Optimal. Leaf size=199 \[ -\frac {d x \left (a+b x^2\right )^{5/2}}{6 c (b c-a d) \left (c+d x^2\right )^3}+\frac {(6 b c-5 a d) x \left (a+b x^2\right )^{3/2}}{24 c^2 (b c-a d) \left (c+d x^2\right )^2}+\frac {a (6 b c-5 a d) x \sqrt {a+b x^2}}{16 c^3 (b c-a d) \left (c+d x^2\right )}+\frac {a^2 (6 b c-5 a d) \tanh ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{16 c^{7/2} (b c-a d)^{3/2}} \]
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Rubi [A]
time = 0.08, antiderivative size = 199, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {390, 386, 385,
214} \begin {gather*} \frac {a^2 (6 b c-5 a d) \tanh ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{16 c^{7/2} (b c-a d)^{3/2}}+\frac {a x \sqrt {a+b x^2} (6 b c-5 a d)}{16 c^3 \left (c+d x^2\right ) (b c-a d)}+\frac {x \left (a+b x^2\right )^{3/2} (6 b c-5 a d)}{24 c^2 \left (c+d x^2\right )^2 (b c-a d)}-\frac {d x \left (a+b x^2\right )^{5/2}}{6 c \left (c+d x^2\right )^3 (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rule 385
Rule 386
Rule 390
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^4} \, dx &=-\frac {d x \left (a+b x^2\right )^{5/2}}{6 c (b c-a d) \left (c+d x^2\right )^3}+\frac {(6 b c-5 a d) \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^3} \, dx}{6 c (b c-a d)}\\ &=-\frac {d x \left (a+b x^2\right )^{5/2}}{6 c (b c-a d) \left (c+d x^2\right )^3}+\frac {(6 b c-5 a d) x \left (a+b x^2\right )^{3/2}}{24 c^2 (b c-a d) \left (c+d x^2\right )^2}+\frac {(a (6 b c-5 a d)) \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^2} \, dx}{8 c^2 (b c-a d)}\\ &=-\frac {d x \left (a+b x^2\right )^{5/2}}{6 c (b c-a d) \left (c+d x^2\right )^3}+\frac {(6 b c-5 a d) x \left (a+b x^2\right )^{3/2}}{24 c^2 (b c-a d) \left (c+d x^2\right )^2}+\frac {a (6 b c-5 a d) x \sqrt {a+b x^2}}{16 c^3 (b c-a d) \left (c+d x^2\right )}+\frac {\left (a^2 (6 b c-5 a d)\right ) \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )} \, dx}{16 c^3 (b c-a d)}\\ &=-\frac {d x \left (a+b x^2\right )^{5/2}}{6 c (b c-a d) \left (c+d x^2\right )^3}+\frac {(6 b c-5 a d) x \left (a+b x^2\right )^{3/2}}{24 c^2 (b c-a d) \left (c+d x^2\right )^2}+\frac {a (6 b c-5 a d) x \sqrt {a+b x^2}}{16 c^3 (b c-a d) \left (c+d x^2\right )}+\frac {\left (a^2 (6 b c-5 a d)\right ) \text {Subst}\left (\int \frac {1}{c-(b c-a d) x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{16 c^3 (b c-a d)}\\ &=-\frac {d x \left (a+b x^2\right )^{5/2}}{6 c (b c-a d) \left (c+d x^2\right )^3}+\frac {(6 b c-5 a d) x \left (a+b x^2\right )^{3/2}}{24 c^2 (b c-a d) \left (c+d x^2\right )^2}+\frac {a (6 b c-5 a d) x \sqrt {a+b x^2}}{16 c^3 (b c-a d) \left (c+d x^2\right )}+\frac {a^2 (6 b c-5 a d) \tanh ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{16 c^{7/2} (b c-a d)^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 10.61, size = 247, normalized size = 1.24 \begin {gather*} \frac {a x \left (1+\frac {b x^2}{a}\right ) \left (c \left (-4 b^3 c^2 x^4 \left (3 c+d x^2\right )-2 a b^2 c x^2 \left (21 c^2+13 c d x^2+4 d^2 x^4\right )+a^3 d \left (33 c^2+40 c d x^2+15 d^2 x^4\right )+a^2 b \left (-30 c^3+11 c^2 d x^2+32 c d^2 x^4+15 d^3 x^6\right )\right )+\frac {3 a^2 (-6 b c+5 a d) \left (c+d x^2\right )^3 \tanh ^{-1}\left (\sqrt {\frac {(b c-a d) x^2}{c \left (a+b x^2\right )}}\right )}{\sqrt {\frac {(b c-a d) x^2}{c \left (a+b x^2\right )}}}\right )}{48 c^4 (-b c+a d) \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^3} \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. \(12815\) vs.
