Optimal. Leaf size=338 \[ -\frac {2 c \text {Li}_2\left (-\frac {2 c f^{c+d x}}{b-\sqrt {b^2-4 a c}}\right )}{d^2 \log ^2(f) \sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )}+\frac {2 c \text {Li}_2\left (-\frac {2 c f^{c+d x}}{b+\sqrt {b^2-4 a c}}\right )}{d^2 \log ^2(f) \sqrt {b^2-4 a c} \left (\sqrt {b^2-4 a c}+b\right )}-\frac {2 c x \log \left (\frac {2 c f^{c+d x}}{b-\sqrt {b^2-4 a c}}+1\right )}{d \log (f) \sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )}+\frac {2 c x \log \left (\frac {2 c f^{c+d x}}{\sqrt {b^2-4 a c}+b}+1\right )}{d \log (f) \sqrt {b^2-4 a c} \left (\sqrt {b^2-4 a c}+b\right )}-\frac {c x^2}{-b \sqrt {b^2-4 a c}-4 a c+b^2}-\frac {c x^2}{b \sqrt {b^2-4 a c}-4 a c+b^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.69, antiderivative size = 338, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2263, 2184, 2190, 2279, 2391} \[ -\frac {2 c \text {PolyLog}\left (2,-\frac {2 c f^{c+d x}}{b-\sqrt {b^2-4 a c}}\right )}{d^2 \log ^2(f) \sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )}+\frac {2 c \text {PolyLog}\left (2,-\frac {2 c f^{c+d x}}{\sqrt {b^2-4 a c}+b}\right )}{d^2 \log ^2(f) \sqrt {b^2-4 a c} \left (\sqrt {b^2-4 a c}+b\right )}-\frac {2 c x \log \left (\frac {2 c f^{c+d x}}{b-\sqrt {b^2-4 a c}}+1\right )}{d \log (f) \sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )}+\frac {2 c x \log \left (\frac {2 c f^{c+d x}}{\sqrt {b^2-4 a c}+b}+1\right )}{d \log (f) \sqrt {b^2-4 a c} \left (\sqrt {b^2-4 a c}+b\right )}-\frac {c x^2}{-b \sqrt {b^2-4 a c}-4 a c+b^2}-\frac {c x^2}{b \sqrt {b^2-4 a c}-4 a c+b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2184
Rule 2190
Rule 2263
Rule 2279
Rule 2391
Rubi steps
\begin {align*} \int \frac {x}{a+b f^{c+d x}+c f^{2 c+2 d x}} \, dx &=\frac {(2 c) \int \frac {x}{b-\sqrt {b^2-4 a c}+2 c f^{c+d x}} \, dx}{\sqrt {b^2-4 a c}}-\frac {(2 c) \int \frac {x}{b+\sqrt {b^2-4 a c}+2 c f^{c+d x}} \, dx}{\sqrt {b^2-4 a c}}\\ &=-\frac {c x^2}{b^2-4 a c-b \sqrt {b^2-4 a c}}-\frac {c x^2}{b^2-4 a c+b \sqrt {b^2-4 a c}}+\frac {\left (4 c^2\right ) \int \frac {f^{c+d x} x}{b-\sqrt {b^2-4 a c}+2 c f^{c+d x}} \, dx}{b^2-4 a c-b \sqrt {b^2-4 a c}}+\frac {\left (4 c^2\right ) \int \frac {f^{c+d x} x}{b+\sqrt {b^2-4 a c}+2 c f^{c+d x}} \, dx}{b^2-4 a c+b \sqrt {b^2-4 a c}}\\ &=-\frac {c x^2}{b^2-4 a c-b \sqrt {b^2-4 a c}}-\frac {c x^2}{b^2-4 a c+b \sqrt {b^2-4 a c}}+\frac {2 c x \log \left (1+\frac {2 c f^{c+d x}}{b-\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c-b \sqrt {b^2-4 a c}\right ) d \log (f)}+\frac {2 c x \log \left (1+\frac {2 c f^{c+d x}}{b+\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c+b \sqrt {b^2-4 a c}\right ) d \log (f)}-\frac {(2 c) \int \log \left (1+\frac {2 c f^{c+d x}}{b-\sqrt {b^2-4 a c}}\right ) \, dx}{\left (b^2-4 a c-b \sqrt {b^2-4 a c}\right ) d \log (f)}-\frac {(2 c) \int \log \left (1+\frac {2 c f^{c+d x}}{b+\sqrt {b^2-4 a c}}\right ) \, dx}{\left (b^2-4 a c+b \sqrt {b^2-4 a c}\right ) d \log (f)}\\ &=-\frac {c x^2}{b^2-4 a c-b \sqrt {b^2-4 a c}}-\frac {c x^2}{b^2-4 a c+b \sqrt {b^2-4 a c}}+\frac {2 c x \log \left (1+\frac {2 c f^{c+d x}}{b-\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c-b \sqrt {b^2-4 a c}\right ) d \log (f)}+\frac {2 c x \log \left (1+\frac {2 c f^{c+d x}}{b+\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c+b \sqrt {b^2-4 a c}\right ) d \log (f)}-\frac {(2 c) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )}{x} \, dx,x,f^{c+d x}\right )}{\left (b^2-4 a c-b \sqrt {b^2-4 a c}\right ) d^2 \log ^2(f)}-\frac {(2 c) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{x} \, dx,x,f^{c+d x}\right )}{\left (b^2-4 a c+b \sqrt {b^2-4 a c}\right ) d^2 \log ^2(f)}\\ &=-\frac {c x^2}{b^2-4 a c-b \sqrt {b^2-4 a c}}-\frac {c x^2}{b^2-4 a c+b \sqrt {b^2-4 a c}}+\frac {2 c x \log \left (1+\frac {2 c f^{c+d x}}{b-\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c-b \sqrt {b^2-4 a c}\right ) d \log (f)}+\frac {2 c x \log \left (1+\frac {2 c f^{c+d x}}{b+\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c+b \sqrt {b^2-4 a c}\right ) d \log (f)}+\frac {2 c \text {Li}_2\left (-\frac {2 c f^{c+d x}}{b-\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c-b \sqrt {b^2-4 a c}\right ) d^2 \log ^2(f)}+\frac {2 c \text {Li}_2\left (-\frac {2 c f^{c+d x}}{b+\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c+b \sqrt {b^2-4 a c}\right ) d^2 \log ^2(f)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [F] time = 5.33, size = 0, normalized size = 0.00 \[ \int \frac {x}{a+b f^{c+d x}+c f^{2 c+2 d x}} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.45, size = 497, normalized size = 1.47 \[ \frac {{\left (b^{2} - 4 \, a c\right )} d^{2} x^{2} \log \relax (f)^{2} - {\left (a b \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} + b^{2} - 4 \, a c\right )} {\rm Li}_2\left (-\frac {{\left (a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} + b\right )} f^{d x + c} + 2 \, a}{2 \, a} + 1\right ) + {\left (a b \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} - b^{2} + 4 \, a c\right )} {\rm Li}_2\left (\frac {{\left (a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} - b\right )} f^{d x + c} - 2 \, a}{2 \, a} + 1\right ) - {\left (a b c \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} \log \relax (f) - {\left (b^{2} c - 4 \, a c^{2}\right )} \log \relax (f)\right )} \log \left (2 \, c f^{d x + c} + a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} + b\right ) + {\left (a b c \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} \log \relax (f) + {\left (b^{2} c - 4 \, a c^{2}\right )} \log \relax (f)\right )} \log \left (2 \, c f^{d x + c} - a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} + b\right ) - {\left ({\left (a b d x + a b c\right )} \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} \log \relax (f) + {\left (b^{2} c - 4 \, a c^{2} + {\left (b^{2} - 4 \, a c\right )} d x\right )} \log \relax (f)\right )} \log \left (\frac {{\left (a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} + b\right )} f^{d x + c} + 2 \, a}{2 \, a}\right ) + {\left ({\left (a b d x + a b c\right )} \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} \log \relax (f) - {\left (b^{2} c - 4 \, a c^{2} + {\left (b^{2} - 4 \, a c\right )} d x\right )} \log \relax (f)\right )} \log \left (-\frac {{\left (a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} - b\right )} f^{d x + c} - 2 \, a}{2 \, a}\right )}{2 \, {\left (a b^{2} - 4 \, a^{2} c\right )} d^{2} \log \relax (f)^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{c f^{2 \, d x + 2 \, c} + b f^{d x + c} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.10, size = 855, normalized size = 2.53 \[ \frac {x^{2}}{2 a}-\frac {b x \ln \left (\frac {-2 c \,f^{c} f^{d x}-b +\sqrt {-4 a c +b^{2}}}{-b +\sqrt {-4 a c +b^{2}}}\right )}{2 \sqrt {-4 a c +b^{2}}\, a d \ln \relax (f )}+\frac {b x \ln \left (\frac {2 c \,f^{c} f^{d x}+b +\sqrt {-4 a c +b^{2}}}{b +\sqrt {-4 a c +b^{2}}}\right )}{2 \sqrt {-4 a c +b^{2}}\, a d \ln \relax (f )}+\frac {c x}{a d}+\frac {b c \arctan \left (\frac {2 c \,f^{c} f^{d x}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, a \,d^{2} \ln \relax (f )}-\frac {b c \ln \left (\frac {-2 c \,f^{c} f^{d x}-b +\sqrt {-4 a c +b^{2}}}{-b +\sqrt {-4 a c +b^{2}}}\right )}{2 \sqrt {-4 a c +b^{2}}\, a \,d^{2} \ln \relax (f )}+\frac {b c \ln \left (\frac {2 c \,f^{c} f^{d x}+b +\sqrt {-4 a c +b^{2}}}{b +\sqrt {-4 a c +b^{2}}}\right )}{2 \sqrt {-4 a c +b^{2}}\, a \,d^{2} \ln \relax (f )}+\frac {c^{2}}{2 a \,d^{2}}-\frac {x \ln \left (\frac {-2 c \,f^{c} f^{d x}-b +\sqrt {-4 a c +b^{2}}}{-b +\sqrt {-4 a c +b^{2}}}\right )}{2 a d \ln \relax (f )}-\frac {x \ln \left (\frac {2 c \,f^{c} f^{d x}+b +\sqrt {-4 a c +b^{2}}}{b +\sqrt {-4 a c +b^{2}}}\right )}{2 a d \ln \relax (f )}-\frac {c \ln \left (f^{c} f^{d x}\right )}{a \,d^{2} \ln \relax (f )}-\frac {c \ln \left (\frac {-2 c \,f^{c} f^{d x}-b +\sqrt {-4 a c +b^{2}}}{-b +\sqrt {-4 a c +b^{2}}}\right )}{2 a \,d^{2} \ln \relax (f )}-\frac {c \ln \left (\frac {2 c \,f^{c} f^{d x}+b +\sqrt {-4 a c +b^{2}}}{b +\sqrt {-4 a c +b^{2}}}\right )}{2 a \,d^{2} \ln \relax (f )}+\frac {c \ln \left (b \,f^{c} f^{d x}+c \,f^{2 c} f^{2 d x}+a \right )}{2 a \,d^{2} \ln \relax (f )}-\frac {b \dilog \left (\frac {-2 c \,f^{c} f^{d x}-b +\sqrt {-4 a c +b^{2}}}{-b +\sqrt {-4 a c +b^{2}}}\right )}{2 \sqrt {-4 a c +b^{2}}\, a \,d^{2} \ln \relax (f )^{2}}+\frac {b \dilog \left (\frac {2 c \,f^{c} f^{d x}+b +\sqrt {-4 a c +b^{2}}}{b +\sqrt {-4 a c +b^{2}}}\right )}{2 \sqrt {-4 a c +b^{2}}\, a \,d^{2} \ln \relax (f )^{2}}-\frac {\dilog \left (\frac {-2 c \,f^{c} f^{d x}-b +\sqrt {-4 a c +b^{2}}}{-b +\sqrt {-4 a c +b^{2}}}\right )}{2 a \,d^{2} \ln \relax (f )^{2}}-\frac {\dilog \left (\frac {2 c \,f^{c} f^{d x}+b +\sqrt {-4 a c +b^{2}}}{b +\sqrt {-4 a c +b^{2}}}\right )}{2 a \,d^{2} \ln \relax (f )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x}{a+b\,f^{c+d\,x}+c\,f^{2\,c+2\,d\,x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{a + b f^{c} f^{d x} + c f^{2 c} f^{2 d x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________