Optimal. Leaf size=193 \[ -\frac {1}{c d x}+\frac {\left (2 a^2 c d+b^3 e-a b (b d+3 c e)\right ) \tanh ^{-1}\left (\frac {b+2 a x}{\sqrt {b^2-4 a c}}\right )}{c^2 \sqrt {b^2-4 a c} \left (a d^2-e (b d-c e)\right )}-\frac {(b d+c e) \log (x)}{c^2 d^2}+\frac {e^3 \log (d+e x)}{d^2 \left (a d^2-e (b d-c e)\right )}+\frac {\left (a b d-b^2 e+a c e\right ) \log \left (c+b x+a x^2\right )}{2 c^2 \left (a d^2-e (b d-c e)\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.23, antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {1583, 907, 648,
632, 212, 642} \begin {gather*} \frac {\left (2 a^2 c d-a b (b d+3 c e)+b^3 e\right ) \tanh ^{-1}\left (\frac {2 a x+b}{\sqrt {b^2-4 a c}}\right )}{c^2 \sqrt {b^2-4 a c} \left (a d^2-e (b d-c e)\right )}+\frac {\left (a b d+a c e+b^2 (-e)\right ) \log \left (a x^2+b x+c\right )}{2 c^2 \left (a d^2-e (b d-c e)\right )}+\frac {e^3 \log (d+e x)}{d^2 \left (a d^2-e (b d-c e)\right )}-\frac {\log (x) (b d+c e)}{c^2 d^2}-\frac {1}{c d x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 212
Rule 632
Rule 642
Rule 648
Rule 907
Rule 1583
Rubi steps
\begin {align*} \int \frac {1}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right ) x^4 (d+e x)} \, dx &=\int \frac {1}{x^2 (d+e x) \left (c+b x+a x^2\right )} \, dx\\ &=\int \left (\frac {1}{c d x^2}+\frac {-b d-c e}{c^2 d^2 x}+\frac {e^4}{d^2 \left (a d^2-e (b d-c e)\right ) (d+e x)}+\frac {-a^2 c d-b^3 e+a b (b d+2 c e)+a \left (a b d-b^2 e+a c e\right ) x}{c^2 \left (a d^2-e (b d-c e)\right ) \left (c+b x+a x^2\right )}\right ) \, dx\\ &=-\frac {1}{c d x}-\frac {(b d+c e) \log (x)}{c^2 d^2}+\frac {e^3 \log (d+e x)}{d^2 \left (a d^2-e (b d-c e)\right )}+\frac {\int \frac {-a^2 c d-b^3 e+a b (b d+2 c e)+a \left (a b d-b^2 e+a c e\right ) x}{c+b x+a x^2} \, dx}{c^2 \left (a d^2-e (b d-c e)\right )}\\ &=-\frac {1}{c d x}-\frac {(b d+c e) \log (x)}{c^2 d^2}+\frac {e^3 \log (d+e x)}{d^2 \left (a d^2-e (b d-c e)\right )}+\frac {\left (a b d-b^2 e+a c e\right ) \int \frac {b+2 a x}{c+b x+a x^2} \, dx}{2 c^2 \left (a d^2-e (b d-c e)\right )}-\frac {\left (2 a^2 c d+b^3 e-a b (b d+3 c e)\right ) \int \frac {1}{c+b x+a x^2} \, dx}{2 c^2 \left (a d^2-e (b d-c e)\right )}\\ &=-\frac {1}{c d x}-\frac {(b d+c e) \log (x)}{c^2 d^2}+\frac {e^3 \log (d+e x)}{d^2 \left (a d^2-e (b d-c e)\right )}+\frac {\left (a b d-b^2 e+a c e\right ) \log \left (c+b x+a x^2\right )}{2 c^2 \left (a d^2-e (b d-c e)\right )}+\frac {\left (2 a^2 c d+b^3 e-a b (b d+3 c