\(2(175)=350\).
time = 0.07, size = 12816, normalized size = 64.40
method | result | size |
default | \(\text {Expression too large to display}\) | \(12816\) |
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. 466 vs.
\(2 (175) = 350\).
time = 0.80, size = 972, normalized size = 4.88 \begin {gather*} \left [\frac {3 \, {\left (6 \, a^{2} b c^{4} - 5 \, a^{3} c^{3} d + {\left (6 \, a^{2} b c d^{3} - 5 \, a^{3} d^{4}\right )} x^{6} + 3 \, {\left (6 \, a^{2} b c^{2} d^{2} - 5 \, a^{3} c d^{3}\right )} x^{4} + 3 \, {\left (6 \, a^{2} b c^{3} d - 5 \, a^{3} c^{2} d^{2}\right )} x^{2}\right )} \sqrt {b c^{2} - a c d} \log \left (\frac {{\left (8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} + 2 \, {\left (4 \, a b c^{2} - 3 \, a^{2} c d\right )} x^{2} + 4 \, {\left ({\left (2 \, b c - a d\right )} x^{3} + a c x\right )} \sqrt {b c^{2} - a c d} \sqrt {b x^{2} + a}}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}\right ) + 4 \, {\left ({\left (4 \, b^{3} c^{4} d + 4 \, a b^{2} c^{3} d^{2} - 23 \, a^{2} b c^{2} d^{3} + 15 \, a^{3} c d^{4}\right )} x^{5} + 2 \, {\left (6 \, b^{3} c^{5} + 5 \, a b^{2} c^{4} d - 31 \, a^{2} b c^{3} d^{2} + 20 \, a^{3} c^{2} d^{3}\right )} x^{3} + 3 \, {\left (10 \, a b^{2} c^{5} - 21 \, a^{2} b c^{4} d + 11 \, a^{3} c^{3} d^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{192 \, {\left (b^{2} c^{9} - 2 \, a b c^{8} d + a^{2} c^{7} d^{2} + {\left (b^{2} c^{6} d^{3} - 2 \, a b c^{5} d^{4} + a^{2} c^{4} d^{5}\right )} x^{6} + 3 \, {\left (b^{2} c^{7} d^{2} - 2 \, a b c^{6} d^{3} + a^{2} c^{5} d^{4}\right )} x^{4} + 3 \, {\left (b^{2} c^{8} d - 2 \, a b c^{7} d^{2} + a^{2} c^{6} d^{3}\right )} x^{2}\right )}}, -\frac {3 \, {\left (6 \, a^{2} b c^{4} - 5 \, a^{3} c^{3} d + {\left (6 \, a^{2} b c d^{3} - 5 \, a^{3} d^{4}\right )} x^{6} + 3 \, {\left (6 \, a^{2} b c^{2} d^{2} - 5 \, a^{3} c d^{3}\right )} x^{4} + 3 \, {\left (6 \, a^{2} b c^{3} d - 5 \, a^{3} c^{2} d^{2}\right )} x^{2}\right )} \sqrt {-b c^{2} + a c d} \arctan \left (\frac {\sqrt {-b c^{2} + a c d} {\left ({\left (2 \, b c - a d\right )} x^{2} + a c\right )} \sqrt {b x^{2} + a}}{2 \, {\left ({\left (b^{2} c^{2} - a b c d\right )} x^{3} + {\left (a b c^{2} - a^{2} c d\right )} x\right )}}\right ) - 2 \, {\left ({\left (4 \, b^{3} c^{4} d + 4 \, a b^{2} c^{3} d^{2} - 23 \, a^{2} b c^{2} d^{3} + 15 \, a^{3} c d^{4}\right )} x^{5} + 2 \, {\left (6 \, b^{3} c^{5} + 5 \, a b^{2} c^{4} d - 31 \, a^{2} b c^{3} d^{2} + 20 \, a^{3} c^{2} d^{3}\right )} x^{3} + 3 \, {\left (10 \, a b^{2} c^{5} - 21 \, a^{2} b c^{4} d + 11 \, a^{3} c^{3} d^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{96 \, {\left (b^{2} c^{9} - 2 \, a b c^{8} d + a^{2} c^{7} d^{2} + {\left (b^{2} c^{6} d^{3} - 2 \, a b c^{5} d^{4} + a^{2} c^{4} d^{5}\right )} x^{6} + 3 \, {\left (b^{2} c^{7} d^{2} - 2 \, a b c^{6} d^{3} + a^{2} c^{5} d^{4}\right )} x^{4} + 3 \, {\left (b^{2} c^{8} d - 2 \, a b c^{7} d^{2} + a^{2} c^{6} d^{3}\right )} x^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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. 919 vs.
