Optimal. Leaf size=362 \[ \frac {d (3 b c+2 a d)}{6 a c (b c-a d)^2 x^3 \left (c+d x^2\right )^{3/2}}+\frac {b}{2 a (b c-a d) x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}+\frac {d \left (b^2 c^2+8 a b c d-4 a^2 d^2\right )}{2 a c^2 (b c-a d)^3 x^3 \sqrt {c+d x^2}}-\frac {\left (5 b^3 c^3-6 a b^2 c^2 d+32 a^2 b c d^2-16 a^3 d^3\right ) \sqrt {c+d x^2}}{6 a^2 c^3 (b c-a d)^3 x^3}+\frac {\left (15 b^4 c^4-20 a b^3 c^3 d-12 a^2 b^2 c^2 d^2+64 a^3 b c d^3-32 a^4 d^4\right ) \sqrt {c+d x^2}}{6 a^3 c^4 (b c-a d)^3 x}+\frac {5 b^4 (b c-2 a d) \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{7/2} (b c-a d)^{7/2}} \]
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Rubi [A]
time = 0.41, antiderivative size = 362, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {483, 593, 597,
12, 385, 211} \begin {gather*} \frac {5 b^4 (b c-2 a d) \text {ArcTan}\left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{7/2} (b c-a d)^{7/2}}+\frac {d \left (-4 a^2 d^2+8 a b c d+b^2 c^2\right )}{2 a c^2 x^3 \sqrt {c+d x^2} (b c-a d)^3}-\frac {\sqrt {c+d x^2} \left (-16 a^3 d^3+32 a^2 b c d^2-6 a b^2 c^2 d+5 b^3 c^3\right )}{6 a^2 c^3 x^3 (b c-a d)^3}+\frac {\sqrt {c+d x^2} \left (-32 a^4 d^4+64 a^3 b c d^3-12 a^2 b^2 c^2 d^2-20 a b^3 c^3 d+15 b^4 c^4\right )}{6 a^3 c^4 x (b c-a d)^3}+\frac {b}{2 a x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} (b c-a d)}+\frac {d (2 a d+3 b c)}{6 a c x^3 \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 483
Rule 593
Rule 597
Rubi steps
\begin {align*} \int \frac {1}{x^4 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx &=\frac {b}{2 a (b c-a d) x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}-\frac {\int \frac {-5 b c+2 a d-8 b d x^2}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}} \, dx}{2 a (b c-a d)}\\ &=\frac {d (3 b c+2 a d)}{6 a c (b c-a d)^2 x^3 \left (c+d x^2\right )^{3/2}}+\frac {b}{2 a (b c-a d) x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}-\frac {\int \frac {-3 \left (5 b^2 c^2-4 a b c d+4 a^2 d^2\right )-6 b d (3 b c+2 a d) x^2}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx}{6 a c (b c-a d)^2}\\ &=\frac {d (3 b c+2 a d)}{6 a c (b c-a d)^2 x^3 \left (c+d x^2\right )^{3/2}}+\frac {b}{2 a (b c-a d) x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}+\frac {d \left (b^2 c^2+8 a b c d-4 a^2 d^2\right )}{2 a c^2 (b c-a d)^3 x^3 \sqrt {c+d x^2}}-\frac {\int \frac {-3 \left (5 b^3 c^3-6 a b^2 c^2 d+32 a^2 b c d^2-16 a^3 d^3\right )-12 b d \left (b^2 c^2+8 a b c d-4 a^2 d^2\right ) x^2}{x^4 \left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{6 a c^2 (b c-a d)^3}\\ &=\frac {d (3 b c+2 a d)}{6 a c (b c-a d)^2 x^3 \left (c+d x^2\right )^{3/2}}+\frac {b}{2 a (b c-a d) x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}+\frac {d \left (b^2 c^2+8 a b c d-4 a^2 d^2\right )}{2 a c^2 (b c-a d)^3 x^3 \sqrt {c+d x^2}}-\frac {\left (5 b^3 c^3-6 a b^2 c^2 d+32 a^2 b c d^2-16 a^3 d^3\right ) \sqrt {c+d x^2}}{6 a^2 c^3 (b c-a d)^3 x^3}+\frac {\int \frac {-3 \left (15 b^4 c^4-20 a b^3 c^3 d-12 a^2 b^2 c^2 d^2+64 a^3 b c d^3-32 a^4 d^4\right )-6 b d \left (5 b^3 c^3-6 a b^2 c^2 d+32 a^2 b c d^2-16 a^3 d^3\right ) x^2}{x^2 \left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{18 a^2 c^3 (b c-a d)^3}\\ &=\frac {d (3 b c+2 a d)}{6 a c (b c-a d)^2 x^3 \left (c+d x^2\right )^{3/2}}+\frac {b}{2 a (b c-a d) x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}+\frac {d \left (b^2 c^2+8 a b c d-4 a^2 d^2\right )}{2 a c^2 (b c-a d)^3 x^3 \sqrt {c+d x^2}}-\frac {\left (5 b^3 c^3-6 a b^2 c^2 d+32 a^2 b c d^2-16 a^3 d^3\right ) \sqrt {c+d x^2}}{6 a^2 c^3 (b c-a d)^3 x^3}+\frac {\left (15 b^4 c^4-20 a b^3 c^3 d-12 a^2 b^2 c^2 d^2+64 a^3 b c d^3-32 a^4 d^4\right ) \sqrt {c+d x^2}}{6 a^3 c^4 (b c-a d)^3 x}-\frac {\int -\frac {45 b^4 c^4 (b c-2 a d)}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{18 a^3 c^4 (b c-a d)^3}\\ &=\frac {d (3 b c+2 a d)}{6 a c (b c-a d)^2 x^3 \left (c+d x^2\right )^{3/2}}+\frac {b}{2 a (b c-a d) x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}+\frac {d \left (b^2 c^2+8 a b c d-4 a^2 d^2\right )}{2 a c^2 (b c-a d)^3 x^3 \sqrt {c+d x^2}}-\frac {\left (5 b^3 c^3-6 a b^2 c^2 d+32 a^2 b c d^2-16 a^3 d^3\right ) \sqrt {c+d x^2}}{6 a^2 c^3 (b c-a d)^3 x^3}+\frac {\left (15 b^4 c^4-20 a b^3 c^3 d-12 a^2 b^2 c^2 d^2+64 a^3 b c d^3-32 a^4 d^4\right ) \sqrt {c+d x^2}}{6 a^3 c^4 (b c-a d)^3 x}+\frac {\left (5 b^4 (b c-2 a d)\right ) \int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{2 a^3 (b c-a d)^3}\\ &=\frac {d (3 b c+2 a d)}{6 a c (b c-a d)^2 x^3 \left (c+d x^2\right )^{3/2}}+\frac {b}{2 a (b c-a d) x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}+\frac {d \left (b^2 c^2+8 a b c d-4 a^2 d^2\right )}{2 a c^2 (b c-a d)^3 x^3 \sqrt {c+d x^2}}-\frac {\left (5 b^3 c^3-6 a b^2 c^2 d+32 a^2 b c d^2-16 a^3 d^3\right ) \sqrt {c+d x^2}}{6 a^2 c^3 (b c-a d)^3 x^3}+\frac {\left (15 b^4 c^4-20 a b^3 c^3 d-12 a^2 b^2 c^2 d^2+64 a^3 b c d^3-32 a^4 d^4\right ) \sqrt {c+d x^2}}{6 a^3 c^4 (b c-a d)^3 x}+\frac {\left (5 b^4 (b c-2 a d)\right ) \text {Subst}\left (\int \frac {1}{a-(-b c+a d) x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{2 a^3 (b c-a d)^3}\\ &=\frac {d (3 b c+2 a d)}{6 a c (b c-a d)^2 x^3 \left (c+d x^2\right )^{3/2}}+\frac {b}{2 a (b c-a d) x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}+\frac {d \left (b^2 