Integrand size = 44, antiderivative size = 770 \[ \int \frac {\left (d+e e^{h+i x}\right ) (f+g x)^3}{a+b e^{h+i x}+c e^{2 h+2 i x}} \, dx=\frac {\left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) (f+g x)^4}{4 \left (b+\sqrt {b^2-4 a c}\right ) g}+\frac {\left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) (f+g x)^4}{4 \left (b-\sqrt {b^2-4 a c}\right ) g}-\frac {\left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) (f+g x)^3 \log \left (1+\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )}{\left (b-\sqrt {b^2-4 a c}\right ) i}-\frac {\left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) (f+g x)^3 \log \left (1+\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) i}-\frac {3 \left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g (f+g x)^2 \operatorname {PolyLog}\left (2,-\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )}{\left (b-\sqrt {b^2-4 a c}\right ) i^2}-\frac {3 \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g (f+g x)^2 \operatorname {PolyLog}\left (2,-\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) i^2}+\frac {6 \left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g^2 (f+g x) \operatorname {PolyLog}\left (3,-\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )}{\left (b-\sqrt {b^2-4 a c}\right ) i^3}+\frac {6 \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g^2 (f+g x) \operatorname {PolyLog}\left (3,-\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) i^3}-\frac {6 \left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g^3 \operatorname {PolyLog}\left (4,-\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )}{\left (b-\sqrt {b^2-4 a c}\right ) i^4}-\frac {6 \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) g^3 \operatorname {PolyLog}\left (4,-\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) i^4} \] Output:
1/4*(e-(-b*e+2*c*d)/(-4*a*c+b^2)^(1/2))*(g*x+f)^4/(b+(-4*a*c+b^2)^(1/2))/g +1/4*(e+(-b*e+2*c*d)/(-4*a*c+b^2)^(1/2))*(g*x+f)^4/(b-(-4*a*c+b^2)^(1/2))/ g-(e+(-b*e+2*c*d)/(-4*a*c+b^2)^(1/2))*(g*x+f)^3*ln(1+2*c*exp(i*x+h)/(b-(-4 *a*c+b^2)^(1/2)))/(b-(-4*a*c+b^2)^(1/2))/i-(e-(-b*e+2*c*d)/(-4*a*c+b^2)^(1 /2))*(g*x+f)^3*ln(1+2*c*exp(i*x+h)/(b+(-4*a*c+b^2)^(1/2)))/(b+(-4*a*c+b^2) ^(1/2))/i-3*(e+(-b*e+2*c*d)/(-4*a*c+b^2)^(1/2))*g*(g*x+f)^2*polylog(2,-2*c *exp(i*x+h)/(b-(-4*a*c+b^2)^(1/2)))/(b-(-4*a*c+b^2)^(1/2))/i^2-3*(e-(-b*e+ 2*c*d)/(-4*a*c+b^2)^(1/2))*g*(g*x+f)^2*polylog(2,-2*c*exp(i*x+h)/(b+(-4*a* c+b^2)^(1/2)))/(b+(-4*a*c+b^2)^(1/2))/i^2+6*(e+(-b*e+2*c*d)/(-4*a*c+b^2)^( 1/2))*g^2*(g*x+f)*polylog(3,-2*c*exp(i*x+h)/(b-(-4*a*c+b^2)^(1/2)))/(b-(-4 *a*c+b^2)^(1/2))/i^3+6*(e-(-b*e+2*c*d)/(-4*a*c+b^2)^(1/2))*g^2*(g*x+f)*pol ylog(3,-2*c*exp(i*x+h)/(b+(-4*a*c+b^2)^(1/2)))/(b+(-4*a*c+b^2)^(1/2))/i^3- 6*(e+(-b*e+2*c*d)/(-4*a*c+b^2)^(1/2))*g^3*polylog(4,-2*c*exp(i*x+h)/(b-(-4 *a*c+b^2)^(1/2)))/(b-(-4*a*c+b^2)^(1/2))/i^4-6*(e-(-b*e+2*c*d)/(-4*a*c+b^2 )^(1/2))*g^3*polylog(4,-2*c*exp(i*x+h)/(b+(-4*a*c+b^2)^(1/2)))/(b+(-4*a*c+ b^2)^(1/2))/i^4
Leaf count is larger than twice the leaf count of optimal. \(2448\) vs. \(2(770)=1540\).
