Integrand size = 28, antiderivative size = 673 \[ \int \frac {b+2 c x}{\sqrt {d+e x} \left (a+b x+c x^2\right )^3} \, dx=-\frac {\sqrt {d+e x} \left (\left (b^2-4 a c\right ) (c d-b e)-c \left (b^2-4 a c\right ) e x\right )}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^2}+\frac {e \sqrt {d+e x} \left (5 a c e (2 c d-b e)+(c d-3 b e) \left (b c d-b^2 e+2 a c e\right )+c \left (2 c^2 d^2+3 b^2 e^2-2 c e (b d+5 a e)\right ) x\right )}{4 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}-\frac {\sqrt {c} e \left (8 c^3 d^3+3 b^2 \left (b+\sqrt {b^2-4 a c}\right ) e^3-2 c^2 d e \left (6 b d-\sqrt {b^2-4 a c} d-16 a e\right )-2 c e^2 \left (b^2 d+b \sqrt {b^2-4 a c} d+8 a b e+5 a \sqrt {b^2-4 a c} e\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{4 \sqrt {2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )^2}+\frac {\sqrt {c} e \left (8 c^3 d^3+3 b^2 \left (b-\sqrt {b^2-4 a c}\right ) e^3-2 c^2 d e \left (6 b d+\sqrt {b^2-4 a c} d-16 a e\right )-2 c e^2 \left (b^2 d-b \sqrt {b^2-4 a c} d+8 a b e-5 a \sqrt {b^2-4 a c} e\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{4 \sqrt {2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )^2} \]
-1/2*((-4*a*c+b^2)*(-b*e+c*d)-c*(-4*a*c+b^2)*e*x)*(e*x+d)^(1/2)/(-4*a*c+b^ 2)/(a*e^2-b*d*e+c*d^2)/(c*x^2+b*x+a)^2+1/4*e*(5*a*c*e*(-b*e+2*c*d)+(-3*b*e +c*d)*(2*a*c*e-b^2*e+b*c*d)+c*(2*c^2*d^2+3*b^2*e^2-2*c*e*(5*a*e+b*d))*x)*( e*x+d)^(1/2)/(-4*a*c+b^2)/(a*e^2-b*d*e+c*d^2)^2/(c*x^2+b*x+a)-1/8*e*arctan h(2^(1/2)*c^(1/2)*(e*x+d)^(1/2)/(2*c*d-e*(b-(-4*a*c+b^2)^(1/2)))^(1/2))*c^ (1/2)*(8*c^3*d^3+3*b^2*e^3*(b+(-4*a*c+b^2)^(1/2))-2*c^2*d*e*(6*b*d-16*a*e- d*(-4*a*c+b^2)^(1/2))-2*c*e^2*(b^2*d+8*a*b*e+b*d*(-4*a*c+b^2)^(1/2)+5*a*e* (-4*a*c+b^2)^(1/2)))/(-4*a*c+b^2)^(3/2)/(a*e^2-b*d*e+c*d^2)^2*2^(1/2)/(2*c *d-e*(b-(-4*a*c+b^2)^(1/2)))^(1/2)+1/8*e*arctanh(2^(1/2)*c^(1/2)*(e*x+d)^( 1/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2)))^(1/2))*c^(1/2)*(8*c^3*d^3+3*b^2*e^3* (b-(-4*a*c+b^2)^(1/2))-2*c^2*d*e*(6*b*d-16*a*e+d*(-4*a*c+b^2)^(1/2))-2*c*e ^2*(b^2*d+8*a*b*e-b*d*(-4*a*c+b^2)^(1/2)-5*a*e*(-4*a*c+b^2)^(1/2)))/(-4*a* c+b^2)^(3/2)/(a*e^2-b*d*e+c*d^2)^2*2^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2)) )^(1/2)
