3.55.66 \(\int \frac {e^{\frac {4 x+20 x^2-4 x^3}{3+4 x^3}} (-18 x-12 x^2-120 x^3-12 x^4+32 x^5+80 x^6-32 x^7)}{9+24 x^3+16 x^6} \, dx\)

Optimal. Leaf size=29 \[ 8-e^{\frac {5+\frac {1}{x}-x}{\frac {3}{4 x^2}+x}} x^2 \]

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Rubi [F]  time = 4.10, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {e^{\frac {4 x+20 x^2-4 x^3}{3+4 x^3}} \left (-18 x-12 x^2-120 x^3-12 x^4+32 x^5+80 x^6-32 x^7\right )}{9+24 x^3+16 x^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((4*x + 20*x^2 - 4*x^3)/(3 + 4*x^3))*(-18*x - 12*x^2 - 120*x^3 - 12*x^4 + 32*x^5 + 80*x^6 - 32*x^7))/(9
+ 24*x^3 + 16*x^6),x]

[Out]

5*Defer[Int][E^((x*(4 + 20*x - 4*x^2))/(3 + 4*x^3)), x] - 2*Defer[Int][E^((x*(4 + 20*x - 4*x^2))/(3 + 4*x^3))*
x, x] + (5*2^(2/3)*Defer[Int][E^((x*(4 + 20*x - 4*x^2))/(3 + 4*x^3))/(6^(1/3) + 2*x)^2, x])/3^(1/3) + (10*2^(1
/3)*Defer[Int][E^((x*(4 + 20*x - 4*x^2))/(3 + 4*x^3))/(6^(1/3) + 2*x), x])/3^(2/3) - ((4*(-2)^(2/3) - 40*3^(1/
3) + 3*(-2)^(1/3)*3^(2/3))*Defer[Int][E^((x*(4 + 20*x - 4*x^2))/(3 + 4*x^3))/(-3^(1/3) - (-2)^(2/3)*x), x])/6
+ (5*2^(2/3)*Defer[Int][E^((x*(4 + 20*x - 4*x^2))/(3 + 4*x^3))/(-6^(1/3) + 2*(-1)^(1/3)*x)^2, x])/3^(1/3) - (3
0*(-3)^(5/6)*2^(1/3)*Defer[Int][E^((x*(4 + 20*x - 4*x^2))/(3 + 4*x^3))/(-6^(1/3) + 2*(-1)^(1/3)*x), x])/(1 + (
-1)^(1/3))^5 + (5*2^(2/3)*Defer[Int][E^((x*(4 + 20*x - 4*x^2))/(3 + 4*x^3))/(6^(1/3) + 2*(-1)^(2/3)*x)^2, x])/
3^(1/3) + ((40*3^(1/3) - 2^(1/3)*(4*2^(1/3) - 3*3^(2/3)))*Defer[Int][E^((x*(4 + 20*x - 4*x^2))/(3 + 4*x^3))/(-
3^(1/3) - 2^(2/3)*x), x])/6 + ((40*3^(1/3) + 2^(1/3)*(3*(-3)^(2/3) + 4*(-2)^(1/3)))*Defer[Int][E^((x*(4 + 20*x
- 4*x^2))/(3 + 4*x^3))/(-3^(1/3) + (-1)^(1/3)*2^(2/3)*x), x])/6 + (10*2^(1/3)*Defer[Int][E^((x*(4 + 20*x - 4*
x^2))/(3 + 4*x^3))/(6^(1/3) - (1 - I*Sqrt[3])*x), x])/3^(2/3) - 27*Defer[Int][(E^((x*(4 + 20*x - 4*x^2))/(3 +
4*x^3))*x)/(3 + 4*x^3)^2, x] - 36*Defer[Int][(E^((x*(4 + 20*x - 4*x^2))/(3 + 4*x^3))*x^2)/(3 + 4*x^3)^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=16 \int \frac {e^{\frac {4 x+20 x^2-4 x^3}{3+4 x^3}} \left (-18 x-12 x^2-120 x^3-12 x^4+32 x^5+80 x^6-32 x^7\right )}{\left (12+16 x^3\right )^2} \, dx\\ &=16 \int \frac {2 e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}} x \left (-9-6 x-60 x^2-6 x^3+16 x^4+40 x^5-16 x^6\right )}{\left (12+16 x^3\right )^2} \, dx\\ &=32 \int \frac {e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}} x \left (-9-6 x-60 x^2-6 x^3+16 x^4+40 x^5-16 x^6\right )}{\left (12+16 x^3\right )^2} \, dx\\ &=32 \int \left (\frac {5}{32} e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}}-\frac {1}{16} e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}} x-\frac {9 e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}} \left (-15+3 x+4 x^2\right )}{32 \left (3+4 x^3\right )^2}+\frac {e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}} \left (-60+9 x+8 x^2\right )}{32 \left (3+4 x^3\right )}\right ) \, dx\\ &=-\left (2 \int e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}} x \, dx\right )+5 \int e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}} \, dx-9 \int \frac {e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}} \left (-15+3 x+4 x^2\right )}{\left (3+4 x^3\right )^2} \, dx+\int \frac {e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}} \left (-60+9 x+8 x^2\right )}{3+4 x^3} \, dx\\ &=-\left (2 \int e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}} x \, dx\right )+5 \int e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}} \, dx-9 \int \left (-\frac {15 e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}}}{\left (3+4 x^3\right )^2}+\frac {3 e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}} x}{\left (3+4 x^3\right )^2}+\frac {4 e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}} x^2}{\left (3+4 x^3\right )^2}\right ) \, dx+\int \left (\frac {\left (-6 (-2)^{2/3}-9 \sqrt [3]{-1} \left (\frac {3}{2}\right )^{2/3}+60 \sqrt [3]{3}\right ) e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}}}{9 \left (-\sqrt [3]{3}-(-2)^{2/3} x\right )}+\frac {\left (9 \left (\frac {3}{2}\right )^{2/3}-6\ 2^{2/3}+60 \sqrt [3]{3}\right ) e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}}}{9 \left (-\sqrt [3]{3}-2^{2/3} x\right )}+\frac {\left (9 \left (-\frac {3}{2}\right )^{2/3}+6 \sqrt [3]{-1} 2^{2/3}+60 \sqrt [3]{3}\right ) e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}}}{9 \left (-\sqrt [3]{3}+\sqrt [3]{-1} 2^{2/3} x\right )}\right ) \, dx\\ &=-\left (2 \int e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}} x \, dx\right )+5 \int e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}} \, dx-27 \int \frac {e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}} x}{\left (3+4 x^3\right )^2} \, dx-36 \int \frac {e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}} x^2}{\left (3+4 x^3\right )^2} \, dx+135 \int \frac {e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}}}{\left (3+4 x^3\right )^2} \, dx+\frac {1}{9} \left (9 \left (\frac {3}{2}\right )^{2/3}-6\ 2^{2/3}+60 \sqrt [3]{3}\right ) \int \frac {e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}}}{-\sqrt [3]{3}-2^{2/3} x} \, dx+\frac {1}{6} \left (-4 (-2)^{2/3}+40 \sqrt [3]{3}-3 \sqrt [3]{-2} 3^{2/3}\right ) \int \frac {e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}}}{-\sqrt [3]{3}-(-2)^{2/3} x} \, dx+\frac {1}{6} \left (40 \sqrt [3]{3}+\sqrt [3]{2} \left (3 (-3)^{2/3}+4 \sqrt [3]{-2}\right )\right ) \int \frac {e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}}}{-\sqrt [3]{3}+\sqrt [3]{-1} 2^{2/3} x} \, dx\\ &=-\left (2 \int e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}} x \, dx\right )+5 \int e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}} \, dx-27 \int \frac {e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}} x}{\left (3+4 x^3\right )^2} \, dx-36 \int \frac {e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}} x^2}{\left (3+4 x^3\right )^2} \, dx+135 \int \left (\frac {2 \sqrt [3]{2} e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}}}{27\ 3^{2/3} \left (\sqrt [3]{6}+2 x\right )}+\frac {(-2)^{2/3} e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}}}{3 \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^4 \left (-\sqrt [3]{6}+2 \sqrt [3]{-1} x\right )^2}-\frac {2 i \sqrt [3]{-2} e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}}}{3 \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^5 \left (-\sqrt [3]{6}+2 \sqrt [3]{-1} x\right )}+\frac {2^{2/3} e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}}}{3 \sqrt [3]{3} \left (-1+\sqrt [3]{-1}\right )^2 \left (1+\sqrt [3]{-1}\right )^4 \left (\sqrt [3]{6}+2 (-1)^{2/3} x\right )^2}+\frac {2 \sqrt [3]{2} e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}}}{27\ 3^{2/3} \left (\sqrt [3]{6}-\left (1-i \sqrt {3}\right ) x\right )}+\frac {2^{2/3} e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}}}{27 \left (3\ 2^{2/3}+4 \sqrt [3]{2} 