Optimal. Leaf size=293 \[ \frac {a e+c d x}{a \left (c d^2+a e^2\right ) (d+e x)^3 \sqrt {a+c x^2}}+\frac {e \left (3 c d^2-4 a e^2\right ) \sqrt {a+c x^2}}{3 a \left (c d^2+a e^2\right )^2 (d+e x)^3}+\frac {c d e \left (6 c d^2-29 a e^2\right ) \sqrt {a+c x^2}}{6 a \left (c d^2+a e^2\right )^3 (d+e x)^2}+\frac {c e \left (6 c^2 d^4-83 a c d^2 e^2+16 a^2 e^4\right ) \sqrt {a+c x^2}}{6 a \left (c d^2+a e^2\right )^4 (d+e x)}-\frac {5 c^2 d e^2 \left (4 c d^2-3 a e^2\right ) \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{2 \left (c d^2+a e^2\right )^{9/2}} \]
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Rubi [A]
time = 0.25, antiderivative size = 293, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {755, 849, 821,
739, 212} \begin {gather*} \frac {c e \sqrt {a+c x^2} \left (16 a^2 e^4-83 a c d^2 e^2+6 c^2 d^4\right )}{6 a (d+e x) \left (a e^2+c d^2\right )^4}-\frac {5 c^2 d e^2 \left (4 c d^2-3 a e^2\right ) \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{2 \left (a e^2+c d^2\right )^{9/2}}+\frac {c d e \sqrt {a+c x^2} \left (6 c d^2-29 a e^2\right )}{6 a (d+e x)^2 \left (a e^2+c d^2\right )^3}+\frac {e \sqrt {a+c x^2} \left (3 c d^2-4 a e^2\right )}{3 a (d+e x)^3 \left (a e^2+c d^2\right )^2}+\frac {a e+c d x}{a \sqrt {a+c x^2} (d+e x)^3 \left (a e^2+c d^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 739
Rule 755
Rule 821
Rule 849
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^4 \left (a+c x^2\right )^{3/2}} \, dx &=\frac {a e+c d x}{a \left (c d^2+a e^2\right ) (d+e x)^3 \sqrt {a+c x^2}}-\frac {\int \frac {-4 a e^2-3 c d e x}{(d+e x)^4 \sqrt {a+c x^2}} \, dx}{a \left (c d^2+a e^2\right )}\\ &=\frac {a e+c d x}{a \left (c d^2+a e^2\right ) (d+e x)^3 \sqrt {a+c x^2}}+\frac {e \left (3 c d^2-4 a e^2\right ) \sqrt {a+c x^2}}{3 a \left (c d^2+a e^2\right )^2 (d+e x)^3}+\frac {\int \frac {21 a c d e^2+2 c e \left (3 c d^2-4 a e^2\right ) x}{(d+e x)^3 \sqrt {a+c x^2}} \, dx}{3 a \left (c d^2+a e^2\right )^2}\\ &=\frac {a e+c d x}{a \left (c d^2+a e^2\right ) (d+e x)^3 \sqrt {a+c x^2}}+\frac {e \left (3 c d^2-4 a e^2\right ) \sqrt {a+c x^2}}{3 a \left (c d^2+a e^2\right )^2 (d+e x)^3}+\frac {c d e \left (6 c d^2-29 a e^2\right ) \sqrt {a+c x^2}}{6 a \left (c d^2+a e^2\right )^3 (d+e x)^2}-\frac {\int \frac {-2 a c e^2 \left (27 c d^2-8 a e^2\right )-c^2 d e \left (6 c d^2-29 a e^2\right ) x}{(d+e x)^2 \sqrt {a+c x^2}} \, dx}{6 a \left (c d^2+a e^2\right )^3}\\ &=\frac {a e+c d x}{a \left (c d^2+a e^2\right ) (d+e x)^3 \sqrt {a+c x^2}}+\frac {e \left (3 c d^2-4 a e^2\right ) \sqrt {a+c x^2}}{3 a \left (c d^2+a e^2\right )^2 (d+e