Time = 8.95 (sec) , antiderivative size = 689, normalized size of antiderivative = 1.02 \[ \int \frac {b+2 c x}{\sqrt {d+e x} \left (a+b x+c x^2\right )^3} \, dx=\frac {e \left (\frac {2 \sqrt {d+e x} \left (b^4 e^2 (-2 d+3 e x)+b^3 \left (5 a e^3+c e \left (4 d^2-6 d e x+6 e^2 x^2\right )\right )-b c e \left (19 a^2 e^2+c^2 d x^2 (-3 d+2 e x)+3 a c \left (5 d^2-6 d e x+7 e^2 x^2\right )\right )+2 c^2 \left (c^2 d^2 e x^3+a^2 e^2 (10 d-9 e x)+a c \left (4 d^3-3 d^2 e x+6 d e^2 x^2-5 e^3 x^3\right )\right )+b^2 c \left (2 a e^2 (d-3 e x)+c \left (-2 d^3+3 d^2 e x-6 d e^2 x^2+3 e^3 x^3\right )\right )\right )}{\left (b^2-4 a c\right ) e (a+x (b+c x))^2}+\frac {\sqrt {2} \sqrt {c} \left (8 c^3 d^3+3 b^2 \left (b+\sqrt {b^2-4 a c}\right ) e^3+2 c^2 d e \left (-6 b d+\sqrt {b^2-4 a c} d+16 a e\right )-2 c e^2 \left (b^2 d+b \sqrt {b^2-4 a c} d+8 a b e+5 a \sqrt {b^2-4 a c} e\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e-\sqrt {b^2-4 a c} e}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt {-2 c d+\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {\sqrt {2} \sqrt {c} \left (-8 c^3 d^3+3 b^2 \left (-b+\sqrt {b^2-4 a c}\right ) e^3+2 c^2 d e \left (6 b d+\sqrt {b^2-4 a c} d-16 a e\right )-2 c e^2 \left (-b^2 d+b \sqrt {b^2-4 a c} d-8 a b e+5 a \sqrt {b^2-4 a c} e\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt {-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{8 \left (c d^2+e (-b d+a e)\right )^2} \]
(e*((2*Sqrt[d + e*x]*(b^4*e^2*(-2*d + 3*e*x) + b^3*(5*a*e^3 + c*e*(4*d^2 - 6*d*e*x + 6*e^2*x^2)) - b*c*e*(19*a^2*e^2 + c^2*d*x^2*(-3*d + 2*e*x) + 3* a*c*(5*d^2 - 6*d*e*x + 7*e^2*x^2)) + 2*c^2*(c^2*d^2*e*x^3 + a^2*e^2*(10*d - 9*e*x) + a*c*(4*d^3 - 3*d^2*e*x + 6*d*e^2*x^2 - 5*e^3*x^3)) + b^2*c*(2*a *e^2*(d - 3*e*x) + c*(-2*d^3 + 3*d^2*e*x - 6*d*e^2*x^2 + 3*e^3*x^3))))/((b ^2 - 4*a*c)*e*(a + x*(b + c*x))^2) + (Sqrt[2]*Sqrt[c]*(8*c^3*d^3 + 3*b^2*( b + Sqrt[b^2 - 4*a*c])*e^3 + 2*c^2*d*e*(-6*b*d + Sqrt[b^2 - 4*a*c]*d + 16* a*e) - 2*c*e^2*(b^2*d + b*Sqrt[b^2 - 4*a*c]*d + 8*a*b*e + 5*a*Sqrt[b^2 - 4 *a*c]*e))*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + b*e - Sqrt[ b^2 - 4*a*c]*e]])/((b^2 - 4*a*c)^(3/2)*Sqrt[-2*c*d + (b - Sqrt[b^2 - 4*a*c ])*e]) + (Sqrt[2]*Sqrt[c]*(-8*c^3*d^3 + 3*b^2*(-b + Sqrt[b^2 - 4*a*c])*e^3 + 2*c^2*d*e*(6*b*d + Sqrt[b^2 - 4*a*c]*d - 16*a*e) - 2*c*e^2*(-(b^2*d) + b*Sqrt[b^2 - 4*a*c]*d - 8*a*b*e + 5*a*Sqrt[b^2 - 4*a*c]*e))*ArcTan[(Sqrt[2 ]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e]])/((b^2 - 4*a*c)^(3/2)*Sqrt[-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e])))/(8*(c*d^2 + e*( -(b*d) + a*e))^2)
Time = 1.41 (sec) , antiderivative size = 647, normalized size of antiderivative = 0.96, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {1235, 27, 1235, 27, 1197, 27, 1480, 221}
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.