c^2+8 a b c d-4 a^2 d^2\right )}{2 a c^2 (b c-a d)^3 x^3 \sqrt {c+d x^2}}-\frac {\left (5 b^3 c^3-6 a b^2 c^2 d+32 a^2 b c d^2-16 a^3 d^3\right ) \sqrt {c+d x^2}}{6 a^2 c^3 (b c-a d)^3 x^3}+\frac {\left (15 b^4 c^4-20 a b^3 c^3 d-12 a^2 b^2 c^2 d^2+64 a^3 b c d^3-32 a^4 d^4\right ) \sqrt {c+d x^2}}{6 a^3 c^4 (b c-a d)^3 x}+\frac {5 b^4 (b c-2 a d) \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{7/2} (b c-a d)^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 1.74, size = 348, normalized size = 0.96 \begin {gather*} \frac {15 b^5 c^4 x^4 \left (c+d x^2\right )^2+10 a b^4 c^3 x^2 \left (c-2 d x^2\right ) \left (c+d x^2\right )^2-2 a^2 b^3 c^2 \left (c+d x^2\right )^3 \left (c+6 d x^2\right )+2 a^5 d^3 \left (c^3-6 c^2 d x^2-24 c d^2 x^4-16 d^3 x^6\right )+2 a^4 b d^2 \left (-3 c^4+13 c^3 d x^2+42 c^2 d^2 x^4+8 c d^3 x^6-16 d^4 x^8\right )+2 a^3 b^2 c d \left (3 c^4-3 c^3 d x^2+3 c^2 d^2 x^4+42 c d^3 x^6+32 d^4 x^8\right )}{6 a^3 c^4 (b c-a d)^3 x^3 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}-\frac {5 b^4 (b c-2 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 a^{7/2} (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. \(3634\) vs.
\(2(330)=660\).
time = 0.35, size = 3635, normalized size = 10.04
method | result | size |
risch | \(\text {Expression too large to display}\) | \(2375\) |
default | \(\text {Expression too large to display}\) | \(3635\) |
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. 925 vs.
\(2 (330) = 660\).
time = 4.01, size = 1890, normalized size = 5.22 \begin {gather*} \left [-\frac {15 \, {\left ({\left (b^{6} c^{5} d^{2} - 2 \, a b^{5} c^{4} d^{3}\right )} x^{9} + {\left (2 \, b^{6} c^{6} d - 3 \, a b^{5} c^{5} d^{2} - 2 \, a^{2} b^{4} c^{4} d^{3}\right )} x^{7} + {\left (b^{6} c^{7} - 4 \, a^{2} b^{4} c^{5} d^{2}\right )} x^{5} + {\left (a b^{5} c^{7} - 2 \, a^{2} b^{4} c^{6} d\right )} x^{3}\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 (2 \, a^{3} b^{4} c^{7} - 8 \, a^{4} b^{3} c^{6} d + 12 \, a^{5} b^{2} c^{5} d^{2} - 8 \, a^{6} b c^{4} d^{3} + 2 \, a^{7} c^{3} d^{4} - {\left (15 \, a b^{6} c^{5} d^{2} - 35 \, a^{2} b^{5} c^{4} d^{3} + 8 \, a^{3} b^{4} c^{3} d^{4} + 76 \, a^{4} b^{3} c^{2} d^{5} - 96 \, a^{5} b^{2} c d^{6} + 32 \, a^{6} b d^{7}\right )} x^{8} - 2 \, {\left (15 \, a b^{6} c^{6} d - 30 \, a^{2} b^{5} c^{5} d^{2} - 4 \, a^{3} b^{4} c^{4} d^{3} + 61 \, a^{4} b^{3} c^{3} d^{4} - 34 \, a^{5} b^{2} c^{2} d^{5} - 24 \, a^{6} b c d^{6} + 16 \, a^{7} d^{7}\right )} x^{6} - 3 \, {\left (5 \, a b^{6} c^{7} - 5 \, a^{2} b^{5} c^{6} d - 14 \, a^{3} b^{4} c^{5} d^{2} + 16 \, a^{4} b^{3} c^{4} d^{3} + 26 \, a^{5} b^{2} c^{3} d^{4} - 44 \, a^{6} b c^{2} d^{5} + 