x)^3}+\frac {c d e \left (6 c d^2-29 a e^2\right ) \sqrt {a+c x^2}}{6 a \left (c d^2+a e^2\right )^3 (d+e x)^2}+\frac {c e \left (6 c^2 d^4-83 a c d^2 e^2+16 a^2 e^4\right ) \sqrt {a+c x^2}}{6 a \left (c d^2+a e^2\right )^4 (d+e x)}+\frac {\left (5 c^2 d e^2 \left (4 c d^2-3 a e^2\right )\right ) \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{2 \left (c d^2+a e^2\right )^4}\\ &=\frac {a e+c d x}{a \left (c d^2+a e^2\right ) (d+e x)^3 \sqrt {a+c x^2}}+\frac {e \left (3 c d^2-4 a e^2\right ) \sqrt {a+c x^2}}{3 a \left (c d^2+a e^2\right )^2 (d+e x)^3}+\frac {c d e \left (6 c d^2-29 a e^2\right ) \sqrt {a+c x^2}}{6 a \left (c d^2+a e^2\right )^3 (d+e x)^2}+\frac {c e \left (6 c^2 d^4-83 a c d^2 e^2+16 a^2 e^4\right ) \sqrt {a+c x^2}}{6 a \left (c d^2+a e^2\right )^4 (d+e x)}-\frac {\left (5 c^2 d e^2 \left (4 c d^2-3 a e^2\right )\right ) \text {Subst}\left (\int \frac {1}{c d^2+a e^2-x^2} \, dx,x,\frac {a e-c d x}{\sqrt {a+c x^2}}\right )}{2 \left (c d^2+a e^2\right )^4}\\ &=\frac {a e+c d x}{a \left (c d^2+a e^2\right ) (d+e x)^3 \sqrt {a+c x^2}}+\frac {e \left (3 c d^2-4 a e^2\right ) \sqrt {a+c x^2}}{3 a \left (c d^2+a e^2\right )^2 (d+e x)^3}+\frac {c d e \left (6 c d^2-29 a e^2\right ) \sqrt {a+c x^2}}{6 a \left (c d^2+a e^2\right )^3 (d+e x)^2}+\frac {c e \left (6 c^2 d^4-83 a c d^2 e^2+16 a^2 e^4\right ) \sqrt {a+c x^2}}{6 a \left (c d^2+a e^2\right )^4 (d+e x)}-\frac {5 c^2 d e^2 \left (4 c d^2-3 a e^2\right ) \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{2 \left (c d^2+a e^2\right )^{9/2}}\\ \end {align*}
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Mathematica [A]
time = 10.54, size = 279, normalized size = 0.95 \begin {gather*} \frac {1}{6} \left (\frac {\sqrt {a+c x^2} \left (-\frac {2 e^3 \left (c d^2+a e^2\right )^2}{(d+e x)^3}-\frac {11 c d e^3 \left (c d^2+a e^2\right )}{(d+e x)^2}+\frac {c e^3 \left (-47 c d^2+10 a e^2\right )}{d+e x}+\frac {6 c^2 \left (c^2 d^4 x+2 a c d^2 e (2 d-3 e x)+a^2 e^3 (-4 d+e x)\right )}{a \left (a+c x^2\right )}\right )}{\left (c d^2+a e^2\right )^4}+\frac {15 c^2 d e^2 \left (4 c d^2-3 a e^2\right ) \log (d+e x)}{\left (c d^2+a e^2\right )^{9/2}}-\frac {15 c^2 d e^2 \left (4 c d^2-3 a e^2\right ) \log \left (a e-c d x+\sqrt {c d^2+a e^2} \sqrt {a+c x^2}\right )}{\left (c d^2+a e^2\right )^{9/2}}\right ) \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. \(1550\) vs.
\(2(271)=542\).
time = 0.44, size = 1551, normalized size = 5.29
method | result | size |
default | \(\text {Expression too large to display}\) | \(1551\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1152 vs.
\(2 (269) = 538\).