Time = 4.87 (sec) , antiderivative size = 2448, normalized size of antiderivative = 3.18 \[ \int \frac {\left (d+e e^{h+i x}\right ) (f+g x)^3}{a+b e^{h+i x}+c e^{2 h+2 i x}} \, dx=\text {Result too large to show} \] Input:
Integrate[((d + e*E^(h + i*x))*(f + g*x)^3)/(a + b*E^(h + i*x) + c*E^(2*h + 2*i*x)),x]
Output:
-1/4*(-6*Sqrt[-(b^2 - 4*a*c)^2]*d*f^2*g*i^4*x^2 - 4*Sqrt[-(b^2 - 4*a*c)^2] *d*f*g^2*i^4*x^3 - Sqrt[-(b^2 - 4*a*c)^2]*d*g^3*i^4*x^4 + 4*b*Sqrt[b^2 - 4 *a*c]*d*f^3*i^3*ArcTan[(b + 2*c*E^(h + i*x))/Sqrt[-b^2 + 4*a*c]] - 8*a*Sqr t[b^2 - 4*a*c]*e*f^3*i^3*ArcTan[(b + 2*c*E^(h + i*x))/Sqrt[-b^2 + 4*a*c]] - 4*Sqrt[-(b^2 - 4*a*c)^2]*d*f^3*i^3*Log[E^(h + i*x)] + 6*Sqrt[-(b^2 - 4*a *c)^2]*d*f^2*g*i^3*x*Log[1 + (2*c*E^(h + i*x))/(b - Sqrt[b^2 - 4*a*c])] + 6*b*Sqrt[-b^2 + 4*a*c]*d*f^2*g*i^3*x*Log[1 + (2*c*E^(h + i*x))/(b - Sqrt[b ^2 - 4*a*c])] - 12*a*Sqrt[-b^2 + 4*a*c]*e*f^2*g*i^3*x*Log[1 + (2*c*E^(h + i*x))/(b - Sqrt[b^2 - 4*a*c])] + 6*Sqrt[-(b^2 - 4*a*c)^2]*d*f*g^2*i^3*x^2* Log[1 + (2*c*E^(h + i*x))/(b - Sqrt[b^2 - 4*a*c])] + 6*b*Sqrt[-b^2 + 4*a*c ]*d*f*g^2*i^3*x^2*Log[1 + (2*c*E^(h + i*x))/(b - Sqrt[b^2 - 4*a*c])] - 12* a*Sqrt[-b^2 + 4*a*c]*e*f*g^2*i^3*x^2*Log[1 + (2*c*E^(h + i*x))/(b - Sqrt[b ^2 - 4*a*c])] + 2*Sqrt[-(b^2 - 4*a*c)^2]*d*g^3*i^3*x^3*Log[1 + (2*c*E^(h + i*x))/(b - Sqrt[b^2 - 4*a*c])] + 2*b*Sqrt[-b^2 + 4*a*c]*d*g^3*i^3*x^3*Log [1 + (2*c*E^(h + i*x))/(b - Sqrt[b^2 - 4*a*c])] - 4*a*Sqrt[-b^2 + 4*a*c]*e *g^3*i^3*x^3*Log[1 + (2*c*E^(h + i*x))/(b - Sqrt[b^2 - 4*a*c])] + 6*Sqrt[- (b^2 - 4*a*c)^2]*d*f^2*g*i^3*x*Log[1 + (2*c*E^(h + i*x))/(b + Sqrt[b^2 - 4 *a*c])] - 6*b*Sqrt[-b^2 + 4*a*c]*d*f^2*g*i^3*x*Log[1 + (2*c*E^(h + i*x))/( b + Sqrt[b^2 - 4*a*c])] + 12*a*Sqrt[-b^2 + 4*a*c]*e*f^2*g*i^3*x*Log[1 + (2 *c*E^(h + i*x))/(b + Sqrt[b^2 - 4*a*c])] + 6*Sqrt[-(b^2 - 4*a*c)^2]*d*f...
