Optimal. Leaf size=145 \[ -\frac {e \sqrt {a+c x^2}}{2 \left (c d^2+a e^2\right ) (d+e x)^2}-\frac {3 c d e \sqrt {a+c x^2}}{2 \left (c d^2+a e^2\right )^2 (d+e x)}-\frac {c \left (2 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{2 \left (c d^2+a e^2\right )^{5/2}} \]
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Rubi [A]
time = 0.04, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {759, 821, 739,
212} \begin {gather*} -\frac {3 c d e \sqrt {a+c x^2}}{2 (d+e x) \left (a e^2+c d^2\right )^2}-\frac {e \sqrt {a+c x^2}}{2 (d+e x)^2 \left (a e^2+c d^2\right )}-\frac {c \left (2 c d^2-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 )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 739
Rule 759
Rule 821
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^3 \sqrt {a+c x^2}} \, dx &=-\frac {e \sqrt {a+c x^2}}{2 \left (c d^2+a e^2\right ) (d+e x)^2}-\frac {c \int \frac {-2 d+e x}{(d+e x)^2 \sqrt {a+c x^2}} \, dx}{2 \left (c d^2+a e^2\right )}\\ &=-\frac {e \sqrt {a+c x^2}}{2 \left (c d^2+a e^2\right ) (d+e x)^2}-\frac {3 c d e \sqrt {a+c x^2}}{2 \left (c d^2+a e^2\right )^2 (d+e x)}+\frac {\left (c \left (2 c d^2-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 )^2}\\ &=-\frac {e \sqrt {a+c x^2}}{2 \left (c d^2+a e^2\right ) (d+e x)^2}-\frac {3 c d e \sqrt {a+c x^2}}{2 \left (c d^2+a e^2\right )^2 (d+e x)}-\frac {\left (c \left (2 c d^2-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 )^2}\\ &=-\frac {e \sqrt {a+c x^2}}{2 \left (c d^2+a e^2\right ) (d+e x)^2}-\frac {3 c d e \sqrt {a+c x^2}}{2 \left (c d^2+a e^2\right )^2 (d+e x)}-\frac {c \left (2 c d^2-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 )^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.77, size = 132, normalized size = 0.91 \begin {gather*} -\frac {e \sqrt {a+c x^2} \left (a e^2+c d (4 d+3 e x)\right )}{2 \left (c d^2+a e^2\right )^2 (d+e x)^2}-\frac {c \left (2 c d^2-a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+c x^2}}{\sqrt {-c d^2-a e^2}}\right )}{\left (-c d^2-a e^2\right )^{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. \(443\) vs.
\(2(129)=258\).
time = 0.44, size = 444, normalized size = 3.06
method | result | size |
default | \(\frac {-\frac {e^{2} \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{2 \left (e^{2} a +c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{2}}+\frac {3 c d e \left (-\frac {e^{2} \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{\left (e^{2} a +c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}-\frac {c d e \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (e^{2} a +c \,d^{2}\right ) \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\right )}{2 \left (e^{2} a +c \,d^{2}\right )}+\frac {c \,e^{2} \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{2 \left (e^{2} a +c \,d^{2}\right ) \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}}{e^{3}}\) | \(444\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.35, size = 241, normalized size = 1.66 \begin {gather*} \frac {3 \, c^{2} d^{2} \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 {5}{2}}} - \frac {3 \, \sqrt {c x^{2} + a} c d}{2 \, {\left (c^{2} d^{5} e^{\left (-1\right )} + c^{2} d^{4} x + 2 \, a c d^{2} x e^{2} + 2 \, a c d^{3} e + a^{2} x e^{4} + a^{2} d e^{3}\right )}} - \frac {c \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 (-3\right )}}{2 \, {\left (c d^{2} e^{\left (-2\right )} + a\right )}^{\frac {3}{2}}} - \frac {\sqrt {c x^{2} + a}}{2 \, {\left (c d^{2} x^{2} e + c d^{4} e^{\left (-1\right )} + 2 \, c d^{3} x + a x^{2} e^{3} + 2 \, a d x e^{2} + a d^{2} 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. 334 vs.