\(2 (175) = 350\).
time = 1.65, size = 919, normalized size = 4.62 \begin {gather*} -\frac {{\left (6 \, a^{2} b^{\frac {3}{2}} c - 5 \, a^{3} \sqrt {b} d\right )} \arctan \left (\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} d + 2 \, b c - a d}{2 \, \sqrt {-b^{2} c^{2} + a b c d}}\right )}{16 \, {\left (b c^{4} - a c^{3} d\right )} \sqrt {-b^{2} c^{2} + a b c d}} - \frac {18 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} a^{2} b^{\frac {3}{2}} c d^{4} - 15 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} a^{3} \sqrt {b} d^{5} - 96 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} b^{\frac {9}{2}} c^{4} d + 96 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a b^{\frac {7}{2}} c^{3} d^{2} + 180 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a^{2} b^{\frac {5}{2}} c^{2} d^{3} - 240 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a^{3} b^{\frac {3}{2}} c d^{4} + 75 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a^{4} \sqrt {b} d^{5} - 128 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} b^{\frac {11}{2}} c^{5} - 64 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a b^{\frac {9}{2}} c^{4} d + 720 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{2} b^{\frac {7}{2}} c^{3} d^{2} - 968 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{3} b^{\frac {5}{2}} c^{2} d^{3} + 620 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{4} b^{\frac {3}{2}} c d^{4} - 150 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{5} \sqrt {b} d^{5} - 96 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{2} b^{\frac {9}{2}} c^{4} d - 288 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{3} b^{\frac {7}{2}} c^{3} d^{2} + 864 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{4} b^{\frac {5}{2}} c^{2} d^{3} - 600 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{5} b^{\frac {3}{2}} c d^{4} + 150 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{6} \sqrt {b} d^{5} - 48 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{4} b^{\frac {7}{2}} c^{3} d^{2} - 72 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{5} b^{\frac {5}{2}} c^{2} d^{3} + 210 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{6} b^{\frac {3}{2}} c d^{4} - 75 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{7} \sqrt {b} d^{5} - 4 \, a^{6} b^{\frac {5}{2}} c^{2} d^{3} - 8 \, a^{7} b^{\frac {3}{2}} c d^{4} + 15 \, a^{8} \sqrt {b} d^{5}}{24 \, {\left (b c^{4} d^{2} - a c^{3} d^{3}\right )} {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} d + 4 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b c - 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a d + a^{2} d\right )}^{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 = -0.01 \begin {gather*} \int \frac {{\left (b\,x^2+a\right )}^{3/2}}{{\left (d\,x^2+c\right )}^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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