Time = 3.56 (sec) , antiderivative size = 518, normalized size of antiderivative = 0.67, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.159, Rules used = {2695, 2615, 2620, 3011, 7163, 2720, 7143}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {(f+g x)^3 \left (d+e e^{h+i x}\right )}{a+b e^{h+i x}+c e^{2 h+2 i x}} \, dx\) |
\(\Big \downarrow \) 2695 |
\(\displaystyle \left (\frac {2 c d-b e}{\sqrt {b^2-4 a c}}+e\right ) \int \frac {(f+g x)^3}{b+2 c e^{h+i x}-\sqrt {b^2-4 a c}}dx+\left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \int \frac {(f+g x)^3}{b+2 c e^{h+i x}+\sqrt {b^2-4 a c}}dx\) |
\(\Big \downarrow \) 2615 |
\(\displaystyle \left (\frac {2 c d-b e}{\sqrt {b^2-4 a c}}+e\right ) \left (\frac {(f+g x)^4}{4 g \left (b-\sqrt {b^2-4 a c}\right )}-\frac {2 c \int \frac {e^{h+i x} (f+g x)^3}{b+2 c e^{h+i x}-\sqrt {b^2-4 a c}}dx}{b-\sqrt {b^2-4 a c}}\right )+\left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \left (\frac {(f+g x)^4}{4 g \left (\sqrt {b^2-4 a c}+b\right )}-\frac {2 c \int \frac {e^{h+i x} (f+g x)^3}{b+2 c e^{h+i x}+\sqrt {b^2-4 a c}}dx}{\sqrt {b^2-4 a c}+b}\right )\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle \left (\frac {2 c d-b e}{\sqrt {b^2-4 a c}}+e\right ) \left (\frac {(f+g x)^4}{4 g \left (b-\sqrt {b^2-4 a c}\right )}-\frac {2 c \left (\frac {(f+g x)^3 \log \left (\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}+1\right )}{2 c i}-\frac {3 g \int (f+g x)^2 \log \left (\frac {2 e^{h+i x} c}{b-\sqrt {b^2-4 a c}}+1\right )dx}{2 c i}\right )}{b-\sqrt {b^2-4 a c}}\right )+\left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \left (\frac {(f+g x)^4}{4 g \left (\sqrt {b^2-4 a c}+b\right )}-\frac {2 c \left (\frac {(f+g x)^3 \log \left (\frac {2 c e^{h+i x}}{\sqrt {b^2-4 a c}+b}+1\right )}{2 c i}-\frac {3 g \int (f+g x)^2 \log \left (\frac {2 e^{h+i x} c}{b+\sqrt {b^2-4 a c}}+1\right )dx}{2 c i}\right )}{\sqrt {b^2-4 a c}+b}\right )\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle \left (\frac {2 c d-b e}{\sqrt {b^2-4 a c}}+e\right ) \left (\frac {(f+g x)^4}{4 g \left (b-\sqrt {b^2-4 a c}\right )}-\frac {2 c \left (\frac {(f+g x)^3 \log \left (\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}+1\right )}{2 c i}-\frac {3 g \left (\frac {2 g \int (f+g x) \operatorname {PolyLog}\left (2,-\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )dx}{i}-\frac {(f+g x)^2 \operatorname {PolyLog}\left (2,-\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )}{i}\right )}{2 c i}\right )}{b-\sqrt {b^2-4 a c}}\right )+\left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \left (\frac {(f+g x)^4}{4 g \left (\sqrt {b^2-4 a c}+b\right )}-\frac {2 c \left (\frac {(f+g x)^3 \log \left (\frac {2 c e^{h+i x}}{\sqrt {b^2-4 a c}+b}+1\right )}{2 c i}-\frac {3 g \left (\frac {2 g \int (f+g x) \operatorname {PolyLog}\left (2,-\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )dx}{i}-\frac {(f+g x)^2 \operatorname {PolyLog}\left (2,-\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )}{i}\right )}{2 c i}\right )}{\sqrt {b^2-4 a c}+b}\right )\) |
\(\Big \downarrow \) 7163 |