\(2 (130) = 260\).
time = 2.04, size = 694, normalized size = 4.79 \begin {gather*} \left [-\frac {{\left (4 \, c^{2} d^{3} x e + 2 \, c^{2} d^{4} - a c x^{2} e^{4} - 2 \, a c d x e^{3} + {\left (2 \, c^{2} d^{2} x^{2} - a c d^{2}\right )} e^{2}\right )} \sqrt {c d^{2} + a e^{2}} \log \left (-\frac {2 \, c^{2} d^{2} x^{2} - 2 \, a c d x e + a c d^{2} - 2 \, \sqrt {c d^{2} + a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a} + {\left (a c x^{2} + 2 \, a^{2}\right )} e^{2}}{x^{2} e^{2} + 2 \, d x e + d^{2}}\right ) + 2 \, {\left (3 \, c^{2} d^{3} x e^{2} + 4 \, c^{2} d^{4} e + 3 \, a c d x e^{4} + 5 \, a c d^{2} e^{3} + a^{2} e^{5}\right )} \sqrt {c x^{2} + a}}{4 \, {\left (2 \, c^{3} d^{7} x e + c^{3} d^{8} + 6 \, a c^{2} d^{5} x e^{3} + 6 \, a^{2} c d^{3} x e^{5} + a^{3} x^{2} e^{8} + 2 \, a^{3} d x e^{7} + {\left (3 \, a^{2} c d^{2} x^{2} + a^{3} d^{2}\right )} e^{6} + 3 \, {\left (a c^{2} d^{4} x^{2} + a^{2} c d^{4}\right )} e^{4} + {\left (c^{3} d^{6} x^{2} + 3 \, a c^{2} d^{6}\right )} e^{2}\right )}}, \frac {{\left (4 \, c^{2} d^{3} x e + 2 \, c^{2} d^{4} - a c x^{2} e^{4} - 2 \, a c d x e^{3} + {\left (2 \, c^{2} d^{2} x^{2} - a c d^{2}\right )} e^{2}\right )} \sqrt {-c d^{2} - a e^{2}} \arctan \left (-\frac {\sqrt {-c d^{2} - a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{c^{2} d^{2} x^{2} + a c d^{2} + {\left (a c x^{2} + a^{2}\right )} e^{2}}\right ) - {\left (3 \, c^{2} d^{3} x e^{2} + 4 \, c^{2} d^{4} e + 3 \, a c d x e^{4} + 5 \, a c d^{2} e^{3} + a^{2} e^{5}\right )} \sqrt {c x^{2} + a}}{2 \, {\left (2 \, c^{3} d^{7} x e + c^{3} d^{8} + 6 \, a c^{2} d^{5} x e^{3} + 6 \, a^{2} c d^{3} x e^{5} + a^{3} x^{2} e^{8} + 2 \, a^{3} d x e^{7} + {\left (3 \, a^{2} c d^{2} x^{2} + a^{3} d^{2}\right )} e^{6} + 3 \, {\left (a c^{2} d^{4} x^{2} + a^{2} c d^{4}\right )} e^{4} + {\left (c^{3} d^{6} x^{2} + 3 \, a c^{2} d^{6}\right )} e^{2}\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}{\sqrt {a + c x^{2}} \left (d + e x\right )^{3}}\, 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. 345 vs.
\(2 (130) = 260\).
time = 0.83, size = 345, normalized size = 2.38 \begin {gather*} -c {\left (\frac {{\left (2 \, c d^{2} - a e^{2}\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^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt {-c d^{2} - a e^{2}}} + \frac {2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} c d^{2} e + 6 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} c^{\frac {3}{2}} d^{3} - 10 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a c d^{2} e - 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a \sqrt {c} d e^{2} - {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} a e^{3} + 3 \, a^{2} \sqrt {c} d e^{2} - {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a^{2} e^{3}}{{\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\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 )}^{2}}\right )} \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 {1}{\sqrt {c\,x^2+a}\,{\left (d+e\,x\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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