Optimal. Leaf size=194 \[ -\frac {e}{a d^2 \sqrt {a+c x^2}}-\frac {1}{a d x \sqrt {a+c x^2}}-\frac {2 c x}{a^2 d \sqrt {a+c x^2}}+\frac {e^2 (a e+c d x)}{a d^2 \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}-\frac {e^4 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{d^2 \left (c d^2+a e^2\right )^{3/2}}+\frac {e \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{a^{3/2} d^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.11, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 11, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {975, 277,
197, 272, 53, 65, 214, 755, 12, 739, 212} \begin {gather*} \frac {e \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{a^{3/2} d^2}-\frac {2 c x}{a^2 d \sqrt {a+c x^2}}+\frac {e^2 (a e+c d x)}{a d^2 \sqrt {a+c x^2} \left (a e^2+c d^2\right )}-\frac {e^4 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{d^2 \left (a e^2+c d^2\right )^{3/2}}-\frac {e}{a d^2 \sqrt {a+c x^2}}-\frac {1}{a d x \sqrt {a+c x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 53
Rule 65
Rule 197
Rule 212
Rule 214
Rule 272
Rule 277
Rule 739
Rule 755
Rule 975
Rubi steps
\begin {align*} \int \frac {1}{x^2 (d+e x) \left (a+c x^2\right )^{3/2}} \, dx &=\int \left (\frac {1}{d x^2 \left (a+c x^2\right )^{3/2}}-\frac {e}{d^2 x \left (a+c x^2\right )^{3/2}}+\frac {e^2}{d^2 (d+e x) \left (a+c x^2\right )^{3/2}}\right ) \, dx\\ &=\frac {\int \frac {1}{x^2 \left (a+c x^2\right )^{3/2}} \, dx}{d}-\frac {e \int \frac {1}{x \left (a+c x^2\right )^{3/2}} \, dx}{d^2}+\frac {e^2 \int \frac {1}{(d+e x) \left (a+c x^2\right )^{3/2}} \, dx}{d^2}\\ &=-\frac {1}{a d x \sqrt {a+c x^2}}+\frac {e^2 (a e+c d x)}{a d^2 \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}-\frac {(2 c) \int \frac {1}{\left (a+c x^2\right )^{3/2}} \, dx}{a d}-\frac {e \text {Subst}\left (\int \frac {1}{x (a+c x)^{3/2}} \, dx,x,x^2\right )}{2 d^2}+\frac {e^2 \int \frac {a e^2}{(d+e x) \sqrt {a+c x^2}} \, dx}{a d^2 \left (c d^2+a e^2\right )}\\ &=-\frac {e}{a d^2 \sqrt {a+c x^2}}-\frac {1}{a d x \sqrt {a+c x^2}}-\frac {2 c x}{a^2 d \sqrt {a+c x^2}}+\frac {e^2 (a e+c d x)}{a d^2 \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}-\frac {e \text {Subst}\left (\int \frac {1}{x \sqrt {a+c x}} \, dx,x,x^2\right )}{2 a d^2}+\frac {e^4 \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{d^2 \left (c d^2+a e^2\right )}\\ &=-\frac {e}{a d^2 \sqrt {a+c x^2}}-\frac {1}{a d x \sqrt {a+c x^2}}-\frac {2 c x}{a^2 d \sqrt {a+c x^2}}+\frac {e^2 (a e+c d x)}{a d^2 \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}-\frac {e \text {Subst}\left (\int \frac {1}{-\frac {a}{c}+\frac {x^2}{c}} \, dx,x,\sqrt {a+c x^2}\right )}{a c d^2}-\frac {e^4 \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 )}{d^2 \left (c d^2+a e^2\right )}\\ &=-\frac {e}{a d^2 \sqrt {a+c x^2}}-\frac {1}{a d x \sqrt {a+c x^2}}-\frac {2 c x}{a^2 d \sqrt {a+c x^2}}+\frac {e^2 (a e+c d x)}{a d^2 \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}-\frac {e^4 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{d^2 \left (c d^2+a e^2\right )^{3/2}}+\frac {e \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{a^{3/2} d^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.71, size = 178, normalized size = 0.92 \begin {gather*} \frac {-\frac {d \left (a^2 e^2+2 c^2 d^2 x^2+a c \left (d^2+d e x+e^2 x^2\right )\right )}{a^2 \left (c d^2+a e^2\right ) x \sqrt {a+c x^2}}+\frac {2 e^4 \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 )^{3/2}}-\frac {2 e \tanh ^{-1}\left (\frac {\sqrt {c} x-\sqrt {a+c x^2}}{\sqrt {a}}\right )}{a^{3/2}}}{d^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(402\) vs.