e)\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 a x\right )}{c^2 \left (a d^2-e (b d-c e)\right )}\\ &=-\frac {1}{c d x}+\frac {\left (2 a^2 c d+b^3 e-a b (b d+3 c e)\right ) \tanh ^{-1}\left (\frac {b+2 a x}{\sqrt {b^2-4 a c}}\right )}{c^2 \sqrt {b^2-4 a c} \left (a d^2-e (b d-c e)\right )}-\frac {(b d+c e) \log (x)}{c^2 d^2}+\frac {e^3 \log (d+e x)}{d^2 \left (a d^2-e (b d-c e)\right )}+\frac {\left (a b d-b^2 e+a c e\right ) \log \left (c+b x+a x^2\right )}{2 c^2 \left (a d^2-e (b d-c e)\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.12, size = 194, normalized size = 1.01 \begin {gather*} -\frac {1}{c d x}+\frac {\left (2 a^2 c d+b^3 e-a b (b d+3 c e)\right ) \tan ^{-1}\left (\frac {b+2 a x}{\sqrt {-b^2+4 a c}}\right )}{c^2 \sqrt {-b^2+4 a c} \left (-a d^2+e (b d-c e)\right )}-\frac {(b d+c e) \log (x)}{c^2 d^2}+\frac {e^3 \log (d+e x)}{a d^4+d^2 e (-b d+c e)}+\frac {\left (a b d-b^2 e+a c e\right ) \log (c+x (b+a x))}{2 c^2 \left (a d^2+e (-b d+c e)\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.28, size = 207, normalized size = 1.07
method | result | size |
default | \(\frac {e^{3} \ln \left (e x +d \right )}{d^{2} \left (a \,d^{2}-d e b +c \,e^{2}\right )}+\frac {\frac {\left (a^{2} b d +a^{2} c e -a \,b^{2} e \right ) \ln \left (a \,x^{2}+b x +c \right )}{2 a}+\frac {2 \left (-a^{2} c d +a \,b^{2} d +2 a b c e -b^{3} e -\frac {\left (a^{2} b d +a^{2} c e -a \,b^{2} e \right ) b}{2 a}\right ) \arctan \left (\frac {2 a x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{\left (a \,d^{2}-d e b +c \,e^{2}\right ) c^{2}}-\frac {1}{c d x}+\frac {\left (-b d -c e \right ) \ln \left (x \right )}{c^{2} d^{2}}\) | \(207\) |
risch | \(\text {Expression too large to display}\) | \(101857\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 4.51, size = 210, normalized size = 1.09 \begin {gather*} \frac {{\left (a b d - b^{2} e + a c e\right )} \log \left (a x^{2} + b x + c\right )}{2 \, {\left (a c^{2} d^{2} - b c^{2} d e + c^{3} e^{2}\right )}} + \frac {e^{4} \log \left ({\left | x e + d \right |}\right )}{a d^{4} e - b d^{3} e^{2} + c d^{2} e^{3}} + \frac {{\left (a b^{2} d - 2 \, a^{2} c d - b^{3} e + 3 \, a b c e\right )} \arctan \left (\frac {2 \, a x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (a c^{2} d^{2} - b c^{2} d e + c^{3} e^{2}\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {{\left (b d + c e\right )} \log \left ({\left | x \right |}\right )}{c^{2} d^{2}} - \frac {1}{c d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 