\(\displaystyle \left (\frac {2 c d-b e}{\sqrt {b^2-4 a c}}+e\right ) \left (\frac {(f+g x)^4}{4 g \left (b-\sqrt {b^2-4 a c}\right )}-\frac {2 c \left (\frac {(f+g x)^3 \log \left (\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}+1\right )}{2 c i}-\frac {3 g \left (\frac {2 g \left (\frac {(f+g x) \operatorname {PolyLog}\left (3,-\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )}{i}-\frac {g \int \operatorname {PolyLog}\left (3,-\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )dx}{i}\right )}{i}-\frac {(f+g x)^2 \operatorname {PolyLog}\left (2,-\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )}{i}\right )}{2 c i}\right )}{b-\sqrt {b^2-4 a c}}\right )+\left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \left (\frac {(f+g x)^4}{4 g \left (\sqrt {b^2-4 a c}+b\right )}-\frac {2 c \left (\frac {(f+g x)^3 \log \left (\frac {2 c e^{h+i x}}{\sqrt {b^2-4 a c}+b}+1\right )}{2 c i}-\frac {3 g \left (\frac {2 g \left (\frac {(f+g x) \operatorname {PolyLog}\left (3,-\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )}{i}-\frac {g \int \operatorname {PolyLog}\left (3,-\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )dx}{i}\right )}{i}-\frac {(f+g x)^2 \operatorname {PolyLog}\left (2,-\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )}{i}\right )}{2 c i}\right )}{\sqrt {b^2-4 a c}+b}\right )\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \left (\frac {2 c d-b e}{\sqrt {b^2-4 a c}}+e\right ) \left (\frac {(f+g x)^4}{4 g \left (b-\sqrt {b^2-4 a c}\right )}-\frac {2 c \left (\frac {(f+g x)^3 \log \left (\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}+1\right )}{2 c i}-\frac {3 g \left (\frac {2 g \left (\frac {(f+g x) \operatorname {PolyLog}\left (3,-\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )}{i}-\frac {g \int e^{-h-i x} \operatorname {PolyLog}\left (3,-\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )de^{h+i x}}{i^2}\right )}{i}-\frac {(f+g x)^2 \operatorname {PolyLog}\left (2,-\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )}{i}\right )}{2 c i}\right )}{b-\sqrt {b^2-4 a c}}\right )+\left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \left (\frac {(f+g x)^4}{4 g \left (\sqrt {b^2-4 a c}+b\right )}-\frac {2 c \left (\frac {(f+g x)^3 \log \left (\frac {2 c e^{h+i x}}{\sqrt {b^2-4 a c}+b}+1\right )}{2 c i}-\frac {3 g \left (\frac {2 g \left (\frac {(f+g x) \operatorname {PolyLog}\left (3,-\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )}{i}-\frac {g \int e^{-h-i x} \operatorname {PolyLog}\left (3,-\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )de^{h+i x}}{i^2}\right )}{i}-\frac {(f+g x)^2 \operatorname {PolyLog}\left (2,-\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )}{i}\right )}{2 c i}\right )}{\sqrt {b^2-4 a c}+b}\right )\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle \left (\frac {2 c d-b e}{\sqrt {b^2-4 a c}}+e\right ) \left (\frac {(f+g x)^4}{4 g \left (b-\sqrt {b^2-4 a c}\right )}-\frac {2 c \left (\frac {(f+g x)^3 \log \left (\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}+1\right )}{2 c i}-\frac {3 g \left (\frac {2 g \left (\frac {(f+g x) \operatorname {PolyLog}\left (3,-\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )}{i}-\frac {g \operatorname {PolyLog}\left (4,-\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )}{i^2}\right )}{i}-\frac {(f+g x)^2 \operatorname {PolyLog}\left (2,-\frac {2 c e^{h+i x}}{b-\sqrt {b^2-4 a c}}\right )}{i}\right )}{2 c i}\right )}{b-\sqrt {b^2-4 a c}}\right )+\left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \left (\frac {(f+g x)^4}{4 g \left (\sqrt {b^2-4 a c}+b\right )}-\frac {2 c \left (\frac {(f+g x)^3 \log \left (\frac {2 c e^{h+i x}}{\sqrt {b^2-4 a c}+b}+1\right )}{2 c i}-\frac {3 g \left (\frac {2 g \left (\frac {(f+g x) \operatorname {PolyLog}\left (3,-\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )}{i}-\frac {g \operatorname {PolyLog}\left (4,-\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )}{i^2}\right )}{i}-\frac {(f+g x)^2 \operatorname {PolyLog}\left (2,-\frac {2 c e^{h+i x}}{b+\sqrt {b^2-4 a c}}\right )}{i}\right )}{2 c i}\right )}{\sqrt {b^2-4 a c}+b}\right )\) |
Input:
Int[((d + e*E^(h + i*x))*(f + g*x)^3)/(a + b*E^(h + i*x) + c*E^(2*h + 2*i* x)),x]
Output:
(e + (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*((f + g*x)^4/(4*(b - Sqrt[b^2 - 4*a* c])*g) - (2*c*(((f + g*x)^3*Log[1 + (2*c*E^(h + i*x))/(b - Sqrt[b^2 - 4*a* c])])/(2*c*i) - (3*g*(-(((f + g*x)^2*PolyLog[2, (-2*c*E^(h + i*x))/(b - Sq rt[b^2 - 4*a*c])])/i) + (2*g*(((f + g*x)*PolyLog[3, (-2*c*E^(h + i*x))/(b - Sqrt[b^2 - 4*a*c])])/i - (g*PolyLog[4, (-2*c*E^(h + i*x))/(b - Sqrt[b^2 - 4*a*c])])/i^2))/i))/(2*c*i)))/(b - Sqrt[b^2 - 4*a*c])) + (e - (2*c*d - b *e)/Sqrt[b^2 - 4*a*c])*((f + g*x)^4/(4*(b + Sqrt[b^2 - 4*a*c])*g) - (2*c*( ((f + g*x)^3*Log[1 + (2*c*E^(h + i*x))/(b + Sqrt[b^2 - 4*a*c])])/(2*c*i) - (3*g*(-(((f + g*x)^2*PolyLog[2, (-2*c*E^(h + i*x))/(b + Sqrt[b^2 - 4*a*c] )])/i) + (2*g*(((f + g*x)*PolyLog[3, (-2*c*E^(h + i*x))/(b + Sqrt[b^2 - 4* a*c])])/i - (g*PolyLog[4, (-2*c*E^(h + i*x))/(b + Sqrt[b^2 - 4*a*c])])/i^2 ))/i))/(2*c*i)))/(b + Sqrt[b^2 - 4*a*c]))
Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x _))))^(n_.)), x_Symbol] :> Simp[(c + d*x)^(m + 1)/(a*d*(m + 1)), x] - Simp[ b/a Int[(c + d*x)^m*((F^(g*(e + f*x)))^n/(a + b*(F^(g*(e + f*x)))^n)), x] , x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[(((i_.)*(F_)^(u_) + (h_))*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F _)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Sim p[(Simplify[(2*c*h - b*i)/q] + i) Int[(f + g*x)^m/(b - q + 2*c*F^u), x], x] - Simp[(Simplify[(2*c*h - b*i)/q] - i) Int[(f + g*x)^m/(b + q + 2*c*F^ u), x], x]] /; FreeQ[{F, a, b, c, f, g, h, i}, x] && EqQ[v, 2*u] && LinearQ [u, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[m, 0]
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) *(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F])) Int[(f + g*x)^( m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e , f, g, n}, x] && GtQ[m, 0]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_. )*(x_))))^(p_.)], x_Symbol] :> Simp[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a + b*x)))^p]/(b*c*p*Log[F])), x] - Simp[f*(m/(b*c*p*Log[F])) Int[(e + f*x) ^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c , d, e, f, n, p}, x] && GtQ[m, 0]
\[\int \frac {\left (d +e \,{\mathrm e}^{i x +h}\right ) \left (g x +f \right )^{3}}{a +b \,{\mathrm e}^{i x +h}+c \,{\mathrm e}^{2 i x +2 h}}d x\]
Input:
int((d+e*exp(i*x+h))*(g*x+f)^3/(a+b*exp(i*x+h)+c*exp(2*i*x+2*h)),x)
Output:
int((d+e*exp(i*x+h))*(g*x+f)^3/(a+b*exp(i*x+h)+c*exp(2*i*x+2*h)),x)
Leaf count of result is larger than twice the leaf count of optimal. 1859 vs. \(2 (700) = 1400\).