16 \, a^{7} c d^{6}\right )} x^{4} - 2 \, {\left (5 \, a^{2} b^{5} c^{7} - 14 \, a^{3} b^{4} c^{6} d + 6 \, a^{4} b^{3} c^{5} d^{2} + 16 \, a^{5} b^{2} c^{4} d^{3} - 19 \, a^{6} b c^{3} d^{4} + 6 \, a^{7} c^{2} d^{5}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{24 \, {\left ({\left (a^{4} b^{5} c^{8} d^{2} - 4 \, a^{5} b^{4} c^{7} d^{3} + 6 \, a^{6} b^{3} c^{6} d^{4} - 4 \, a^{7} b^{2} c^{5} d^{5} + a^{8} b c^{4} d^{6}\right )} x^{9} + {\left (2 \, a^{4} b^{5} c^{9} d - 7 \, a^{5} b^{4} c^{8} d^{2} + 8 \, a^{6} b^{3} c^{7} d^{3} - 2 \, a^{7} b^{2} c^{6} d^{4} - 2 \, a^{8} b c^{5} d^{5} + a^{9} c^{4} d^{6}\right )} x^{7} + {\left (a^{4} b^{5} c^{10} - 2 \, a^{5} b^{4} c^{9} d - 2 \, a^{6} b^{3} c^{8} d^{2} + 8 \, a^{7} b^{2} c^{7} d^{3} - 7 \, a^{8} b c^{6} d^{4} + 2 \, a^{9} c^{5} d^{5}\right )} x^{5} + {\left (a^{5} b^{4} c^{10} - 4 \, a^{6} b^{3} c^{9} d + 6 \, a^{7} b^{2} c^{8} d^{2} - 4 \, a^{8} b c^{7} d^{3} + a^{9} c^{6} d^{4}\right )} x^{3}\right )}}, \frac {15 \, {\left ({\left (b^{6} c^{5} d^{2} - 2 \, a b^{5} c^{4} d^{3}\right )} x^{9} + {\left (2 \, b^{6} c^{6} d - 3 \, a b^{5} c^{5} d^{2} - 2 \, a^{2} b^{4} c^{4} d^{3}\right )} x^{7} + {\left (b^{6} c^{7} - 4 \, a^{2} b^{4} c^{5} d^{2}\right )} x^{5} + {\left (a b^{5} c^{7} - 2 \, a^{2} b^{4} c^{6} d\right )} x^{3}\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 (2 \, a^{3} b^{4} c^{7} - 8 \, a^{4} b^{3} c^{6} d + 12 \, a^{5} b^{2} c^{5} d^{2} - 8 \, a^{6} b c^{4} d^{3} + 2 \, a^{7} c^{3} d^{4} - {\left (15 \, a b^{6} c^{5} d^{2} - 35 \, a^{2} b^{5} c^{4} d^{3} + 8 \, a^{3} b^{4} c^{3} d^{4} + 76 \, a^{4} b^{3} c^{2} d^{5} - 96 \, a^{5} b^{2} c d^{6} + 32 \, a^{6} b d^{7}\right )} x^{8} - 2 \, {\left (15 \, a b^{6} c^{6} d - 30 \, a^{2} b^{5} c^{5} d^{2} - 4 \, a^{3} b^{4} c^{4} d^{3} + 61 \, a^{4} b^{3} c^{3} d^{4} - 34 \, a^{5} b^{2} c^{2} d^{5} - 24 \, a^{6} b c d^{6} + 16 \, a^{7} d^{7}\right )} x^{6} - 3 \, {\left (5 \, a b^{6} c^{7} - 5 \, a^{2} b^{5} c^{6} d - 14 \, a^{3} b^{4} c^{5} d^{2} + 16 \, a^{4} b^{3} c^{4} d^{3} + 26 \, a^{5} b^{2} c^{3} d^{4} - 44 \, a^{6} b c^{2} d^{5} + 16 \, a^{7} c d^{6}\right )} x^{4} - 2 \, {\left (5 \, a^{2} b^{5} c^{7} - 14 \, a^{3} b^{4} c^{6} d + 6 \, a^{4} b^{3} c^{5} d^{2} + 16 \, a^{5} b^{2} c^{4} d^{3} - 19 \, a^{6} b c^{3} d^{4} + 6 \, a^{7} c^{2} d^{5}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{12 \, {\left ({\left (a^{4} b^{5} c^{8} d^{2} - 4 \, a^{5} b^{4} c^{7} d^{3} + 6 \, a^{6} b^{3} c^{6} d^{4} - 4 \, a^{7} b^{2} c^{5} d^{5} + a^{8} b c^{4} d^{6}\right )} x^{9} + {\left (2 \, a^{4} b^{5} c^{9} d - 7 \, a^{5} b^{4} c^{8} d^{2} + 8 \, a^{6} b^{3} c^{7} d^{3} - 2 \, a^{7} b^{2} c^{6} d^{4} - 