time = 0.37, size = 1152, normalized size = 3.93 \begin {gather*} \frac {35 \, c^{4} d^{4} x}{2 \, {\left (\sqrt {c x^{2} + a} a c^{4} d^{8} + 4 \, \sqrt {c x^{2} + a} a^{2} c^{3} d^{6} e^{2} + 6 \, \sqrt {c x^{2} + a} a^{3} c^{2} d^{4} e^{4} + 4 \, \sqrt {c x^{2} + a} a^{4} c d^{2} e^{6} + \sqrt {c x^{2} + a} a^{5} e^{8}\right )}} + \frac {35 \, c^{3} d^{3}}{2 \, {\left (\sqrt {c x^{2} + a} c^{4} d^{8} e^{\left (-1\right )} + 4 \, \sqrt {c x^{2} + a} a c^{3} d^{6} e + 6 \, \sqrt {c x^{2} + a} a^{2} c^{2} d^{4} e^{3} + 4 \, \sqrt {c x^{2} + a} a^{3} c d^{2} e^{5} + \sqrt {c x^{2} + a} a^{4} e^{7}\right )}} - \frac {115 \, c^{3} d^{2} x}{6 \, {\left (\sqrt {c x^{2} + a} a c^{3} d^{6} + 3 \, \sqrt {c x^{2} + a} a^{2} c^{2} d^{4} e^{2} + 3 \, \sqrt {c x^{2} + a} a^{3} c d^{2} e^{4} + \sqrt {c x^{2} + a} a^{4} e^{6}\right )}} + \frac {35 \, c^{3} d^{3} \operatorname {arsinh}\left (\frac {c d x}{\sqrt {a c} {\left | x e + d \right |}} - \frac {a e}{\sqrt {a c} {\left | x e + d \right |}}\right ) e^{\left (-7\right )}}{2 \, {\left (c d^{2} e^{\left (-2\right )} + a\right )}^{\frac {9}{2}}} - \frac {35 \, c^{2} d^{2}}{6 \, {\left (\sqrt {c x^{2} + a} c^{3} d^{7} e^{\left (-1\right )} + \sqrt {c x^{2} + a} c^{3} d^{6} x + 3 \, \sqrt {c x^{2} + a} a c^{2} d^{4} x e^{2} + 3 \, \sqrt {c x^{2} + a} a c^{2} d^{5} e + 3 \, \sqrt {c x^{2} + a} a^{2} c d^{2} x e^{4} + 3 \, \sqrt {c x^{2} + a} a^{2} c d^{3} e^{3} + \sqrt {c x^{2} + a} a^{3} x e^{6} + \sqrt {c x^{2} + a} a^{3} d e^{5}\right )}} - \frac {15 \, c^{2} d}{2 \, {\left (\sqrt {c x^{2} + a} c^{3} d^{6} e^{\left (-1\right )} + 3 \, \sqrt {c x^{2} + a} a c^{2} d^{4} e + 3 \, \sqrt {c x^{2} + a} a^{2} c d^{2} e^{3} + \sqrt {c x^{2} + a} a^{3} e^{5}\right )}} + \frac {8 \, c^{2} x}{3 \, {\left (\sqrt {c x^{2} + a} a c^{2} d^{4} + 2 \, \sqrt {c x^{2} + a} a^{2} c d^{2} e^{2} + \sqrt {c x^{2} + a} a^{3} e^{4}\right )}} - \frac {15 \, c^{2} d \operatorname {arsinh}\left (\frac {c d x}{\sqrt {a c} {\left | x e + d \right |}} - \frac {a e}{\sqrt {a c} {\left | x e + d \right |}}\right ) e^{\left (-5\right )}}{2 \, {\left (c d^{2} e^{\left (-2\right )} + a\right )}^{\frac {7}{2}}} - \frac {7 \, c d}{6 \, {\left (\sqrt {c x^{2} + a} c^{2} d^{4} x^{2} e + \sqrt {c x^{2} + a} c^{2} d^{6} e^{\left (-1\right )} + 2 \, \sqrt {c x^{2} + a} c^{2} d^{5} x + 2 \, \sqrt {c x^{2} + a} a c d^{2} x^{2} e^{3} + 4 \, \sqrt {c x^{2} + a} a c d^{3} x e^{2} + 2 \, \sqrt {c x^{2} + a} a c d^{4} e + \sqrt {c x^{2} + a} a^{2} x^{2} e^{5} + 2 \, \sqrt {c x^{2} + a} a^{2} d x e^{4} + \sqrt {c x^{2} + a} a^{2} d^{2} e^{3}\right )}} + \frac {4 \, c}{3 \, {\left (\sqrt {c x^{2} + a} c^{2} d^{5} e^{\left (-1\right )} + \sqrt {c x^{2} + a} c^{2} d^{4} x + 2 \, \sqrt {c x^{2} + a} a c d^{2} x e^{2} + 2 \, \sqrt {c x^{2} + a} a c d^{3} e + \sqrt {c x^{2} + a} a^{2} x e^{4} + \sqrt {c x^{2} + a} a^{2} d e^{3}\right )}} - \frac {1}{3 \, {\left (\sqrt {c x^{2} + a} c d^{2} x^{3} e^{2} + 3 \, \sqrt {c x^{2} + a} c d^{3} x^{2} e + \sqrt {c x^{2} + a} c d^{5} e^{\left (-1\right )} + 3 \, \sqrt {c x^{2} + a} c d^{4} x + \sqrt {c x^{2} + a} a x^{3} e^{4} + 3 \, \sqrt {c x^{2} + a} a d x^{2} e^{3} + 3 \, \sqrt {c x^{2} + a} a d^{2} x e^{2} + \sqrt {c x^{2} + a} a d^{3} e\right )}} \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. 1101 vs.