Time = 0.15 (sec) , antiderivative size = 1859, normalized size of antiderivative = 2.41 \[ \int \frac {\left (d+e e^{h+i x}\right ) (f+g x)^3}{a+b e^{h+i x}+c e^{2 h+2 i x}} \, dx=\text {Too large to display} \] Input:
integrate((d+e*exp(i*x+h))*(g*x+f)^3/(a+b*exp(i*x+h)+c*exp(2*i*x+2*h)),x, algorithm="fricas")
Output:
1/4*((b^2 - 4*a*c)*d*g^3*i^4*x^4 + 4*(b^2 - 4*a*c)*d*f*g^2*i^4*x^3 + 6*(b^ 2 - 4*a*c)*d*f^2*g*i^4*x^2 + 4*(b^2 - 4*a*c)*d*f^3*i^4*x - 6*((b^2 - 4*a*c )*d*g^3*i^2*x^2 + 2*(b^2 - 4*a*c)*d*f*g^2*i^2*x + (b^2 - 4*a*c)*d*f^2*g*i^ 2 + ((a*b*d - 2*a^2*e)*g^3*i^2*x^2 + 2*(a*b*d - 2*a^2*e)*f*g^2*i^2*x + (a* b*d - 2*a^2*e)*f^2*g*i^2)*sqrt((b^2 - 4*a*c)/a^2))*dilog(-1/2*(a*sqrt((b^2 - 4*a*c)/a^2)*e^(i*x + h) + b*e^(i*x + h) + 2*a)/a + 1) - 6*((b^2 - 4*a*c )*d*g^3*i^2*x^2 + 2*(b^2 - 4*a*c)*d*f*g^2*i^2*x + (b^2 - 4*a*c)*d*f^2*g*i^ 2 - ((a*b*d - 2*a^2*e)*g^3*i^2*x^2 + 2*(a*b*d - 2*a^2*e)*f*g^2*i^2*x + (a* b*d - 2*a^2*e)*f^2*g*i^2)*sqrt((b^2 - 4*a*c)/a^2))*dilog(1/2*(a*sqrt((b^2 - 4*a*c)/a^2)*e^(i*x + h) - b*e^(i*x + h) - 2*a)/a + 1) + 2*((b^2 - 4*a*c) *d*g^3*h^3 - 3*(b^2 - 4*a*c)*d*f*g^2*h^2*i + 3*(b^2 - 4*a*c)*d*f^2*g*h*i^2 - (b^2 - 4*a*c)*d*f^3*i^3 - ((a*b*d - 2*a^2*e)*g^3*h^3 - 3*(a*b*d - 2*a^2 *e)*f*g^2*h^2*i + 3*(a*b*d - 2*a^2*e)*f^2*g*h*i^2 - (a*b*d - 2*a^2*e)*f^3* i^3)*sqrt((b^2 - 4*a*c)/a^2))*log(2*c*e^(i*x + h) + a*sqrt((b^2 - 4*a*c)/a ^2) + b) + 2*((b^2 - 4*a*c)*d*g^3*h^3 - 3*(b^2 - 4*a*c)*d*f*g^2*h^2*i + 3* (b^2 - 4*a*c)*d*f^2*g*h*i^2 - (b^2 - 4*a*c)*d*f^3*i^3 + ((a*b*d - 2*a^2*e) *g^3*h^3 - 3*(a*b*d - 2*a^2*e)*f*g^2*h^2*i + 3*(a*b*d - 2*a^2*e)*f^2*g*h*i ^2 - (a*b*d - 2*a^2*e)*f^3*i^3)*sqrt((b^2 - 4*a*c)/a^2))*log(2*c*e^(i*x + h) - a*sqrt((b^2 - 4*a*c)/a^2) + b) - 2*((b^2 - 4*a*c)*d*g^3*i^3*x^3 + 3*( b^2 - 4*a*c)*d*f*g^2*i^3*x^2 + 3*(b^2 - 4*a*c)*d*f^2*g*i^3*x + (b^2 - 4...