| \(\displaystyle \int \frac {b+2 c x}{\sqrt {d+e x} \left (a+b x+c x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle -\frac {\int \frac {\left (b^2-4 a c\right ) e (c d-3 b e-5 c e x)}{2 \sqrt {d+e x} \left (c x^2+b x+a\right )^2}dx}{2 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {d+e x} \left (\left (b^2-4 a c\right ) (c d-b e)-c e x \left (b^2-4 a c\right )\right )}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {e \int \frac {c d-3 b e-5 c e x}{\sqrt {d+e x} \left (c x^2+b x+a\right )^2}dx}{4 \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {d+e x} \left (\left (b^2-4 a c\right ) (c d-b e)-c e x \left (b^2-4 a c\right )\right )}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle -\frac {e \left (-\frac {\int \frac {4 c^3 d^3-c^2 e (5 b d-16 a e) d+3 b^3 e^3-b c e^2 (2 b d+13 a e)+c e \left (2 c^2 d^2+3 b^2 e^2-2 c e (b d+5 a e)\right ) x}{2 \sqrt {d+e x} \left (c x^2+b x+a\right )}dx}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {d+e x} \left (c x \left (-2 c e (5 a e+b d)+3 b^2 e^2+2 c^2 d^2\right )+(c d-3 b e) \left (2 a c e+b^2 (-e)+b c d\right )+5 a c e (2 c d-b e)\right )}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}\right )}{4 \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {d+e x} \left (\left (b^2-4 a c\right ) (c d-b e)-c e x \left (b^2-4 a c\right )\right )}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {e \left (-\frac {\int \frac {4 c^3 d^3-c^2 e (5 b d-16 a e) d+3 b^3 e^3-b c e^2 (2 b d+13 a e)+c e \left (2 c^2 d^2+3 b^2 e^2-2 c e (b d+5 a e)\right ) x}{\sqrt {d+e x} \left (c x^2+b x+a\right )}dx}{2 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {d+e x} \left (c x \left (-2 c e (5 a e+b d)+3 b^2 e^2+2 c^2 d^2\right )+(c d-3 b e) \left (2 a c e+b^2 (-e)+b c d\right )+5 a c e (2 c d-b e)\right )}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}\right )}{4 \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {d+e x} \left (\left (b^2-4 a c\right ) (c d-b e)-c e x \left (b^2-4 a c\right )\right )}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 1197 |
| \(\displaystyle -\frac {e \left (-\frac {\int \frac {e \left ((2 c d-b e) \left (c^2 d^2-3 b^2 e^2-c e (b d-13 a e)\right )+c \left (2 c^2 d^2+3 b^2 e^2-2 c e (b d+5 a e)\right ) (d+e x)\right )}{c d^2-b e d+a e^2+c (d+e x)^2-(2 c d-b e) (d+e x)}d\sqrt {d+e x}}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {d+e x} \left (c x \left (-2 c e (5 a e+b d)+3 b^2 e^2+2 c^2 d^2\right )+(c d-3 b e) \left (2 a c e+b^2 (-e)+b c d\right )+5 a c e (2 c d-b e)\right )}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}\right )}{4 \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {d+e x} \left (\left (b^2-4 a c\right ) (c d-b e)-c e x \left (b^2-4 a c\right )\right )}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {e \left (-\frac {e \int \frac {(2 c d-b e) \left (c^2 d^2-3 b^2 e^2-c e (b d-13 a e)\right )+c \left (2 c^2 d^2+3 b^2 e^2-2 c e (b d+5 a e)\right ) (d+e x)}{c d^2-b e d+a e^2+c (d+e x)^2-(2 c d-b e) (d+e x)}d\sqrt {d+e x}}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {d+e x} \left (c x \left (-2 c e (5 a e+b d)+3 b^2 e^2+2 c^2 d^2\right )+(c d-3 b e) \left (2 a c e+b^2 (-e)+b c d\right )+5 a c e (2 c d-b e)\right )}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}\right )}{4 \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {d+e x} \left (\left (b^2-4 a c\right ) (c d-b e)-c e x \left (b^2-4 a c\right )\right )}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle -\frac {e \left (-\frac {e \left (\frac {1}{2} c \left (-\frac {(2 c d-b e) \left (-4 c e (b d-4 a e)-3 b^2 e^2+4 c^2 d^2\right )}{e \sqrt {b^2-4 a c}}-2 c e (5 a e+b d)+3 b^2 e^2+2 c^2 d^2\right ) \int \frac {1}{\frac {1}{2} \left (\left (b+\sqrt {b^2-4 a c}\right ) e-2 c d\right )+c (d+e x)}d\sqrt {d+e x}+\frac {c \left (-2 c^2 d e \left (-d \sqrt {b^2-4 a c}-16 a e+6 b d\right )-2 c e^2 \left (b d \sqrt {b^2-4 a c}+5 a e \sqrt {b^2-4 a c}+8 a b e+b^2 d\right )+3 b^2 e^3 \left (\sqrt {b^2-4 a c}+b\right )+8 c^3 d^3\right ) \int \frac {1}{\frac {1}{2} \left (\left (b-\sqrt {b^2-4 a c}\right ) e-2 c d\right )+c (d+e x)}d\sqrt {d+e x}}{2 e \sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {d+e x} \left (c x \left (-2 c e (5 a e+b d)+3 b^2 e^2+2 c^2 d^2\right )+(c d-3 b e) \left (2 a c e+b^2 (-e)+b c d\right )+5 a c e (2 c d-b e)\right )}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}\right )}{4 \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {d+e x} \left (\left (b^2-4 a c\right ) (c d-b e)-c e x \left (b^2-4 a c\right )\right )}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {e \left (-\frac {e \left (-\frac {\sqrt {c} \left (-\frac {(2 c d-b e) \left (-4 c e (b d-4 a e)-3 b^2 e^2+4 c^2 d^2\right )}{e \sqrt {b^2-4 a c}}-2 c e (5 a e+b d)+3 b^2 e^2+2 c^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {2} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {\sqrt {c} \left (-2 c^2 d e \left (-d \sqrt {b^2-4 a c}-16 a e+6 b d\right )-2 c e^2 \left (b d \sqrt {b^2-4 a c}+5 a e \sqrt {b^2-4 a c}+8 a b e+b^2 d\right )+3 b^2 e^3 \left (\sqrt {b^2-4 a c}+b\right )+8 c^3 d^3\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {2} e \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {d+e x} \left (c x \left (-2 c e (5 a e+b d)+3 b^2 e^2+2 c^2 d^2\right )+(c d-3 b e) \left (2 a c e+b^2 (-e)+b c d\right )+5 a c e (2 c d-b e)\right )}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}\right )}{4 \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {d+e x} \left (\left (b^2-4 a c\right ) (c d-b e)-c e x \left (b^2-4 a c\right )\right )}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 \left (a e^2-b d e+c d^2\right )}\) |
-1/2*(Sqrt[d + e*x]*((b^2 - 4*a*c)*(c*d - b*e) - c*(b^2 - 4*a*c)*e*x))/((b ^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2)^2) - (e*(-((Sqrt[d + e*x]*(5*a*c*e*(2*c*d - b*e) + (c*d - 3*b*e)*(b*c*d - b^2*e + 2*a*c*e) + c *(2*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(b*d + 5*a*e))*x))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2))) - (e*(-((Sqrt[c]*(8*c^3*d^3 + 3*b^2*(b + Sqrt[b^2 - 4*a*c])*e^3 - 2*c^2*d*e*(6*b*d - Sqrt[b^2 - 4*a*c]*d - 16*a* e) - 2*c*e^2*(b^2*d + b*Sqrt[b^2 - 4*a*c]*d + 8*a*b*e + 5*a*Sqrt[b^2 - 4*a *c]*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[2]*Sqrt[b^2 - 4*a*c]*e*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e])) - (Sqrt[c]*(2*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(b*d + 5*a*e) - ((2 *c*d - b*e)*(4*c^2*d^2 - 3*b^2*e^2 - 4*c*e*(b*d - 4*a*e)))/(Sqrt[b^2 - 4*a *c]*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[2]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e])))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))))/(4*(c*d^2 - b*d*e + a*e^2))