\(2(174)=348\).
time = 0.10, size = 403, normalized size = 2.08
method | result | size |
risch | \(-\frac {\sqrt {c \,x^{2}+a}}{a^{2} d x}+\frac {c \,e^{3} \ln \left (\frac {\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{d^{2} \left (\sqrt {-a c}\, e +c d \right ) \left (\sqrt {-a c}\, e -c d \right ) \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}+\frac {e \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {c \,x^{2}+a}}{x}\right )}{d^{2} a^{\frac {3}{2}}}+\frac {c \sqrt {c \left (x +\frac {\sqrt {-a c}}{c}\right )^{2}-2 \sqrt {-a c}\, \left (x +\frac {\sqrt {-a c}}{c}\right )}}{2 a^{2} \left (\sqrt {-a c}\, e -c d \right ) \left (x +\frac {\sqrt {-a c}}{c}\right )}-\frac {c \sqrt {c \left (x -\frac {\sqrt {-a c}}{c}\right )^{2}+2 \sqrt {-a c}\, \left (x -\frac {\sqrt {-a c}}{c}\right )}}{2 a^{2} \left (\sqrt {-a c}\, e +c d \right ) \left (x -\frac {\sqrt {-a c}}{c}\right )}\) | \(361\) |
default | \(\frac {e \left (\frac {e^{2}}{\left (a \,e^{2}+c \,d^{2}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}+\frac {2 c d e \left (2 c \left (x +\frac {d}{e}\right )-\frac {2 c d}{e}\right )}{\left (a \,e^{2}+c \,d^{2}\right ) \left (\frac {4 c \left (a \,e^{2}+c \,d^{2}\right )}{e^{2}}-\frac {4 c^{2} d^{2}}{e^{2}}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}-\frac {e^{2} \ln \left (\frac {\frac {2 a \,e^{2}+2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (a \,e^{2}+c \,d^{2}\right ) \sqrt {\frac {a \,e^{2}+c \,d^{2}}{e^{2}}}}\right )}{d^{2}}+\frac {-\frac {1}{a x \sqrt {c \,x^{2}+a}}-\frac {2 c x}{a^{2} \sqrt {c \,x^{2}+a}}}{d}-\frac {e \left (\frac {1}{a \sqrt {c \,x^{2}+a}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {c \,x^{2}+a}}{x}\right )}{a^{\frac {3}{2}}}\right )}{d^{2}}\) | \(403\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 363 vs.
\(2 (174) = 348\).
time = 3.82, size = 1527, normalized size = 7.87 \begin {gather*} \left [\frac {{\left (a^{2} c x^{3} + a^{3} x\right )} \sqrt {c d^{2} + a e^{2}} e^{4} \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 ) + {\left ({\left (a^{2} c x^{3} + a^{3} x\right )} e^{5} + 2 \, {\left (a c^{2} d^{2} x^{3} + a^{2} c d^{2} x\right )} e^{3} + {\left (c^{3} d^{4} x^{3} + a c^{2} d^{4} x\right )} e\right )} \sqrt {a} \log \left (-\frac {c x^{2} + 2 \, \sqrt {c x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (2 \, c^{3} d^{5} x^{2} + a c^{2} d^{4} x e + a c^{2} d^{5} + a^{2} c d^{2} x e^{3} + {\left (a^{2} c d x^{2} + a^{3} d\right )} e^{4} + {\left (3 \, a c^{2} d^{3} x^{2} + 2 \, a^{2} c d^{3}\right )} e^{2}\right )} \sqrt {c x^{2} + a}}{2 \, {\left (a^{2} c^{3} d^{6} x^{3} + a^{3} c^{2} d^{6} x + {\left (a^{4} c d^{2} x^{3} + a^{5} d^{2} x\right )} e^{4} + 2 \, {\left (a^{3} c^{2} d^{4} x^{3} + a^{4} c d^{4} x\right )} e^{2}\right )}}, \frac {2 \, {\left (a^{2} c x^{3} + a^{3} x\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 ) e^{4} + {\left ({\left (a^{2} c x^{3} + a^{3} x\right )} e^{5} + 2 \, {\left (a c^{2} d^{2} x^{3} + a^{2} c d^{2} x\right )} e^{3} + {\left (c^{3} d^{4} x^{3} + a c^{2} d^{4} x\right )} e\right )} \sqrt {a} \log \left (-\frac {c x^{2} + 