Optimal. Leaf size=163 \[ -\frac {5 d x}{6 (b c-a d)^2 \left (c+d x^2\right )^{3/2}}-\frac {x}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}-\frac {d (13 b c+2 a d) x}{6 c (b c-a d)^3 \sqrt {c+d x^2}}+\frac {b (b c+4 a d) \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 \sqrt {a} (b c-a d)^{7/2}} \]
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Rubi [A]
time = 0.11, antiderivative size = 163, 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 = {482, 541, 12,
385, 211} \begin {gather*} \frac {b (4 a d+b c) \text {ArcTan}\left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 \sqrt {a} (b c-a d)^{7/2}}-\frac {d x (2 a d+13 b c)}{6 c \sqrt {c+d x^2} (b c-a d)^3}-\frac {x}{2 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} (b c-a d)}-\frac {5 d x}{6 \left (c+d x^2\right )^{3/2} (b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 211
Rule 385
Rule 482
Rule 541
Rubi steps
\begin {align*} \int \frac {x^2}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx &=-\frac {x}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}+\frac {\int \frac {c-4 d x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}} \, dx}{2 (b c-a d)}\\ &=-\frac {5 d x}{6 (b c-a d)^2 \left (c+d x^2\right )^{3/2}}-\frac {x}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}+\frac {\int \frac {c (3 b c+2 a d)-10 b c d x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx}{6 c (b c-a d)^2}\\ &=-\frac {5 d x}{6 (b c-a d)^2 \left (c+d x^2\right )^{3/2}}-\frac {x}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}-\frac {d (13 b c+2 a d) x}{6 c (b c-a d)^3 \sqrt {c+d x^2}}+\frac {\int \frac {3 b c^2 (b c+4 a d)}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{6 c^2 (b c-a d)^3}\\ &=-\frac {5 d x}{6 (b c-a d)^2 \left (c+d x^2\right )^{3/2}}-\frac {x}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}-\frac {d (13 b c+2 a d) x}{6 c (b c-a d)^3 \sqrt {c+d x^2}}+\frac {(b (b c+4 a d)) \int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{2 (b c-a d)^3}\\ &=-\frac {5 d x}{6 (b c-a d)^2 \left (c+d x^2\right )^{3/2}}-\frac {x}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}-\frac {d (13 b c+2 a d) x}{6 c (b c-a d)^3 \sqrt {c+d x^2}}+\frac {(b (b c+4 a d)) \text {Subst}\left (\int \frac {1}{a-(-b c+a d) x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{2 (b c-a d)^3}\\ &=-\frac {5 d x}{6 (b c-a d)^2 \left (c+d x^2\right )^{3/2}}-\frac {x}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}-\frac {d (13 b c+2 a d) x}{6 c (b c-a d)^3 \sqrt {c+d x^2}}+\frac {b (b c+4 a d) \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 \sqrt {a} (b c-a d)^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 1.02, size = 183, normalized size = 1.12 \begin {gather*} -\frac {x \left (2 a^2 d^3 x^2+2 a b d \left (6 c^2+5 c d x^2+d^2 x^4\right )+b^2 c \left (3 c^2+18 c d x^2+13 d^2 x^4\right )\right )}{6 c (b c-a d)^3 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}-\frac {b (b c+4 a d) \tan ^{-1}\left (\frac {a \sqrt {d}+b x \left (\sqrt {d} x-\sqrt {c+d x^2}\right )}{\sqrt {a} \sqrt {b c-a d}}\right )}{2 \sqrt {a} (b c-a d)^{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. \(3482\) vs.
\(2(139)=278\).
time = 0.10, size = 3483, normalized size = 21.37
method | result | size |
default | \(\text {Expression too large to display}\) | \(3483\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 626 vs.
\(2 (139) = 278\).