20.39, size = 2388, normalized size = 12.37 \begin {gather*} \frac {e^3\,\ln \left (d+e\,x\right )}{a\,d^4-b\,d^3\,e+c\,d^2\,e^2}+\frac {\ln \left (\frac {a^4\,e^4\,x}{c^2\,d^2}-\frac {\left (\frac {a\,e\,x\,\left (a^4\,d^4+2\,a^3\,c\,d^2\,e^2+2\,a^2\,b\,c\,d\,e^3+2\,a^2\,c^2\,e^4-4\,a\,b^2\,c\,e^4+b^4\,e^4\right )}{c^2\,d^2}-\frac {\left (\frac {a\,e\,\left (-a^3\,c\,d^4+a^2\,b^2\,d^4+5\,a^2\,b\,c\,d^3\,e+4\,a^2\,c^2\,d^2\,e^2-2\,a\,b^3\,d^3\,e-5\,a\,b^2\,c\,d^2\,e^2-4\,a\,b\,c^2\,d\,e^3-4\,a\,c^3\,e^4+b^4\,d^2\,e^2+b^3\,c\,d\,e^3+b^2\,c^2\,e^4\right )}{c\,d}+\frac {a\,e\,x\,\left (2\,a^3\,b\,d^4+a^3\,c\,d^3\,e-2\,a^2\,b^2\,d^3\,e+8\,a^2\,b\,c\,d^2\,e^2+12\,a^2\,c^2\,d\,e^3-2\,a\,b^3\,d^2\,e^2-11\,a\,b^2\,c\,d\,e^3-8\,a\,b\,c^2\,e^4+2\,b^4\,d\,e^3+2\,b^3\,c\,e^4\right )}{c\,d}+\frac {a\,e\,\left (b^4\,e+b^3\,e\,\sqrt {b^2-4\,a\,c}+4\,a^2\,c^2\,e-a\,b^3\,d+4\,a^2\,b\,c\,d-5\,a\,b^2\,c\,e-a\,b^2\,d\,\sqrt {b^2-4\,a\,c}+2\,a^2\,c\,d\,\sqrt {b^2-4\,a\,c}-3\,a\,b\,c\,e\,\sqrt {b^2-4\,a\,c}\right )\,\left (-6\,x\,a^3\,c\,d^4+2\,x\,a^2\,b^2\,d^4+a^2\,b\,c\,d^4+14\,x\,a^2\,b\,c\,d^3\,e+4\,a^2\,c^2\,d^3\,e-6\,x\,a^2\,c^2\,d^2\,e^2-4\,x\,a\,b^3\,d^3\,e-2\,a\,b^2\,c\,d^3\,e-6\,x\,a\,b^2\,c\,d^2\,e^2-3\,a\,b\,c^2\,d^2\,e^2+8\,x\,a\,b\,c^2\,d\,e^3-4\,a\,c^3\,d\,e^3-8\,x\,a\,c^3\,e^4+2\,x\,b^4\,d^2\,e^2+b^3\,c\,d^2\,e^2-2\,x\,b^3\,c\,d\,e^3+b^2\,c^2\,d\,e^3+2\,x\,b^2\,c^2\,e^4\right )}{2\,c^2\,\left (4\,a\,c-b^2\right )\,\left (a\,d^2-b\,d\,e+c\,e^2\right )}\right )\,\left (b^4\,e+b^3\,e\,\sqrt {b^2-4\,a\,c}+4\,a^2\,c^2\,e-a\,b^3\,d+4\,a^2\,b\,c\,d-5\,a\,b^2\,c\,e-a\,b^2\,d\,\sqrt {b^2-4\,a\,c}+2\,a^2\,c\,d\,\sqrt {b^2-4\,a\,c}-3\,a\,b\,c\,e\,\sqrt {b^2-4\,a\,c}\right )}{2\,c^2\,\left (4\,a\,c-b^2\right )\,\left (a\,d^2-b\,d\,e+c\,e^2\right )}+\frac {a\,e\,\left (b\,d+c\,e\right )\,\left (a^3\,d^3-3\,c\,a\,b\,e^3+b^3\,e^3\right )}{c^2\,d^2}\right )\,\left (b^4\,e+b^3\,e\,\sqrt {b^2-4\,a\,c}+4\,a^2\,c^2\,e-a\,b^3\,d+4\,a^2\,b\,c\,d-5\,a\,b^2\,c\,e-a\,b^2\,d\,\sqrt {b^2-4\,a\,c}+2\,a^2\,c\,d\,\sqrt {b^2-4\,a\,c}-3\,a\,b\,c\,e\,\sqrt {b^2-4\,a\,c}\right )}{2\,c^2\,\left (4\,a\,c-b^2\right )\,\left (a\,d^2-b\,d\,e+c\,e^2\right )}\right )\,\left (b^4\,e+b^3\,e\,\sqrt {b^2-4\,a\,c}+4\,a^2\,c^2\,e-a\,b^3\,d+4\,a^2\,b\,c\,d-5\,a\,b^2\,c\,e-a\,b^2\,d\,\sqrt {b^2-4\,a\,c}+2\,a^2\,c\,d\,\sqrt {b^2-4\,a\,c}-3\,a\,b\,c\,e\,\sqrt {b^2-4\,a\,c}\right )}{2\,\left (4\,a^2\,c^3\,d^2-a\,b^2\,c^2\,d^2-4\,a\,b\,c^3\,d\,e+4\,a\,c^4\,e^2+b^3\,c^2\,d\,e-b^2\,c^3\,e^2\right )}+\frac {\ln \left (\frac {a^4\,e^4\,x}{c^2\,d^2}-\frac {\left (\frac {a\,e\,x\,\left (a^4\,d^4+2\,a^3\,c\,d^2\,e^2+2\,a^2\,b\,c\,d\,e^3+2\,a^2\,c^2\,e^4-4\,a\,b^2\,c\,e^4+b^4\,e^4\right )}{c^2\,d^2}-\frac {\left (\frac {a\,e\,\left (-a^3\,c\,d^4+a^2\,b^2\,d^4+5\,a^2\,b\,c\,d^3\,e+4\,a^2\,c^2\,d^2\,e^2-2\,a\,b^3\,d^3\,e-5\,a\,b^2\,c\,d^2\,e^2-4\,a\,b\,c^2\,d\,e^3-4\,a\,c^3\,e^4+b^4\,d^2\,e^2+b^3\,c\,d\,e^3+b^2\,c^2\,e^4\right )}{c\,d}+\frac {a\,e\,x\,\left (2\,a^3\,b\,d^4+a^3\,c\,d^3\,e-2\,a^2\,b^2\,d^3\,e+8\,a^2\,b\,c\,d^2\,e^2+12\,a^2\,c^2\,d\,e^3-2\,a\,b^3\,d^2\,e^2-11\,a\,b^2\,c\,d\,e^3-8\,a\,b\,c^2\,e^4+2\,b^4\,d\,e^3+2\,b^3\,c\,e^4\right )}{c\,d}+\frac {a\,e\,\left (b^4\,e-b^3\,e\,\sqrt {b^2-4\,a\,c}+4\,a^2\,c^2\,e-a\,b^3\,d+4\,a^2\,b\,c\,d-5\,a\,b^2\,c\,e+a\,b^2\,d\,\sqrt {b^2-4\,a\,c}-2\,a^2\,c\,d\,\sqrt {b^2-4\,a\,c}+3\,a\,b\,c\,e\,\sqrt {b^2-4\,a\,c}\right )\,\left (-6\,x\,a^3\,c\,d^4+2\,x\,a^2\,b^2\,d^4+a^2\,b\,c\,d^4+14\,x\,a^2\,b\,c\,d^3\,e+4\,a^2\,c^2\,d^3\,e-6\,x\,a^2\,c^2\,d^2\,e^2-4\,x\,a\,b^3\,d^3\,e-2\,a\,b^2\,c\,d^3\,e-6\,x\,a\,b^2\,c\,d^2\,e^2-3\,a\,b\,c^2\,d^2\,e^2+8\,x\,a\,b\,c^2\,d\,e^3-4\,a\,c^3\,d\,e^3-8\,x\,a\,c^3\,e^4+2\,x\,b^4\,d^2\,e^2+b^3\,c\,d^2\,e^2-2\,x\,b^3\,c\,d\,e^3+b^2\,c^2\,d\,e^3+2\,x\,b^2\,c^2\,e^4\right )}{2\,c^2\,\left (4\,a\,c-b^2\right )\,\left (a\,d^2-b\,d\,e+c\,e^2\right )}\right )\,\left (b^4\,e-b^3\,e\,\sqrt {b^2-4\,a\,c}+4\,a^2\,c^2\,e-a\,b^3\,d+4\,a^2\,b\,c\,d-5\,a\,b^2\,c\,e+a\,b^2\,d\,\sqrt {b^2-4\,a\,c}-2\,a^2\,c\,d\,\sqrt {b^2-4\,a\,c}+3\,a\,b\,c\,e\,\sqrt {b^2-4\,a\,c}\right )}{2\,c^2\,\left (4\,a\,c-b^2\right )\,\left (a\,d^2-b\,d\,e+c\,e^2\right )}+\frac {a\,e\,\left (b\,d+c\,e\right )\,\left (a^3\,d^3-3\,c\,a\,b\,e^3+b^3\,e^3\right )}{c^2\,d^2}\right )\,\left (b^4\,e-b^3\,e\,\sqrt {b^2-4\,a\,c}+4\,a^2\,c^2\,e-a\,b^3\,d+4\,a^2\,b\,c\,d-5\,a\,b^2\,c\,e+a\,b^2\,d\,\sqrt {b^2-4\,a\,c}-2\,a^2\,c\,d\,\sqrt {b^2-4\,a\,c}+3\,a\,b\,c\,e\,\sqrt {b^2-4\,a\,c}\right )}{2\,c^2\,\left (4\,a\,c-b^2\right )\,\left (a\,d^2-b\,d\,e+c\,e^2\right )}\right )\,\left (b^4\,e-b^3\,e\,\sqrt {b^2-4\,a\,c}+4\,a^2\,c^2\,e-a\,b^3\,d+4\,a^2\,b\,c\,d-5\,a\,b^2\,c\,e+a\,b^2\,d\,\sqrt {b^2-4\,a\,c}-2\,a^2\,c\,d\,\sqrt {b^2-4\,a\,c}+3\,a\,b\,c\,e\,\sqrt {b^2-4\,a\,c}\right )}{2\,\left (4\,a^2\,c^3\,d^2-a\,b^2\,c^2\,d^2-4\,a\,b\,c^3\,d\,e+4\,a\,c^4\,e^2+b^3\,c^2\,d\,e-b^2\,c^3\,e^2\right )}-\frac {1}{c\,d\,x}-\frac {\ln \left (x\right )\,\left (b\,d+c\,e\right )}{c^2\,d^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________