Optimal. Leaf size=245 \[ -\frac {d}{3 c (b c-a d) x^3 \left (c+d x^2\right )^{3/2}}-\frac {d (3 b c-2 a d)}{c^2 (b c-a d)^2 x^3 \sqrt {c+d x^2}}-\frac {\left (b^2 c^2-12 a b c d+8 a^2 d^2\right ) \sqrt {c+d x^2}}{3 a c^3 (b c-a d)^2 x^3}+\frac {(b c-2 a d) \left (3 b^2 c^2+8 a b c d-8 a^2 d^2\right ) \sqrt {c+d x^2}}{3 a^2 c^4 (b c-a d)^2 x}+\frac {b^4 \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{a^{5/2} (b c-a d)^{5/2}} \]
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Rubi [A]
time = 0.23, antiderivative size = 245, normalized size of antiderivative = 1.00, number of steps
used = 7, 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 {b^4 \text {ArcTan}\left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{a^{5/2} (b c-a d)^{5/2}}+\frac {\sqrt {c+d x^2} (b c-2 a d) \left (-8 a^2 d^2+8 a b c d+3 b^2 c^2\right )}{3 a^2 c^4 x (b c-a d)^2}-\frac {\sqrt {c+d x^2} \left (8 a^2 d^2-12 a b c d+b^2 c^2\right )}{3 a c^3 x^3 (b c-a d)^2}-\frac {d (3 b c-2 a d)}{c^2 x^3 \sqrt {c+d x^2} (b c-a d)^2}-\frac {d}{3 c x^3 \left (c+d x^2\right )^{3/2} (b c-a d)} \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 ) \left (c+d x^2\right )^{5/2}} \, dx &=-\frac {d}{3 c (b c-a d) x^3 \left (c+d x^2\right )^{3/2}}+\frac {\int \frac {3 (b c-2 a d)-6 b d x^2}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx}{3 c (b c-a d)}\\ &=-\frac {d}{3 c (b c-a d) x^3 \left (c+d x^2\right )^{3/2}}-\frac {d (3 b c-2 a d)}{c^2 (b c-a d)^2 x^3 \sqrt {c+d x^2}}+\frac {\int \frac {3 \left (b^2 c^2-12 a b c d+8 a^2 d^2\right )-12 b d (3 b c-2 a d) x^2}{x^4 \left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{3 c^2 (b c-a d)^2}\\ &=-\frac {d}{3 c (b c-a d) x^3 \left (c+d x^2\right )^{3/2}}-\frac {d (3 b c-2 a d)}{c^2 (b c-a d)^2 x^3 \sqrt {c+d x^2}}-\frac {\left (\frac {b^2 c}{a}-12 b d+\frac {8 a d^2}{c}\right ) \sqrt {c+d x^2}}{3 c^2 (b c-a d)^2 x^3}-\frac {\int \frac {3 (b c-2 a d) \left (3 b^2 c^2+8 a b c d-8 a^2 d^2\right )+6 b d \left (b^2 c^2-12 a b c d+8 a^2 d^2\right ) x^2}{x^2 \left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{9 a c^3 (b c-a d)^2}\\ &=-\frac {d}{3 c (b c-a d) x^3 \left (c+d x^2\right )^{3/2}}-\frac {d (3 b c-2 a d)}{c^2 (b c-a d)^2 x^3 \sqrt {c+d x^2}}-\frac {\left (\frac {b^2 c}{a}-12 b d+\frac {8 a d^2}{c}\right ) \sqrt {c+d x^2}}{3 c^2 (b c-a d)^2 x^3}+\frac {(b c-2 a d) \left (3 b^2 c^2+8 a b c d-8 a^2 d^2\right ) \sqrt {c+d x^2}}{3 a^2 c^4 (b c-a d)^2 x}+\frac {\int \frac {9 b^4 c^4}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{9 a^2 c^4 (b c-a d)^2}\\ &=-\frac {d}{3 c (b c-a d) x^3 \left (c+d x^2\right )^{3/2}}-\frac {d (3 b c-2 a d)}{c^2 (b c-a d)^2 x^3 \sqrt {c+d x^2}}-\frac {\left (\frac {b^2 c}{a}-12 b d+\frac {8 a d^2}{c}\right ) \sqrt {c+d x^2}}{3 c^2 (b c-a d)^2 x^3}+\frac {(b c-2 a d) \left (3 b^2 c^2+8 a b c d-8 a^2 d^2\right ) \sqrt {c+d x^2}}{3 a^2 c^4 (b c-a d)^2 x}+\frac {b^4 \int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{a^2 (b c-a d)^2}\\ &=-\frac {d}{3 c (b c-a d) x^3 \left (c+d x^2\right )^{3/2}}-\frac {d (3 b c-2 a d)}{c^2 (b c-a d)^2 x^3 \sqrt {c+d x^2}}-\frac {\left (\frac {b^2 c}{a}-12 b d+\frac {8 a d^2}{c}\right ) \sqrt {c+d x^2}}{3 c^2 (b c-a d)^2 x^3}+\frac {(b c-2 a d) \left (3 b^2 c^2+8 a b c d-8 a^2 d^2\right ) \sqrt {c+d x^2}}{3 a^2 c^4 (b c-a d)^2 x}+\frac {b^4 \text {Subst}\left (\int \frac {1}{a-(-b c+a d) x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{a^2 (b c-a d)^2}\\ &=-\frac {d}{3 c (b c-a d) x^3 \left (c+d x^2\right )^{3/2}}-\frac {d (3 b c-2 a d)}{c^2 (b c-a d)^2 x^3 \sqrt {c+d x^2}}-\frac {\left (\frac {b^2 c}{a}-12 b d+\frac {8 a d^2}{c}\right ) \sqrt {c+d x^2}}{3 c^2 (b c-a d)^2 x^3}+\frac {(b c-2 a d) \left (3 b^2 c^2+8 a b c d-8 a^2 d^2\right ) \sqrt {c+d x^2}}{3 a^2 c^4 (b c-a d)^2 x}+\frac {b^4 \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{a^{5/2} (b c-a d)^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.80, size = 233, normalized size = 0.95 \begin {gather*} \frac {3 b^3 c^3 x^2 \left (c+d x^2\right )^2-a b^2 c^2 \left (c-2 d x^2\right ) \left (c+d x^2\right )^2+a^2 b c d \left (2 c^3-9 c^2 d x^2-36 c d^2 x^4-24 d^3 x^6\right )+a^3 d^2 \left (-c^3+6 c^2 d x^2+24 c d^2 x^4+16 d^3 x^6\right )}{3 a^2 c^4 (b c-a d)^2 x^3 \left (c+d x^2\right )^{3/2}}-\frac {b^4 \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 )}{a^{5/2} (b c-a d)^{5/2}} \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. \(1551\) vs.
\(2(221)=442\).
time = 0.18, size = 1552, normalized size = 6.33
method | result | size |
risch | \(-\frac {\sqrt {d \,x^{2}+c}\, \left (-8 a d \,x^{2}-3 c \,x^{2} b +a c \right )}{3 c^{4} a^{2} x^{3}}+\frac {5 a \,b^{2} d^{5} \sqrt {\left (x -\frac {\sqrt {-c d}}{d}\right )^{2} d +2 \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}}{4 c^{4} \left (b \sqrt {-c d}+\sqrt {-a b}\, d \right )^{2} \left (b \sqrt {-c d}-\sqrt {-a b}\, d \right )^{2} \left (x -\frac {\sqrt {-c d}}{d}\right )}-\frac {7 b^{3} d^{4} \sqrt {\left (x -\frac {\sqrt {-c d}}{d}\right )^{2} d +2 \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}}{4 c^{3} \left (b \sqrt {-c d}+\sqrt {-a b}\, d \right )^{2} \left (b \sqrt {-c d}-\sqrt {-a b}\, d \right )^{2} \left (x -\frac {\sqrt {-c d}}{d}\right )}-\frac {b \,d^{3} \sqrt {\left (x +\frac {\sqrt {-c d}}{d}\right )^{2} d -2 \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}}{12 c^{3} \left (b \sqrt {-c d}+\sqrt {-a b}\, d \right ) \left (b \sqrt {-c d}-\sqrt {-a b}\, d \right ) \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )^{2}}+\frac {b \,d^{3} \sqrt {\left (x +\frac {\sqrt {-c d}}{d}\right )^{2} d -2 \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}}{12 c^{4} \left (b \sqrt {-c d}+\sqrt {-a