2 \, \sqrt {c x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (2 \, c^{3} d^{5} x^{2} + a c^{2} d^{4} x e + a c^{2} d^{5} + a^{2} c d^{2} x e^{3} + {\left (a^{2} c d x^{2} + a^{3} d\right )} e^{4} + {\left (3 \, a c^{2} d^{3} x^{2} + 2 \, a^{2} c d^{3}\right )} e^{2}\right )} \sqrt {c x^{2} + a}}{2 \, {\left (a^{2} c^{3} d^{6} x^{3} + a^{3} c^{2} d^{6} x + {\left (a^{4} c d^{2} x^{3} + a^{5} d^{2} x\right )} e^{4} + 2 \, {\left (a^{3} c^{2} d^{4} x^{3} + a^{4} c d^{4} x\right )} e^{2}\right )}}, \frac {{\left (a^{2} c x^{3} + a^{3} x\right )} \sqrt {c d^{2} + a e^{2}} e^{4} \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 ({\left (a^{2} c x^{3} + a^{3} x\right )} e^{5} + 2 \, {\left (a c^{2} d^{2} x^{3} + a^{2} c d^{2} x\right )} e^{3} + {\left (c^{3} d^{4} x^{3} + a c^{2} d^{4} x\right )} e\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x^{2} + a}}\right ) - 2 \, {\left (2 \, c^{3} d^{5} x^{2} + a c^{2} d^{4} x e + a c^{2} d^{5} + a^{2} c d^{2} x e^{3} + {\left (a^{2} c d x^{2} + a^{3} d\right )} e^{4} + {\left (3 \, a c^{2} d^{3} x^{2} + 2 \, a^{2} c d^{3}\right )} e^{2}\right )} \sqrt {c x^{2} + a}}{2 \, {\left (a^{2} c^{3} d^{6} x^{3} + a^{3} c^{2} d^{6} x + {\left (a^{4} c d^{2} x^{3} + a^{5} d^{2} x\right )} e^{4} + 2 \, {\left (a^{3} c^{2} d^{4} x^{3} + a^{4} c d^{4} x\right )} e^{2}\right )}}, \frac {{\left (a^{2} c x^{3} + a^{3} x\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 ) e^{4} - {\left ({\left (a^{2} c x^{3} + a^{3} x\right )} e^{5} + 2 \, {\left (a c^{2} d^{2} x^{3} + a^{2} c d^{2} x\right )} e^{3} + {\left (c^{3} d^{4} x^{3} + a c^{2} d^{4} x\right )} e\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x^{2} + a}}\right ) - {\left (2 \, c^{3} d^{5} x^{2} + a c^{2} d^{4} x e + a c^{2} d^{5} + a^{2} c d^{2} x e^{3} + {\left (a^{2} c d x^{2} + a^{3} d\right )} e^{4} + {\left (3 \, a c^{2} d^{3} x^{2} + 2 \, a^{2} c d^{3}\right )} e^{2}\right )} \sqrt {c x^{2} + a}}{a^{2} c^{3} d^{6} x^{3} + a^{3} c^{2} d^{6} x + {\left (a^{4} c d^{2} x^{3} + a^{5} d^{2} x\right )} e^{4} + 2 \, {\left (a^{3} c^{2} d^{4} x^{3} + a^{4} c d^{4} x\right )} e^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{2} \left (a + c x^{2}\right )^{\frac {3}{2}} \left (d + e x\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.69, size = 266, normalized size = 1.37 \begin {gather*} -\frac {\frac {{\left (a c^{3} d^{3} + a^{2} c^{2} d e^{2}\right )} x}{a^{3} c^{2} d^{4} + 2 \, a^{4} c d^{2} e^{2} + a^{5} e^{4}} + \frac {a^{2} c^{2} d^{2} e + a^{3} c e^{3}}{a^{3} c^{2} d^{4} + 2 \, a^{4} c d^{2} e^{2} + a^{5} e^{4}}}{\sqrt {c x^{2} + a}} - \frac {2 \, \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 ) e^{4}}{{\left (c d^{4} + a d^{2} e^{2}\right )} \sqrt {-c d^{2} - a e^{2}}} - \frac {2 \, \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + a}}{\sqrt {-a}}\right ) e}{\sqrt {-a} a d^{2}} + \frac {2 \, \sqrt {c}}{{\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} - a\right )} a d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{x^2\,{\left (c\,x^2+a\right )}^{3/2}\,\left (d+e\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________