Optimal. Leaf size=208 \[ \frac {x \sqrt {a+b x^2}}{6 c \left (c+d x^2\right )^3}+\frac {(4 b c-5 a d) x \sqrt {a+b x^2}}{24 c^2 (b c-a d) \left (c+d x^2\right )^2}+\frac {(2 b c-5 a d) (4 b c-3 a d) x \sqrt {a+b x^2}}{48 c^3 (b c-a d)^2 \left (c+d x^2\right )}+\frac {a \left (8 b^2 c^2-12 a b c d+5 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{16 c^{7/2} (b c-a d)^{5/2}} \]
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Rubi [A]
time = 0.15, antiderivative size = 208, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {423, 541, 12,
385, 214} \begin {gather*} \frac {a \left (5 a^2 d^2-12 a b c d+8 b^2 c^2\right ) \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)^{5/2}}+\frac {x \sqrt {a+b x^2} (2 b c-5 a d) (4 b c-3 a d)}{48 c^3 \left (c+d x^2\right ) (b c-a d)^2}+\frac {x \sqrt {a+b x^2} (4 b c-5 a d)}{24 c^2 \left (c+d x^2\right )^2 (b c-a d)}+\frac {x \sqrt {a+b x^2}}{6 c \left (c+d x^2\right )^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 214
Rule 385
Rule 423
Rule 541
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^4} \, dx &=\frac {x \sqrt {a+b x^2}}{6 c \left (c+d x^2\right )^3}-\frac {\int \frac {-5 a-4 b x^2}{\sqrt {a+b x^2} \left (c+d x^2\right )^3} \, dx}{6 c}\\ &=\frac {x \sqrt {a+b x^2}}{6 c \left (c+d x^2\right )^3}+\frac {(4 b c-5 a d) x \sqrt {a+b x^2}}{24 c^2 (b c-a d) \left (c+d x^2\right )^2}-\frac {\int \frac {-a (16 b c-15 a d)-2 b (4 b c-5 a d) x^2}{\sqrt {a+b x^2} \left (c+d x^2\right )^2} \, dx}{24 c^2 (b c-a d)}\\ &=\frac {x \sqrt {a+b x^2}}{6 c \left (c+d x^2\right )^3}+\frac {(4 b c-5 a d) x \sqrt {a+b x^2}}{24 c^2 (b c-a d) \left (c+d x^2\right )^2}+\frac {(2 b c-5 a d) (4 b c-3 a d) x \sqrt {a+b x^2}}{48 c^3 (b c-a d)^2 \left (c+d x^2\right )}-\frac {\int -\frac {3 a \left (8 b^2 c^2-12 a b c d+5 a^2 d^2\right )}{\sqrt {a+b x^2} \left (c+d x^2\right )} \, dx}{48 c^3 (b c-a d)^2}\\ &=\frac {x \sqrt {a+b x^2}}{6 c \left (c+d x^2\right )^3}+\frac {(4 b c-5 a d) x \sqrt {a+b x^2}}{24 c^2 (b c-a d) \left (c+d x^2\right )^2}+\frac {(2 b c-5 a d) (4 b c-3 a d) x \sqrt {a+b x^2}}{48 c^3 (b c-a d)^2 \left (c+d x^2\right )}+\frac {\left (a \left (8 b^2 c^2-12 a b c d+5 a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )} \, dx}{16 c^3 (b c-a d)^2}\\ &=\frac {x \sqrt {a+b x^2}}{6 c \left (c+d x^2\right )^3}+\frac {(4 b c-5 a d) x \sqrt {a+b x^2}}{24 c^2 (b c-a d) \left (c+d x^2\right )^2}+\frac {(2 b c-5 a d) (4 b c-3 a d) x \sqrt {a+b x^2}}{48 c^3 (b c-a d)^2 \left (c+d x^2\right )}+\frac {\left (a \left (8 b^2 c^2-12 a b c d+5 a^2 d^2\right )\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)^2}\\ &=\frac {x \sqrt {a+b x^2}}{6 c \left (c+d x^2\right )^3}+\frac {(4 b c-5 a d) x \sqrt {a+b x^2}}{24 c^2 (b c-a d) \left (c+d x^2\right )^2}+\frac {(2 b c-5 a d) (4 b c-3 a d) x \sqrt {a+b x^2}}{48 c^3 (b c-a d)^2 \left (c+d x^2\right )}+\frac {a \left (8 b^2 c^2-12 a b c d+5 a^2 d^2\right ) \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)^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 10.77, size = 227, normalized size = 1.09 \begin {gather*} \frac {x \sqrt {a+b x^2} \left ((b c-a d) \left (8 b^2 c^2 \left (3 c^2+3 c d x^2+d^2 x^4\right )-2 a b c d \left (30 c^2+35 c d x^2+13 d^2 x^4\right )+a^2 d^2 \left (33 c^2+40 c d x^2+15 d^2 x^4\right )\right )+\frac {3 a \left (8 b^2 c^2-12 a b c d+5 a^2 d^2\right ) \sqrt {\frac {(b c-a d) x^2}{c \left (a+b x^2\right )}} \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 )}{x^2}\right )}{48 c^3 (b c-a d)^3 \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. \(6474\) vs.
