2} x + {\left (2 \, b c d + a d^{2}\right )} x^{3}\right )} \sqrt {d x^{2} + c}}{12 \, {\left (b^{2} c^{5} - 2 \, a b c^{4} d + a^{2} c^{3} d^{2} + {\left (b^{2} c^{3} d^{2} - 2 \, a b c^{2} d^{3} + a^{2} c d^{4}\right )} x^{4} + 2 \, {\left (b^{2} c^{4} d - 2 \, a b c^{3} d^{2} + a^{2} c^{2} d^{3}\right )} x^{2}\right )}}, \frac {3 \, {\left (b c d^{2} x^{4} + 2 \, b c^{2} d x^{2} + b c^{3}\right )} \sqrt {\frac {a}{b c - a d}} \arctan \left (-\frac {{\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c} \sqrt {\frac {a}{b c - a d}}}{2 \, {\left (a d x^{3} + a c x\right )}}\right ) + 2 \, {\left (3 \, b c^{2} x + {\left (2 \, b c d + a d^{2}\right )} x^{3}\right )} \sqrt {d x^{2} + c}}{6 \, {\left (b^{2} c^{5} - 2 \, a b c^{4} d + a^{2} c^{3} d^{2} + {\left (b^{2} c^{3} d^{2} - 2 \, a b c^{2} d^{3} + a^{2} c d^{4}\right )} x^{4} + 2 \, {\left (b^{2} c^{4} d - 2 \, a b c^{3} d^{2} + a^{2} c^{2} d^{3}\right )} x^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.45, size = 291, normalized size = 2.53 \begin {gather*} \frac {a b \sqrt {d} \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 )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {a b c d - a^{2} d^{2}}} + \frac {{\left (\frac {{\left (2 \, b^{3} c^{3} d^{2} - 3 \, a b^{2} c^{2} d^{3} + a^{3} d^{5}\right )} x^{2}}{b^{4} c^{5} d - 4 \, a b^{3} c^{4} d^{2} + 6 \, a^{2} b^{2} c^{3} d^{3} - 4 \, a^{3} b c^{2} d^{4} + a^{4} c d^{5}} + \frac {3 \, {\left (b^{3} c^{4} d - 2 \, a b^{2} c^{3} d^{2} + a^{2} b c^{2} d^{3}\right )}}{b^{4} c^{5} d - 4 \, a b^{3} c^{4} d^{2} + 6 \, a^{2} b^{2} c^{3} d^{3} - 4 \, a^{3} b c^{2} d^{4} + a^{4} c d^{5}}\right )} x}{3 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 1134, normalized size = 9.86 \begin {gather*} \frac {a b \ln \left (\frac {\frac {2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {2 \left (a d -b c \right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{2 \sqrt {-a b}\, \left (a d -b c \right )^{2} \sqrt {-\frac {a d -b c}{b}}}-\frac {a b \ln \left (\frac {-\frac {2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {2 \left (a d -b c \right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{2 \sqrt {-a b}\, \left (a d -b c \right )^{2} \sqrt {-\frac {a d -b c}{b}}}+\frac {a b}{2 \sqrt {-a b}\, \left (a d -b c \right )^{2} \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}}-\frac {a b}{2 \sqrt {-a b}\, \left (a d -b c \right )^{2} \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}}+\frac {a d x}{2 \left (a d -b c \right )^{2} \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}\, c}+\frac {a d x}{2 \left (a d -b c \right )^{2} \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}\, c}-\frac {a d x}{6 \left (a d -b c \right ) \left (\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}\right )^{\frac {3}{2}} b c}-\frac {a d x}{6 \left (a d -b c \right ) \left (\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}\right )^{\frac {3}{2}} b c}-\frac {a}{6 \sqrt {-a b}\, \left (a d -b c \right ) \left (\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}\right )^{\frac {3}{2}}}+\frac {a}{6 \sqrt {-a b}\, \left (a d -b c \right ) \left (\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}\right )^{\frac {3}{2}}}-\frac {a d x}{3 \left (a d -b c \right ) \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}\, b \,c^{2}}-\frac {a d x}{3 \left (a d -b c \right ) \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}\, b \,c^{2}}+\frac {x}{3 \left (d \,x^{2}+c \right )^{\frac {3}{2}} b c}+\frac {2 x}{3 \sqrt {d \,x^{2}+c}\, b \,c^{2}} \end {gather*}
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 {x^{2}}{{\left (b x^{2} + a\right )} {\left (d x^{2} + c\right )}^{\frac {5}{2}}}\,{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 (b\,x^2+a\right )\,{\left (d\,x^2+c\right )}^{5/2}} \,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 {x^{2}}{\left (a + b x^{2}\right ) \left (c + d x^{2}\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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