\[ \int \frac {\left (d+e e^{h+i x}\right ) (f+g x)^3}{a+b e^{h+i x}+c e^{2 h+2 i x}} \, dx=\int \frac {\left (d + e e^{h} e^{i x}\right ) \left (f + g x\right )^{3}}{a + b e^{h} e^{i x} + c e^{2 h} e^{2 i x}}\, dx \] Input:
integrate((d+e*exp(i*x+h))*(g*x+f)**3/(a+b*exp(i*x+h)+c*exp(2*i*x+2*h)),x)
Output:
Integral((d + e*exp(h)*exp(i*x))*(f + g*x)**3/(a + b*exp(h)*exp(i*x) + c*e xp(2*h)*exp(2*i*x)), x)
Exception generated. \[ \int \frac {\left (d+e e^{h+i x}\right ) (f+g x)^3}{a+b e^{h+i x}+c e^{2 h+2 i x}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((d+e*exp(i*x+h))*(g*x+f)^3/(a+b*exp(i*x+h)+c*exp(2*i*x+2*h)),x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for more deta
\[ \int \frac {\left (d+e e^{h+i x}\right ) (f+g x)^3}{a+b e^{h+i x}+c e^{2 h+2 i x}} \, dx=\int { \frac {{\left (g x + f\right )}^{3} {\left (e e^{\left (i x + h\right )} + d\right )}}{c e^{\left (2 \, i x + 2 \, h\right )} + b e^{\left (i x + h\right )} + a} \,d x } \] Input:
integrate((d+e*exp(i*x+h))*(g*x+f)^3/(a+b*exp(i*x+h)+c*exp(2*i*x+2*h)),x, algorithm="giac")
Output:
integrate((g*x + f)^3*(e*e^(i*x + h) + d)/(c*e^(2*i*x + 2*h) + b*e^(i*x + h) + a), x)
Timed out. \[ \int \frac {\left (d+e e^{h+i x}\right ) (f+g x)^3}{a+b e^{h+i x}+c e^{2 h+2 i x}} \, dx=\int \frac {{\left (f+g\,x\right )}^3\,\left (d+e\,{\mathrm {e}}^{h+i\,x}\right )}{a+b\,{\mathrm {e}}^{h+i\,x}+c\,{\mathrm {e}}^{2\,h+2\,i\,x}} \,d x \] Input:
int(((f + g*x)^3*(d + e*exp(h + i*x)))/(a + b*exp(h + i*x) + c*exp(2*h + 2 *i*x)),x)
Output:
int(((f + g*x)^3*(d + e*exp(h + i*x)))/(a + b*exp(h + i*x) + c*exp(2*h + 2 *i*x)), x)
\[ \int \frac {\left (d+e e^{h+i x}\right ) (f+g x)^3}{a+b e^{h+i x}+c e^{2 h+2 i x}} \, dx=\frac {-4 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 e^{i x +h} c +b}{\sqrt {4 a c -b^{2}}}\right ) a e \,f^{3} i +2 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 e^{i x +h} c +b}{\sqrt {4 a c -b^{2}}}\right ) b d \,f^{3} i +8 e^{h} \left (\int \frac {e^{i x} x^{3}}{e^{2 i x +2 h} c +e^{i x +h} b +a}d x \right ) a^{2} c e \,g^{3}-2 e^{h} \left (\int \frac {e^{i x} x^{3}}{e^{2 i x +2 h} c +e^{i x +h} b +a}d x \right ) a \,b^{2} e \,g^{3}+24 e^{h} \left (\int \frac {e^{i x} x^{2}}{e^{2 i x +2 h} c +e^{i x +h} b +a}d x \right ) a^{2} c e f \,g^{2}-6 e^{h} \left (\int \frac {e^{i x} x^{2}}{e^{2 i x +2 h} c +e^{i x +h} b +a}d x \right ) a \,b^{2} e f \,g^{2}+24 e^{h} \left (\int \frac {e^{i x} x}{e^{2 i x +2 h} c +e^{i x +h} b +a}d x \right ) a^{2} c e \,f^{2} g -6 e^{h} \left (\int \frac {e^{i x} x}{e^{2 i x +2 h} c +e^{i x +h} b +a}d x \right ) a \,b^{2} e \,f^{2} g +8 \left (\int \frac {x^{3}}{e^{2 i x +2 h} c +e^{i x +h} b +a}d x \right ) a^{2} c d \,g^{3}-2 \left (\int \frac {x^{3}}{e^{2 i x +2 h} c +e^{i x +h} b +a}d x \right ) a \,b^{2} d \,g^{3}+24 \left (\int \frac {x^{2}}{e^{2 i x +2 h} c +e^{i x +h} b +a}d x \right ) a^{2} c d f \,g^{2}-6 \left (\int \frac {x^{2}}{e^{2 i x +2 h} c +e^{i x +h} b +a}d x \right ) a \,b^{2} d f \,g^{2}+24 \left (\int \frac {x}{e^{2 i x +2 h} c +e^{i x +h} b +a}d x \right ) a^{2} c d \,f^{2} g -6 \left (\int \frac {x}{e^{2 i x +2 h} c +e^{i x +h} b +a}d x \right ) a \,b^{2} d \,f^{2} g +4 \,\mathrm {log}\left (e^{2 i x +2 h} c +e^{i x +h} b +a \right ) a c d \,f^{3} i -\mathrm {log}\left (e^{2 i x +2 h} c +e^{i x +h} b +a \right ) b^{2} d \,f^{3} i +8 a c d \,f^{3} x -2 b^{2} d \,f^{3} x}{2 a \left (4 a c -b^{2}\right )} \] Input:
int((d+e*exp(i*x+h))*(g*x+f)^3/(a+b*exp(i*x+h)+c*exp(2*i*x+2*h)),x)
Output:
( - 4*sqrt(4*a*c - b**2)*atan((2*e**(h + i*x)*c + b)/sqrt(4*a*c - b**2))*a *e*f**3*i + 2*sqrt(4*a*c - b**2)*atan((2*e**(h + i*x)*c + b)/sqrt(4*a*c - b**2))*b*d*f**3*i + 8*e**h*int((e**(i*x)*x**3)/(e**(2*h + 2*i*x)*c + e**(h + i*x)*b + a),x)*a**2*c*e*g**3 - 2*e**h*int((e**(i*x)*x**3)/(e**(2*h + 2* i*x)*c + e**(h + i*x)*b + a),x)*a*b**2*e*g**3 + 24*e**h*int((e**(i*x)*x**2 )/(e**(2*h + 2*i*x)*c + e**(h + i*x)*b + a),x)*a**2*c*e*f*g**2 - 6*e**h*in t((e**(i*x)*x**2)/(e**(2*h + 2*i*x)*c + e**(h + i*x)*b + a),x)*a*b**2*e*f* g**2 + 24*e**h*int((e**(i*x)*x)/(e**(2*h + 2*i*x)*c + e**(h + i*x)*b + a), x)*a**2*c*e*f**2*g - 6*e**h*int((e**(i*x)*x)/(e**(2*h + 2*i*x)*c + e**(h + i*x)*b + a),x)*a*b**2*e*f**2*g + 8*int(x**3/(e**(2*h + 2*i*x)*c + e**(h + i*x)*b + a),x)*a**2*c*d*g**3 - 2*int(x**3/(e**(2*h + 2*i*x)*c + e**(h + i *x)*b + a),x)*a*b**2*d*g**3 + 24*int(x**2/(e**(2*h + 2*i*x)*c + e**(h + i* x)*b + a),x)*a**2*c*d*f*g**2 - 6*int(x**2/(e**(2*h + 2*i*x)*c + e**(h + i* x)*b + a),x)*a*b**2*d*f*g**2 + 24*int(x/(e**(2*h + 2*i*x)*c + e**(h + i*x) *b + a),x)*a**2*c*d*f**2*g - 6*int(x/(e**(2*h + 2*i*x)*c + e**(h + i*x)*b + a),x)*a*b**2*d*f**2*g + 4*log(e**(2*h + 2*i*x)*c + e**(h + i*x)*b + a)*a *c*d*f**3*i - log(e**(2*h + 2*i*x)*c + e**(h + i*x)*b + a)*b**2*d*f**3*i + 8*a*c*d*f**3*x - 2*b**2*d*f**3*x)/(2*a*(4*a*c - b**2))