\(2 (269) = 538\).
time = 8.40, size = 2228, normalized size = 7.60 \begin {gather*} \text {Too large to display} \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}{\left (a + c x^{2}\right )^{\frac {3}{2}} \left (d + e x\right )^{4}}\, 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. 1025 vs.
\(2 (269) = 538\).
time = 0.88, size = 1025, normalized size = 3.50 \begin {gather*} \frac {\frac {{\left (c^{8} d^{12} - 2 \, a c^{7} d^{10} e^{2} - 17 \, a^{2} c^{6} d^{8} e^{4} - 28 \, a^{3} c^{5} d^{6} e^{6} - 17 \, a^{4} c^{4} d^{4} e^{8} - 2 \, a^{5} c^{3} d^{2} e^{10} + a^{6} c^{2} e^{12}\right )} x}{a c^{8} d^{16} + 8 \, a^{2} c^{7} d^{14} e^{2} + 28 \, a^{3} c^{6} d^{12} e^{4} + 56 \, a^{4} c^{5} d^{10} e^{6} + 70 \, a^{5} c^{4} d^{8} e^{8} + 56 \, a^{6} c^{3} d^{6} e^{10} + 28 \, a^{7} c^{2} d^{4} e^{12} + 8 \, a^{8} c d^{2} e^{14} + a^{9} e^{16}} + \frac {4 \, {\left (a c^{7} d^{11} e + 3 \, a^{2} c^{6} d^{9} e^{3} + 2 \, a^{3} c^{5} d^{7} e^{5} - 2 \, a^{4} c^{4} d^{5} e^{7} - 3 \, a^{5} c^{3} d^{3} e^{9} - a^{6} c^{2} d e^{11}\right )}}{a c^{8} d^{16} + 8 \, a^{2} c^{7} d^{14} e^{2} + 28 \, a^{3} c^{6} d^{12} e^{4} + 56 \, a^{4} c^{5} d^{10} e^{6} + 70 \, a^{5} c^{4} d^{8} e^{8} + 56 \, a^{6} c^{3} d^{6} e^{10} + 28 \, a^{7} c^{2} d^{4} e^{12} + 8 \, a^{8} c d^{2} e^{14} + a^{9} e^{16}}}{\sqrt {c x^{2} + a}} + \frac {5 \, {\left (4 \, c^{3} d^{3} e^{2} - 3 \, a c^{2} d e^{4}\right )} \arctan \left (-\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} - a e^{2}}}\right )}{{\left (c^{4} d^{8} + 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} + 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} \sqrt {-c d^{2} - a e^{2}}} - \frac {188 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} c^{4} d^{5} e^{2} + 162 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} c^{\frac {7}{2}} d^{4} e^{3} + 36 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{5} c^{3} d^{3} e^{4} - 402 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a c^{\frac {7}{2}} d^{4} e^{3} - 322 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} a c^{3} d^{3} e^{4} - 117 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} a c^{\frac {5}{2}} d^{2} e^{5} - 21 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{5} a c^{2} d e^{6} + 246 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a^{2} c^{3} d^{3} e^{4} + 144 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a^{2} c^{\frac {5}{2}} d^{2} e^{5} + 60 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} a^{2} c^{2} d e^{6} + 6 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} a^{2} c^{\frac {3}{2}} e^{7} - 47 \, a^{3} c^{\frac {5}{2}} d^{2} e^{5} - 39 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a^{3} c^{2} d e^{6} - 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a^{3} c^{\frac {3}{2}} e^{7} + 10 \, a^{4} c^{\frac {3}{2}} e^{7}}{3 \, {\left (c^{4} d^{8} + 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} + 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} e + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} \sqrt {c} d - a e\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 {1}{{\left (c\,x^2+a\right )}^{3/2}\,{\left (d+e\,x\right )}^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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