time = 2.75, size = 1292, normalized size = 7.93 \begin {gather*} \left [\frac {3 \, {\left (a b^{2} c^{4} + 4 \, a^{2} b c^{3} d + {\left (b^{3} c^{2} d^{2} + 4 \, a b^{2} c d^{3}\right )} x^{6} + {\left (2 \, b^{3} c^{3} d + 9 \, a b^{2} c^{2} d^{2} + 4 \, a^{2} b c d^{3}\right )} x^{4} + {\left (b^{3} c^{4} + 6 \, a b^{2} c^{3} d + 8 \, a^{2} b c^{2} d^{2}\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 (13 \, a b^{3} c^{2} d^{2} - 11 \, a^{2} b^{2} c d^{3} - 2 \, a^{3} b d^{4}\right )} x^{5} + 2 \, {\left (9 \, a b^{3} c^{3} d - 4 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} x^{3} + 3 \, {\left (a b^{3} c^{4} + 3 \, a^{2} b^{2} c^{3} d - 4 \, a^{3} b c^{2} d^{2}\right )} x\right )} \sqrt {d x^{2} + c}}{24 \, {\left (a^{2} b^{4} c^{7} - 4 \, a^{3} b^{3} c^{6} d + 6 \, a^{4} b^{2} c^{5} d^{2} - 4 \, a^{5} b c^{4} d^{3} + a^{6} c^{3} d^{4} + {\left (a b^{5} c^{5} d^{2} - 4 \, a^{2} b^{4} c^{4} d^{3} + 6 \, a^{3} b^{3} c^{3} d^{4} - 4 \, a^{4} b^{2} c^{2} d^{5} + a^{5} b c d^{6}\right )} x^{6} + {\left (2 \, a b^{5} c^{6} d - 7 \, a^{2} b^{4} c^{5} d^{2} + 8 \, a^{3} b^{3} c^{4} d^{3} - 2 \, a^{4} b^{2} c^{3} d^{4} - 2 \, a^{5} b c^{2} d^{5} + a^{6} c d^{6}\right )} x^{4} + {\left (a b^{5} c^{7} - 2 \, a^{2} b^{4} c^{6} d - 2 \, a^{3} b^{3} c^{5} d^{2} + 8 \, a^{4} b^{2} c^{4} d^{3} - 7 \, a^{5} b c^{3} d^{4} + 2 \, a^{6} c^{2} d^{5}\right )} x^{2}\right )}}, \frac {3 \, {\left (a b^{2} c^{4} + 4 \, a^{2} b c^{3} d + {\left (b^{3} c^{2} d^{2} + 4 \, a b^{2} c d^{3}\right )} x^{6} + {\left (2 \, b^{3} c^{3} d + 9 \, a b^{2} c^{2} d^{2} + 4 \, a^{2} b c d^{3}\right )} x^{4} + {\left (b^{3} c^{4} + 6 \, a b^{2} c^{3} d + 8 \, a^{2} b c^{2} d^{2}\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 (13 \, a b^{3} c^{2} d^{2} - 11 \, a^{2} b^{2} c d^{3} - 2 \, a^{3} b d^{4}\right )} x^{5} + 2 \, {\left (9 \, a b^{3} c^{3} d - 4 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} x^{3} + 3 \, {\left (a b^{3} c^{4} + 3 \, a^{2} b^{2} c^{3} d - 4 \, a^{3} b c^{2} d^{2}\right )} x\right )} \sqrt {d x^{2} + c}}{12 \, {\left (a^{2} b^{4} c^{7} - 4 \, a^{3} b^{3} c^{6} d + 6 \, a^{4} b^{2} c^{5} d^{2} - 4 \, a^{5} b c^{4} d^{3} + a^{6} c^{3} d^{4} + {\left (a b^{5} c^{5} d^{2} - 4 \, a^{2} b^{4} c^{4} d^{3} + 6 \, a^{3} b^{3} c^{3} d^{4} - 4 \, a^{4} b^{2} c^{2} d^{5} + a^{5} b c d^{6}\right )} x^{6} + {\left (2 \, a b^{5} c^{6} d - 7 \, a^{2} b^{4} c^{5} d^{2} + 8 \, a^{3} b^{3} c^{4} d^{3} - 2 \, a^{4} b^{2} c^{3} d^{4} - 2 \, a^{5} b c^{2} d^{5} + a^{6} c d^{6}\right )} x^{4} + {\left (a b^{5} c^{7} - 2 \, a^{2} b^{4} c^{6} d - 2 \, a^{3} b^{3} c^{5} d^{2} + 8 \, a^{4} b^{2} c^{4} d^{3} - 7 \, a^{5} b c^{3} d^{4} + 2 \, a^{6} c^{2} d^{5}\right )} x^{2}\right )}}\right ] \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 )^{2} \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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 595 vs.
\(2 (139) = 278\).
time = 1.45, size = 595, normalized size = 3.65 \begin {gather*} -\frac {{\left (\frac {{\left (5 \, b^{4} c^{4} d^{3} - 14 \, a b^{3} c^{3} d^{4} + 12 \, a^{2} b^{2} c^{2} d^{5} - 2 \, a^{3} b c d^{6} - a^{4} d^{7}\right )} x^{2}}{b^{6} c^{7} d - 6 \, a b^{5} c^{6} d^{2} + 15 \, a^{2} b^{4} c^{5} d^{3} - 20 \, a^{3} b^{3} c^{4} d^{4} + 15 \, a^{4} b^{2} c^{3} d^{5} - 6 \, a^{5} b c^{2} d^{6} + a^{6} c d^{7}} + \frac {6 \, {\left (b^{4} c^{5} d^{2} - 3 \, a b^{3} c^{4} d^{3} + 3 \, a^{2} b^{2} c^{3} d^{4} - a^{3} b c^{2} d^{5}\right )}}{b^{6} c^{7} d - 6 \, a b^{5} c^{6} d^{2} + 15 \, a^{2} b^{4} c^{5} d^{3} - 20 \, a^{3} b^{3} c^{4} d^{4} + 15 \, a^{4} b^{2} c^{3} d^{5} - 6 \, a^{5} b c^{2} d^{6} + a^{6} c d^{7}}\right )} x}{3 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}}} - \frac {{\left (b^{2} c \sqrt {d} + 4 \, a b 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 )}{2 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {a b c d - a^{2} d^{2}}} + \frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b^{2} c \sqrt {d} - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a b d^{\frac {3}{2}} - b^{2} c^{2} \sqrt {d}}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} {\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 )}} \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 )}^2\,{\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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