2 \, a^{8} b c^{5} d^{5} + a^{9} c^{4} d^{6}\right )} x^{7} + {\left (a^{4} b^{5} c^{10} - 2 \, a^{5} b^{4} c^{9} d - 2 \, a^{6} b^{3} c^{8} d^{2} + 8 \, a^{7} b^{2} c^{7} d^{3} - 7 \, a^{8} b c^{6} d^{4} + 2 \, a^{9} c^{5} d^{5}\right )} x^{5} + {\left (a^{5} b^{4} c^{10} - 4 \, a^{6} b^{3} c^{9} d + 6 \, a^{7} b^{2} c^{8} d^{2} - 4 \, a^{8} b c^{7} d^{3} + a^{9} c^{6} d^{4}\right )} x^{3}\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 {1}{x^{4} \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. 789 vs.
\(2 (330) = 660\).
time = 4.54, size = 789, normalized size = 2.18 \begin {gather*} \frac {{\left (\frac {2 \, {\left (7 \, b^{4} c^{7} d^{6} - 25 \, a b^{3} c^{6} d^{7} + 33 \, a^{2} b^{2} c^{5} d^{8} - 19 \, a^{3} b c^{4} d^{9} + 4 \, a^{4} c^{3} d^{10}\right )} x^{2}}{b^{6} c^{13} d - 6 \, a b^{5} c^{12} d^{2} + 15 \, a^{2} b^{4} c^{11} d^{3} - 20 \, a^{3} b^{3} c^{10} d^{4} + 15 \, a^{4} b^{2} c^{9} d^{5} - 6 \, a^{5} b c^{8} d^{6} + a^{6} c^{7} d^{7}} + \frac {3 \, {\left (5 \, b^{4} c^{8} d^{5} - 18 \, a b^{3} c^{7} d^{6} + 24 \, a^{2} b^{2} c^{6} d^{7} - 14 \, a^{3} b c^{5} d^{8} + 3 \, a^{4} c^{4} d^{9}\right )}}{b^{6} c^{13} d - 6 \, a b^{5} c^{12} d^{2} + 15 \, a^{2} b^{4} c^{11} d^{3} - 20 \, a^{3} b^{3} c^{10} d^{4} + 15 \, a^{4} b^{2} c^{9} d^{5} - 6 \, a^{5} b c^{8} d^{6} + a^{6} c^{7} d^{7}}\right )} x}{3 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}}} - \frac {5 \, {\left (b^{5} c \sqrt {d} - 2 \, a b^{4} 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 (a^{3} b^{3} c^{3} - 3 \, a^{4} b^{2} c^{2} d + 3 \, a^{5} b c d^{2} - a^{6} 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^{5} c \sqrt {d} - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a b^{4} d^{\frac {3}{2}} - b^{5} c^{2} \sqrt {d}}{{\left (a^{3} b^{3} c^{3} - 3 \, a^{4} b^{2} c^{2} d + 3 \, a^{5} b c d^{2} - a^{6} 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 )}} - \frac {4 \, {\left (3 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b c \sqrt {d} + 3 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a d^{\frac {3}{2}} - 6 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c^{2} \sqrt {d} - 9 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a c d^{\frac {3}{2}} + 3 \, b c^{3} \sqrt {d} + 4 \, a c^{2} d^{\frac {3}{2}}\right )}}{3 \, {\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} - c\right )}^{3} a^{3} c^{3}} \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 = -0.00 \begin {gather*} \int \frac {1}{x^4\,{\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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[Out]
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