3^{2/3} x+4 \sqrt [3]{3} x^2\right )}\right ) \, dx+\frac {1}{9} \left (9 \left (\frac {3}{2}\right )^{2/3}-6\ 2^{2/3}+60 \sqrt [3]{3}\right ) \int \frac {e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}}}{-\sqrt [3]{3}-2^{2/3} x} \, dx+\frac {1}{6} \left (-4 (-2)^{2/3}+40 \sqrt [3]{3}-3 \sqrt [3]{-2} 3^{2/3}\right ) \int \frac {e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}}}{-\sqrt [3]{3}-(-2)^{2/3} x} \, dx+\frac {1}{6} \left (40 \sqrt [3]{3}+\sqrt [3]{2} \left (3 (-3)^{2/3}+4 \sqrt [3]{-2}\right )\right ) \int \frac {e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}}}{-\sqrt [3]{3}+\sqrt [3]{-1} 2^{2/3} x} \, dx\\ &=-\left (2 \int e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}} x \, dx\right )+5 \int e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}} \, dx-27 \int \frac {e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}} x}{\left (3+4 x^3\right )^2} \, dx-36 \int \frac {e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}} x^2}{\left (3+4 x^3\right )^2} \, dx+\left (5\ 2^{2/3}\right ) \int \frac {e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}}}{3\ 2^{2/3}+4 \sqrt [3]{2} 3^{2/3} x+4 \sqrt [3]{3} x^2} \, dx+\frac {\left (10 \sqrt [3]{2}\right ) \int \frac {e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}}}{\sqrt [3]{6}+2 x} \, dx}{3^{2/3}}+\frac {\left (10 \sqrt [3]{2}\right ) \int \frac {e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}}}{\sqrt [3]{6}-\left (1-i \sqrt {3}\right ) x} \, dx}{3^{2/3}}+\frac {\left (5\ 2^{2/3}\right ) \int \frac {e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}}}{\left (-\sqrt [3]{6}+2 \sqrt [3]{-1} x\right )^2} \, dx}{\sqrt [3]{3}}+\frac {\left (5\ 2^{2/3}\right ) \int \frac {e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}}}{\left (\sqrt [3]{6}+2 (-1)^{2/3} x\right )^2} \, dx}{\sqrt [3]{3}}-\frac {\left (30 i \sqrt [3]{-2} 3^{5/6}\right ) \int \frac {e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}}}{-\sqrt [3]{6}+2 \sqrt [3]{-1} x} \, dx}{\left (1+\sqrt [3]{-1}\right )^5}+\frac {1}{9} \left (9 \left (\frac {3}{2}\right )^{2/3}-6\ 2^{2/3}+60 \sqrt [3]{3}\right ) \int \frac {e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}}}{-\sqrt [3]{3}-2^{2/3} x} \, dx+\frac {1}{6} \left (-4 (-2)^{2/3}+40 \sqrt [3]{3}-3 \sqrt [3]{-2} 3^{2/3}\right ) \int \frac {e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}}}{-\sqrt [3]{3}-(-2)^{2/3} x} \, dx+\frac {1}{6} \left (40 \sqrt [3]{3}+\sqrt [3]{2} \left (3 (-3)^{2/3}+4 \sqrt [3]{-2}\right )\right ) \int \frac {e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}}}{-\sqrt [3]{3}+\sqrt [3]{-1} 2^{2/3} x} \, dx\\ &=-\left (2 \int e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}} x \, dx\right )+5 \int e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}} \, dx-27 \int \frac {e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}} x}{\left (3+4 x^3\right )^2} \, dx-36 \int \frac {e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}} x^2}{\left (3+4 x^3\right )^2} \, dx+\left (5\ 2^{2/3}\right ) \int \frac {e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}}}{\sqrt [3]{3} \left (\sqrt [3]{6}+2 x\right )^2} \, dx+\frac {\left (10 \sqrt [3]{2}\right ) \int \frac {e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}}}{\sqrt [3]{6}+2 x} \, dx}{3^{2/3}}+\frac {\left (10 \sqrt [3]{2}\right ) \int \frac {e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}}}{\sqrt [3]{6}-\left (1-i \sqrt {3}\right ) x} \, dx}{3^{2/3}}+\frac {\left (5\ 2^{2/3}\right ) \int \frac {e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}}}{\left (-\sqrt [3]{6}+2 \sqrt [3]{-1} x\right )^2} \, dx}{\sqrt [3]{3}}+\frac {\left (5\ 2^{2/3}\right ) \int \frac {e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}}}{\left (\sqrt [3]{6}+2 (-1)^{2/3} x\right )^2} \, dx}{\sqrt [3]{3}}-\frac {\left (30 i \sqrt [3]{-2} 3^{5/6}\right ) \int \frac {e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}}}{-\sqrt [3]{6}+2 \sqrt [3]{-1} x} \, dx}{\left (1+\sqrt [3]{-1}\right )^5}+\frac {1}{9} \left (9 \left (\frac {3}{2}\right )^{2/3}-6\ 2^{2/3}+60 \sqrt [3]{3}\right ) \int \frac {e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}}}{-\sqrt [3]{3}-2^{2/3} x} \, dx+\frac {1}{6} \left (-4 (-2)^{2/3}+40 \sqrt [3]{3}-3 \sqrt [3]{-2} 3^{2/3}\right ) \int \frac {e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}}}{-\sqrt [3]{3}-(-2)^{2/3} x} \, dx+\frac {1}{6} \left (40 \sqrt [3]{3}+\sqrt [3]{2} \left (3 (-3)^{2/3}+4 \sqrt [3]{-2}\right )\right ) \int \frac {e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}}}{-\sqrt [3]{3}+\sqrt [3]{-1} 2^{2/3} x} \, dx\\ &=-\left (2 \int e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}} x \, dx\right )+5 \int e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}} \, dx-27 \int \frac {e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}} x}{\left (3+4 x^3\right )^2} \, dx-36 \int \frac {e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}} x^2}{\left (3+4 x^3\right )^2} \, dx+\frac {\left (10 \sqrt [3]{2}\right ) \int \frac {e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}}}{\sqrt [3]{6}+2 x} \, dx}{3^{2/3}}+\frac {\left (10 \sqrt [3]{2}\right ) \int \frac {e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}}}{\sqrt [3]{6}-\left (1-i \sqrt {3}\right ) x} \, dx}{3^{2/3}}+\frac {\left (5\ 2^{2/3}\right ) \int \frac {e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}}}{\left (\sqrt [3]{6}+2 x\right )^2} \, dx}{\sqrt [3]{3}}+\frac {\left (5\ 2^{2/3}\right ) \int \frac {e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}}}{\left (-\sqrt [3]{6}+2 \sqrt [3]{-1} x\right )^2} \, dx}{\sqrt [3]{3}}+\frac {\left (5\ 2^{2/3}\right ) \int \frac {e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}}}{\left (\sqrt [3]{6}+2 (-1)^{2/3} x\right )^2} \, dx}{\sqrt [3]{3}}-\frac {\left (30 i \sqrt [3]{-2} 3^{5/6}\right ) \int \frac {e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}}}{-\sqrt [3]{6}+2 \sqrt [3]{-1} x} \, dx}{\left (1+\sqrt [3]{-1}\right )^5}+\frac {1}{9} \left (9 \left (\frac {3}{2}\right )^{2/3}-6\ 2^{2/3}+60 \sqrt [3]{3}\right ) \int \frac {e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}}}{-\sqrt [3]{3}-2^{2/3} x} \, dx+\frac {1}{6} \left (-4 (-2)^{2/3}+40 \sqrt [3]{3}-3 \sqrt [3]{-2} 3^{2/3}\right ) \int \frac {e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}}}{-\sqrt [3]{3}-(-2)^{2/3} x} \, dx+\frac {1}{6} \left (40 \sqrt [3]{3}+\sqrt [3]{2} \left (3 (-3)^{2/3}+4 \sqrt [3]{-2}\right )\right ) \int \frac {e^{\frac {x \left (4+20 x-4 x^2\right )}{3+4 x^3}}}{-\sqrt [3]{3}+\sqrt [3]{-1} 2^{2/3} x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.07, size = 29, normalized size = 1.00 \begin {gather*} -e^{-1+\frac {3+4 x+20 x^2}{3+4 x^3}} x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((4*x + 20*x^2 - 4*x^3)/(3 + 4*x^3))*(-18*x - 12*x^2 - 120*x^3 - 12*x^4 + 32*x^5 + 80*x^6 - 32*x^
7))/(9 + 24*x^3 + 16*x^6),x]