b}\, d \right ) \left (b \sqrt {-c d}-\sqrt {-a b}\, d \right ) \left (x +\frac {\sqrt {-c d}}{d}\right )}+\frac {b \,d^{3} \sqrt {\left (x -\frac {\sqrt {-c d}}{d}\right )^{2} d +2 \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}}{12 c^{3} \left (b \sqrt {-c d}+\sqrt {-a b}\, d \right ) \left (b \sqrt {-c d}-\sqrt {-a b}\, d \right ) \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )^{2}}+\frac {b \,d^{3} \sqrt {\left (x -\frac {\sqrt {-c d}}{d}\right )^{2} d +2 \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}}{12 c^{4} \left (b \sqrt {-c d}+\sqrt {-a b}\, d \right ) \left (b \sqrt {-c d}-\sqrt {-a b}\, d \right ) \left (x -\frac {\sqrt {-c d}}{d}\right )}-\frac {b^{6} d^{2} \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{2 a^{2} \left (b \sqrt {-c d}+\sqrt {-a b}\, d \right )^{2} \left (b \sqrt {-c d}-\sqrt {-a b}\, d \right )^{2} \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}+\frac {5 a \,b^{2} d^{5} \sqrt {\left (x +\frac {\sqrt {-c d}}{d}\right )^{2} d -2 \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}}{4 c^{4} \left (b \sqrt {-c d}+\sqrt {-a b}\, d \right )^{2} \left (b \sqrt {-c d}-\sqrt {-a b}\, d \right )^{2} \left (x +\frac {\sqrt {-c d}}{d}\right )}-\frac {7 b^{3} d^{4} \sqrt {\left (x +\frac {\sqrt {-c d}}{d}\right )^{2} d -2 \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}}{4 c^{3} \left (b \sqrt {-c d}+\sqrt {-a b}\, d \right )^{2} \left (b \sqrt {-c d}-\sqrt {-a b}\, d \right )^{2} \left (x +\frac {\sqrt {-c d}}{d}\right )}+\frac {b^{6} d^{2} \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{2 a^{2} \left (b \sqrt {-c d}+\sqrt {-a b}\, d \right )^{2} \left (b \sqrt {-c d}-\sqrt {-a b}\, d \right )^{2} \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}\) | \(1282\) |
default | \(\text {Expression too large to display}\) | \(1552\) |
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. 544 vs.
\(2 (221) = 442\).
time = 1.90, size = 1128, normalized size = 4.60 \begin {gather*} \left [-\frac {3 \, {\left (b^{4} c^{4} d^{2} x^{7} + 2 \, b^{4} c^{5} d x^{5} + b^{4} c^{6} 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 (a^{2} b^{3} c^{6} - 3 \, a^{3} b^{2} c^{5} d + 3 \, a^{4} b c^{4} d^{2} - a^{5} c^{3} d^{3} - {\left (3 \, a b^{4} c^{4} d^{2} - a^{2} b^{3} c^{3} d^{3} - 26 \, a^{3} b^{2} c^{2} d^{4} + 40 \, a^{4} b c d^{5} - 16 \, a^{5} d^{6}\right )} x^{6} - 3 \, {\left (2 \, a b^{4} c^{5} d - a^{2} b^{3} c^{4} d^{2} - 13 \, a^{3} b^{2} c^{3} d^{3} + 20 \, a^{4} b c^{2} d^{4} - 8 \, a^{5} c d^{5}\right )} x^{4} - 3 \, {\left (a b^{4} c^{6} - a^{2} b^{3} c^{5} d - 3 \, a^{3} b^{2} c^{4} d^{2} + 5 \, a^{4} b c^{3} d^{3} - 2 \, a^{5} c^{2} d^{4}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{12 \, {\left ({\left (a^{3} b^{3} c^{7} d^{2} - 3 \, a^{4} b^{2} c^{6} d^{3} + 3 \, a^{5} b c^{5} d^{4} - a^{6} c^{4} d^{5}\right )} x^{7} + 2 \, {\left (a^{3} b^{3} c^{8} d - 3 \, a^{4} b^{2} c^{7} d^{2} + 3 \, a^{5} b c^{6} d^{3} - a^{6} c^{5} d^{4}\right )} x^{5} + {\left (a^{3} b^{3} c^{9} - 3 \, a^{4} b^{2} c^{8} d + 3 \, a^{5} b c^{7} d^{2} - a^{6} c^{6} d^{3}\right )} x^{3}\right )}}, \frac {3 \, {\left (b^{4} c^{4} d^{2} x^{7} + 2 \, b^{4} c^{5} d x^{5} + b^{4} c^{6} 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 (a^{2} b^{3} c^{6} - 3 \, a^{3} b^{2} c^{5} d + 3 \, a^{4} b c^{4} d^{2} - a^{5} c^{3} d^{3} - {\left (3 \, a b^{4} c^{4} d^{2} - a^{2} b^{3} c^{3} d^{3} - 26 \, a^{3} b^{2} c^{2} d^{4} + 40 \, a^{4} b c d^{5} - 16 \, a^{5} d^{6}\right )} x^{6} - 3 \, {\left (2 \, a b^{4} c^{5} d - a^{2} b^{3} c^{4} d^{2} - 13 \, a^{3} b^{2} c^{3} d^{3} + 20 \, a^{4} b c^{2} d^{4} - 8 \, a^{5} c d^{5}\right )} x^{4} - 3 \, {\left (a b^{4} c^{6} - a^{2} b^{3} c^{5} d - 3 \, a^{3} b^{2} c^{4} d^{2} + 5 \, a^{4} b c^{3} d^{3} - 2 \, a^{5} c^{2} d^{4}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{6 \, {\left ({\left (a^{3} b^{3} c^{7} d^{2} - 3 \, a^{4} b^{2} c^{6} d^{3} + 3 \, a^{5} b c^{5} d^{4} - a^{6} c^{4} d^{5}\right )} x^{7} + 2 \, {\left (a^{3} b^{3} c^{8} d - 3 \, a^{4} b^{2} c^{7} d^{2} + 3 \, a^{5} b c^{6} d^{3} - a^{6} c^{5} d^{4}\right )} x^{5} + {\left (a^{3} b^{3} c^{9} - 3 \, a^{4} b^{2} c^{8} d + 3 \, a^{5} b c^{7} d^{2} - a^{6} c^{6} d^{3}\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 ) \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. 490 vs.
\(2 (221) = 442\).
time = 1.30, size = 490, normalized size = 2.00 \begin {gather*} -\frac {b^{4} \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 (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )} \sqrt {a b c d - a^{2} d^{2}}} - \frac {{\left (\frac {{\left (11 \, b^{3} c^{6} d^{5} - 30 \, a b^{2} c^{5} d^{6} + 27 \, a^{2} b c^{4} d^{7} - 8 \, a^{3} c^{3} d^{8}\right )} x^{2}}{b^{4} c^{11} d - 4 \, a b^{3} c^{10} d^{2} + 6 \, a^{2} b^{2} c^{9} d^{3} - 4 \, a^{3} b c^{8} d^{4} + a^{4} c^{7} d^{5}} + \frac {3 \, {\left (4 \, b^{3} c^{7} d^{4} - 11 \, a b^{2} c^{6} d^{5} + 10 \, a^{2} b c^{5} d^{6} - 3 \, a^{3} c^{4} d^{7}\right )}}{b^{4} c^{11} d - 4 \, a b^{3} c^{10} d^{2} + 6 \, a^{2} b^{2} c^{9} d^{3} - 4 \, a^{3} b c^{8} d^{4} + a^{4} c^{7} d^{5}}\right )} x}{3 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}}} - \frac {2 \, {\left (3 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b c \sqrt {d} + 6 \, {\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} - 18 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a c d^{\frac {3}{2}} + 3 \, b c^{3} \sqrt {d} + 8 \, 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^{2} 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 )\,{\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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