\(2(184)=368\).
time = 0.08, size = 6475, normalized size = 31.13
method | result | size |
default | \(\text {Expression too large to display}\) | \(6475\) |
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. 590 vs.
\(2 (184) = 368\).
time = 1.33, size = 1220, normalized size = 5.87 \begin {gather*} \left [\frac {3 \, {\left (8 \, a b^{2} c^{5} - 12 \, a^{2} b c^{4} d + 5 \, a^{3} c^{3} d^{2} + {\left (8 \, a b^{2} c^{2} d^{3} - 12 \, a^{2} b c d^{4} + 5 \, a^{3} d^{5}\right )} x^{6} + 3 \, {\left (8 \, a b^{2} c^{3} d^{2} - 12 \, a^{2} b c^{2} d^{3} + 5 \, a^{3} c d^{4}\right )} x^{4} + 3 \, {\left (8 \, a b^{2} c^{4} d - 12 \, a^{2} b c^{3} d^{2} + 5 \, a^{3} c^{2} d^{3}\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 (8 \, b^{3} c^{4} d^{2} - 34 \, a b^{2} c^{3} d^{3} + 41 \, a^{2} b c^{2} d^{4} - 15 \, a^{3} c d^{5}\right )} x^{5} + 2 \, {\left (12 \, b^{3} c^{5} d - 47 \, a b^{2} c^{4} d^{2} + 55 \, a^{2} b c^{3} d^{3} - 20 \, a^{3} c^{2} d^{4}\right )} x^{3} + 3 \, {\left (8 \, b^{3} c^{6} - 28 \, a b^{2} c^{5} d + 31 \, a^{2} b c^{4} d^{2} - 11 \, a^{3} c^{3} d^{3}\right )} x\right )} \sqrt {b x^{2} + a}}{192 \, {\left (b^{3} c^{10} - 3 \, a b^{2} c^{9} d + 3 \, a^{2} b c^{8} d^{2} - a^{3} c^{7} d^{3} + {\left (b^{3} c^{7} d^{3} - 3 \, a b^{2} c^{6} d^{4} + 3 \, a^{2} b c^{5} d^{5} - a^{3} c^{4} d^{6}\right )} x^{6} + 3 \, {\left (b^{3} c^{8} d^{2} - 3 \, a b^{2} c^{7} d^{3} + 3 \, a^{2} b c^{6} d^{4} - a^{3} c^{5} d^{5}\right )} x^{4} + 3 \, {\left (b^{3} c^{9} d - 3 \, a b^{2} c^{8} d^{2} + 3 \, a^{2} b c^{7} d^{3} - a^{3} c^{6} d^{4}\right )} x^{2}\right )}}, -\frac {3 \, {\left (8 \, a b^{2} c^{5} - 12 \, a^{2} b c^{4} d + 5 \, a^{3} c^{3} d^{2} + {\left (8 \, a b^{2} c^{2} d^{3} - 12 \, a^{2} b c d^{4} + 5 \, a^{3} d^{5}\right )} x^{6} + 3 \, {\left (8 \, a b^{2} c^{3} d^{2} - 12 \, a^{2} b c^{2} d^{3} + 5 \, a^{3} c d^{4}\right )} x^{4} + 3 \, {\left (8 \, a b^{2} c^{4} d - 12 \, a^{2} b c^{3} d^{2} + 5 \, a^{3} c^{2} d^{3}\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 (8 \, b^{3} c^{4} d^{2} - 34 \, a b^{2} c^{3} d^{3} + 41 \, a^{2} b c^{2} d^{4} - 15 \, a^{3} c d^{5}\right )} x^{5} + 2 \, {\left (12 \, b^{3} c^{5} d - 47 \, a b^{2} c^{4} d^{2} + 55 \, a^{2} b c^{3} d^{3} - 20 \, a^{3} c^{2} d^{4}\right )} x^{3} + 3 \, {\left (8 \, b^{3} c^{6} - 28 \, a b^{2} c^{5} d + 31 \, a^{2} b c^{4} d^{2} - 11 \, a^{3} c^{3} d^{3}\right )} x\right )} \sqrt {b x^{2} + a}}{96 \, {\left (b^{3} c^{10} - 3 \, a b^{2} c^{9} d + 3 \, a^{2} b c^{8} d^{2} - a^{3} c^{7} d^{3} + {\left (b^{3} c^{7} d^{3} - 3 \, a b^{2} c^{6} d^{4} + 3 \, a^{2} b c^{5} d^{5} - a^{3} c^{4} d^{6}\right )} x^{6} + 3 \, {\left (b^{3} c^{8} d^{2} - 3 \, a b^{2} c^{7} d^{3} + 3 \, a^{2} b c^{6} d^{4} - a^{3} c^{5} d^{5}\right )} x^{4} + 3 \, {\left (b^{3} c^{9} d - 3 \, a b^{2} c^{8} d^{2} + 3 \, a^{2} b c^{7} d^{3} - a^{3} c^{6} d^{4}\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. 958 vs.
\(2 (184) = 368\).
time = 2.21, size = 958, normalized size = 4.61 \begin {gather*} -\frac {{\left (8 \, a b^{\frac {5}{2}} c^{2} - 12 \, a^{2} b^{\frac {3}{2}} c d + 5 \, a^{3} \sqrt {b} d^{2}\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^{2} c^{5} - 2 \, a b c^{4} d + a^{2} c^{3} d^{2}\right )} \sqrt {-b^{2} c^{2} + a b c d}} - \frac {24 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} a b^{\frac {5}{2}} c^{2} d^{3} - 36 \, {\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} + 240 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a b^{\frac {7}{2}} c^{3} d^{2} - 480 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a^{2} b^{\frac {5}{2}} c^{2} d^{3} + 330 \, {\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} - 256 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} b^{\frac {11}{2}} c^{5} + 1216 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a b^{\frac {9}{2}} c^{4} d - 2016 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{2} b^{\frac {7}{2}} c^{3} d^{2} + 1736 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{3} b^{\frac {5}{2}} c^{2} d^{3} - 800 \, {\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} - 384 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{2} b^{\frac {9}{2}} c^{4} d + 1392 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{3} b^{\frac {7}{2}} c^{3} d^{2} - 1608 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{4} b^{\frac {5}{2}} c^{2} d^{3} + 780 \, {\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} - 96 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{4} b^{\frac {7}{2}} c^{3} d^{2} + 336 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{5} b^{\frac {5}{2}} c^{2} d^{3} - 300 \, {\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} - 8 \, a^{6} b^{\frac {5}{2}} c^{2} d^{3} + 26 \, a^{7} b^{\frac {3}{2}} c d^{4} - 15 \, a^{8} \sqrt {b} d^{5}}{24 \, {\left (b^{2} c^{5} d - 2 \, a b c^{4} d^{2} + a^{2} 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 = -0.00 \begin {gather*} \int \frac {\sqrt {b\,x^2+a}}{{\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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