Optimal. Leaf size=195 \[ \frac {\sqrt {d} \left (24 a^2 d^2-40 a b c d+15 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{8 b^4}+\frac {(b c-6 a d) (b c-a d)^{3/2} \tan ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 \sqrt {a} b^4}+\frac {d x \sqrt {c+d x^2} (11 b c-12 a d)}{8 b^3}-\frac {x \left (c+d x^2\right )^{5/2}}{2 b \left (a+b x^2\right )}+\frac {3 d x \left (c+d x^2\right )^{3/2}}{4 b^2} \]
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Rubi [A] time = 0.24, antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {467, 528, 523, 217, 206, 377, 205} \begin {gather*} \frac {\sqrt {d} \left (24 a^2 d^2-40 a b c d+15 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{8 b^4}+\frac {d x \sqrt {c+d x^2} (11 b c-12 a d)}{8 b^3}+\frac {(b c-6 a d) (b c-a d)^{3/2} \tan ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 \sqrt {a} b^4}-\frac {x \left (c+d x^2\right )^{5/2}}{2 b \left (a+b x^2\right )}+\frac {3 d x \left (c+d x^2\right )^{3/2}}{4 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 206
Rule 217
Rule 377
Rule 467
Rule 523
Rule 528
Rubi steps
\begin {align*} \int \frac {x^2 \left (c+d x^2\right )^{5/2}}{\left (a+b x^2\right )^2} \, dx &=-\frac {x \left (c+d x^2\right )^{5/2}}{2 b \left (a+b x^2\right )}+\frac {\int \frac {\left (c+d x^2\right )^{3/2} \left (c+6 d x^2\right )}{a+b x^2} \, dx}{2 b}\\ &=\frac {3 d x \left (c+d x^2\right )^{3/2}}{4 b^2}-\frac {x \left (c+d x^2\right )^{5/2}}{2 b \left (a+b x^2\right )}+\frac {\int \frac {\sqrt {c+d x^2} \left (2 c (2 b c-3 a d)+2 d (11 b c-12 a d) x^2\right )}{a+b x^2} \, dx}{8 b^2}\\ &=\frac {d (11 b c-12 a d) x \sqrt {c+d x^2}}{8 b^3}+\frac {3 d x \left (c+d x^2\right )^{3/2}}{4 b^2}-\frac {x \left (c+d x^2\right )^{5/2}}{2 b \left (a+b x^2\right )}+\frac {\int \frac {2 c \left (4 b^2 c^2-17 a b c d+12 a^2 d^2\right )+2 d \left (15 b^2 c^2-40 a b c d+24 a^2 d^2\right ) x^2}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{16 b^3}\\ &=\frac {d (11 b c-12 a d) x \sqrt {c+d x^2}}{8 b^3}+\frac {3 d x \left (c+d x^2\right )^{3/2}}{4 b^2}-\frac {x \left (c+d x^2\right )^{5/2}}{2 b \left (a+b x^2\right )}+\frac {\left ((b c-6 a d) (b c-a d)^2\right ) \int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{2 b^4}+\frac {\left (d \left (15 b^2 c^2-40 a b c d+24 a^2 d^2\right )\right ) \int \frac {1}{\sqrt {c+d x^2}} \, dx}{8 b^4}\\ &=\frac {d (11 b c-12 a d) x \sqrt {c+d x^2}}{8 b^3}+\frac {3 d x \left (c+d x^2\right )^{3/2}}{4 b^2}-\frac {x \left (c+d x^2\right )^{5/2}}{2 b \left (a+b x^2\right )}+\frac {\left ((b c-6 a d) (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 )}{2 b^4}+\frac {\left (d \left (15 b^2 c^2-40 a b c d+24 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-d x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{8 b^4}\\ &=\frac {d (11 b c-12 a d) x \sqrt {c+d x^2}}{8 b^3}+\frac {3 d x \left (c+d x^2\right )^{3/2}}{4 b^2}-\frac {x \left (c+d x^2\right )^{5/2}}{2 b \left (a+b x^2\right )}+\frac {(b c-6 a d) (b c-a d)^{3/2} \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 \sqrt {a} b^4}+\frac {\sqrt {d} \left (15 b^2 c^2-40 a b c d+24 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{8 b^4}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 173, normalized size = 0.89 \begin {gather*} \frac {\sqrt {d} \left (24 a^2 d^2-40 a b c d+15 b^2 c^2\right ) \log \left (\sqrt {d} \sqrt {c+d x^2}+d x\right )+b x \sqrt {c+d x^2} \left (-\frac {4 (b c-a d)^2}{a+b x^2}+d (9 b c-8 a d)+2 b d^2 x^2\right )+\frac {4 (b c-6 a d) (b c-a d)^{3/2} \tan ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{\sqrt {a}}}{8 b^4} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.02, size = 245, normalized size = 1.26 \begin {gather*} \frac {\left (-24 a^2 d^{5/2}+40 a b c d^{3/2}-15 b^2 c^2 \sqrt {d}\right ) \log \left (\sqrt {c+d x^2}-\sqrt {d} x\right )}{8 b^4}-\frac {\sqrt {b c-a d} \left (6 a^2 d^2-7 a b c d+b^2 c^2\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 )}{2 \sqrt {a} b^4}+\frac {\sqrt {c+d x^2} \left (-12 a^2 d^2 x+17 a b c d x-6 a b d^2 x^3-4 b^2 c^2 x+9 b^2 c d x^3+2 b^2 d^2 x^5\right )}{8 b^3 \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 3.14, size = 1379, normalized size = 7.07
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.57, size = 446, normalized size = 2.29 \begin {gather*} \frac {1}{8} \, \sqrt {d x^{2} + c} {\left (\frac {2 \, d^{2} x^{2}}{b^{2}} + \frac {9 \, b^{7} c d^{3} - 8 \, a b^{6} d^{4}}{b^{9} d^{2}}\right )} x - \frac {{\left (15 \, b^{2} c^{2} \sqrt {d} - 40 \, a b c d^{\frac {3}{2}} + 24 \, a^{2} d^{\frac {5}{2}}\right )} \log \left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2}\right )}{16 \, b^{4}} - \frac {{\left (b^{3} c^{3} \sqrt {d} - 8 \, a b^{2} c^{2} d^{\frac {3}{2}} + 13 \, a^{2} b c d^{\frac {5}{2}} - 6 \, a^{3} d^{\frac {7}{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 )}{2 \, \sqrt {a b c d - a^{2} d^{2}} b^{4}} + \frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b^{3} c^{3} \sqrt {d} - 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a b^{2} c^{2} d^{\frac {3}{2}} + 5 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a^{2} b c d^{\frac {5}{2}} - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a^{3} d^{\frac {7}{2}} - b^{3} c^{4} \sqrt {d} + 2 \, a b^{2} c^{3} d^{\frac {3}{2}} - a^{2} b c^{2} d^{\frac {5}{2}}}{{\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 )} b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 7459, normalized size = 38.25 \begin {gather*} \text {output too large to display} \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 {{\left (d x^{2} + c\right )}^{\frac {5}{2}} x^{2}}{{\left (b x^{2} + a\right )}^{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 (d\,x^2+c\right )}^{5/2}}{{\left (b\,x^2+a\right )}^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 (c + d x^{2}\right )^{\frac {5}{2}}}{\left (a + b x^{2}\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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