[Out]

-(E^(-1 + (3 + 4*x + 20*x^2)/(3 + 4*x^3))*x^2)

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fricas [A]  time = 0.56, size = 29, normalized size = 1.00 \begin {gather*} -x^{2} e^{\left (-\frac {4 \, {\left (x^{3} - 5 \, x^{2} - x\right )}}{4 \, x^{3} + 3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-32*x^7+80*x^6+32*x^5-12*x^4-120*x^3-12*x^2-18*x)*exp((-4*x^3+20*x^2+4*x)/(4*x^3+3))/(16*x^6+24*x^3
+9),x, algorithm="fricas")

[Out]

-x^2*e^(-4*(x^3 - 5*x^2 - x)/(4*x^3 + 3))

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giac [A]  time = 0.26, size = 29, normalized size = 1.00 \begin {gather*} -x^{2} e^{\left (-\frac {4 \, {\left (x^{3} - 5 \, x^{2} - x\right )}}{4 \, x^{3} + 3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-32*x^7+80*x^6+32*x^5-12*x^4-120*x^3-12*x^2-18*x)*exp((-4*x^3+20*x^2+4*x)/(4*x^3+3))/(16*x^6+24*x^3
+9),x, algorithm="giac")

[Out]

-x^2*e^(-4*(x^3 - 5*x^2 - x)/(4*x^3 + 3))

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maple [A]  time = 0.10, size = 27, normalized size = 0.93

method	result	size
gosper	\(-x^{2} {\mathrm e}^{-\frac {4 x \left (x^{2}-5 x -1\right )}{4 x^{3}+3}}\)	\(27\)
risch	\(-x^{2} {\mathrm e}^{-\frac {4 x \left (x^{2}-5 x -1\right )}{4 x^{3}+3}}\)	\(27\)
norman	\(\frac {-3 x^{2} {\mathrm e}^{\frac {-4 x^{3}+20 x^{2}+4 x}{4 x^{3}+3}}-4 x^{5} {\mathrm e}^{\frac {-4 x^{3}+20 x^{2}+4 x}{4 x^{3}+3}}}{4 x^{3}+3}\)	\(72\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-32*x^7+80*x^6+32*x^5-12*x^4-120*x^3-12*x^2-18*x)*exp((-4*x^3+20*x^2+4*x)/(4*x^3+3))/(16*x^6+24*x^3+9),x,
method=_RETURNVERBOSE)

[Out]

-x^2*exp(-4*x*(x^2-5*x-1)/(4*x^3+3))

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maxima [A]  time = 0.56, size = 45, normalized size = 1.55 \begin {gather*} -x^{2} e^{\left (\frac {20 \, x^{2}}{4 \, x^{3} + 3} + \frac {4 \, x}{4 \, x^{3} + 3} + \frac {3}{4 \, x^{3} + 3} - 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-32*x^7+80*x^6+32*x^5-12*x^4-120*x^3-12*x^2-18*x)*exp((-4*x^3+20*x^2+4*x)/(4*x^3+3))/(16*x^6+24*x^3
+9),x, algorithm="maxima")

[Out]

-x^2*e^(20*x^2/(4*x^3 + 3) + 4*x/(4*x^3 + 3) + 3/(4*x^3 + 3) - 1)

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mupad [B]  time = 3.57, size = 30, normalized size = 1.03 \begin {gather*} -x^2\,{\mathrm {e}}^{\frac {-4\,x^3+20\,x^2+4\,x}{4\,x^3+3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((4*x + 20*x^2 - 4*x^3)/(4*x^3 + 3))*(18*x + 12*x^2 + 120*x^3 + 12*x^4 - 32*x^5 - 80*x^6 + 32*x^7))/(
24*x^3 + 16*x^6 + 9),x)

[Out]

-x^2*exp((4*x + 20*x^2 - 4*x^3)/(4*x^3 + 3))

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sympy [A]  time = 0.62, size = 26, normalized size = 0.90 \begin {gather*} - x^{2} e^{\frac {- 4 x^{3} + 20 x^{2} + 4 x}{4 x^{3} + 3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-32*x**7+80*x**6+32*x**5-12*x**4-120*x**3-12*x**2-18*x)*exp((-4*x**3+20*x**2+4*x)/(4*x**3+3))/(16*x
**6+24*x**3+9),x)

[Out]

-x**2*exp((-4*x**3 + 20*x**2 + 4*x)/(4*x**3 + 3))

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