3.17.27.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)), x_Symbol] :> Simp[2 Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; Fr eeQ[{a, b, c, d, e, f, g}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2 *a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^m *(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d* m + b*e*m) - b*d*(3*c*d - b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p] )
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) Int[1/( b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q)) Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
Time = 14.52 (sec) , antiderivative size = 871, normalized size of antiderivative = 1.29
| method | result | size |
| derivativedivides | \(128 e^{4} c^{3} \left (-\frac {\frac {-\frac {\left (-2 b e +4 c d -5 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right ) \left (e x +d \right )^{\frac {3}{2}}}{16 c \left (-2 a c \,e^{2}+b^{2} e^{2}-2 b c d e +2 c^{2} d^{2}+b e \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}-2 d \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\, c \right )}+\frac {\left (-2 b e +4 c d -7 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right ) \sqrt {e x +d}}{16 c^{2} \left (-b e +2 c d -\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right )}}{{\left (-e x -\frac {b e}{2 c}-\frac {\sqrt {e^{2} \left (-4 a c +b^{2}\right )}}{2 c}\right )}^{2}}+\frac {\left (2 b e -4 c d +5 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \left (-8 a c \,e^{2}+4 b^{2} e^{2}-8 b c d e +8 c^{2} d^{2}+4 b e \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}-8 d \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\, c \right ) \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}}{16 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, e^{2} \left (4 a c -b^{2}\right )}+\frac {\frac {-\frac {\left (-2 b e +4 c d +5 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right ) \left (e x +d \right )^{\frac {3}{2}}}{16 c \left (-2 a c \,e^{2}+b^{2} e^{2}-2 b c d e +2 c^{2} d^{2}-b e \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}+2 d \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\, c \right )}+\frac {\left (-2 b e +4 c d +7 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right ) \sqrt {e x +d}}{16 c^{2} \left (-b e +2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right )}}{{\left (-e x -\frac {b e}{2 c}+\frac {\sqrt {e^{2} \left (-4 a c +b^{2}\right )}}{2 c}\right )}^{2}}-\frac {\left (-2 b e +4 c d +5 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \left (8 a c \,e^{2}-4 b^{2} e^{2}+8 b c d e -8 c^{2} d^{2}+4 b e \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}-8 d \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\, c \right ) \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}}{16 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, e^{2} \left (4 a c -b^{2}\right )}\right )\) | \(871\) |
| default | \(128 e^{4} c^{3} \left (-\frac {\frac {-\frac {\left (-2 b e +4 c d -5 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right ) \left (e x +d \right )^{\frac {3}{2}}}{16 c \left (-2 a c \,e^{2}+b^{2} e^{2}-2 b c d e +2 c^{2} d^{2}+b e \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}-2 d \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\, c \right )}+\frac {\left (-2 b e +4 c d -7 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right ) \sqrt {e x +d}}{16 c^{2} \left (-b e +2 c d -\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right )}}{{\left (-e x -\frac {b e}{2 c}-\frac {\sqrt {e^{2} \left (-4 a c +b^{2}\right )}}{2 c}\right )}^{2}}+\frac {\left (2 b e -4 c d +5 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \left (-8 a c \,e^{2}+4 b^{2} e^{2}-8 b c d e +8 c^{2} d^{2}+4 b e \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}-8 d \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\, c \right ) \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}}{16 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, e^{2} \left (4 a c -b^{2}\right )}+\frac {\frac {-\frac {\left (-2 b e +4 c d +5 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right ) \left (e x +d \right )^{\frac {3}{2}}}{16 c \left (-2 a c \,e^{2}+b^{2} e^{2}-2 b c d e +2 c^{2} d^{2}-b e \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}+2 d \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\, c \right )}+\frac {\left (-2 b e +4 c d +7 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right ) \sqrt {e x +d}}{16 c^{2} \left (-b e +2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right )}}{{\left (-e x -\frac {b e}{2 c}+\frac {\sqrt {e^{2} \left (-4 a c +b^{2}\right )}}{2 c}\right )}^{2}}-\frac {\left (-2 b e +4 c d +5 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \left (8 a c \,e^{2}-4 b^{2} e^{2}+8 b c d e -8 c^{2} d^{2}+4 b e \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}-8 d \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\, c \right ) \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}}{16 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, e^{2} \left (4 a c -b^{2}\right )}\right )\) | \(871\) |
| pseudoelliptic | \(\text {Expression too large to display}\) | \(1540\) |
128*e^4*c^3*(-1/16/(-e^2*(4*a*c-b^2))^(1/2)/e^2/(4*a*c-b^2)*((-1/16/c/(-2* a*c*e^2+b^2*e^2-2*b*c*d*e+2*c^2*d^2+b*e*(-4*a*c*e^2+b^2*e^2)^(1/2)-2*d*(-4 *a*c*e^2+b^2*e^2)^(1/2)*c)*(-2*b*e+4*c*d-5*(-4*a*c*e^2+b^2*e^2)^(1/2))*(e* x+d)^(3/2)+1/16/c^2*(-2*b*e+4*c*d-7*(-4*a*c*e^2+b^2*e^2)^(1/2))/(-b*e+2*c* d-(-4*a*c*e^2+b^2*e^2)^(1/2))*(e*x+d)^(1/2))/(-e*x-1/2*b*e/c-1/2/c*(e^2*(- 4*a*c+b^2))^(1/2))^2+1/4*(2*b*e-4*c*d+5*(-4*a*c*e^2+b^2*e^2)^(1/2))/(-8*a* c*e^2+4*b^2*e^2-8*b*c*d*e+8*c^2*d^2+4*b*e*(-4*a*c*e^2+b^2*e^2)^(1/2)-8*d*( -4*a*c*e^2+b^2*e^2)^(1/2)*c)*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2)) *c)^(1/2)*arctan(c*(e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1 /2))*c)^(1/2)))+1/16/(-e^2*(4*a*c-b^2))^(1/2)/e^2/(4*a*c-b^2)*((-1/16/c/(- 2*a*c*e^2+b^2*e^2-2*b*c*d*e+2*c^2*d^2-b*e*(-4*a*c*e^2+b^2*e^2)^(1/2)+2*d*( -4*a*c*e^2+b^2*e^2)^(1/2)*c)*(-2*b*e+4*c*d+5*(-4*a*c*e^2+b^2*e^2)^(1/2))*( e*x+d)^(3/2)+1/16/c^2*(-2*b*e+4*c*d+7*(-4*a*c*e^2+b^2*e^2)^(1/2))/(-b*e+2* c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))*(e*x+d)^(1/2))/(-e*x-1/2*b*e/c+1/2/c*(e^2* (-4*a*c+b^2))^(1/2))^2-1/4*(-2*b*e+4*c*d+5*(-4*a*c*e^2+b^2*e^2)^(1/2))/(8* a*c*e^2-4*b^2*e^2+8*b*c*d*e-8*c^2*d^2+4*b*e*(-4*a*c*e^2+b^2*e^2)^(1/2)-8*d *(-4*a*c*e^2+b^2*e^2)^(1/2)*c)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/ 2))*c)^(1/2)*arctanh(c*(e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2 ))^(1/2))*c)^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 41435 vs. \(2 (605) = 1210\).
Time = 35.49 (sec) , antiderivative size = 41435, normalized size of antiderivative = 61.57 \[ \int \frac {b+2 c x}{\sqrt {d+e x} \left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {b+2 c x}{\sqrt {d+e x} \left (a+b x+c x^2\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {b+2 c x}{\sqrt {d+e x} \left (a+b x+c x^2\right )^3} \, dx=\int { \frac {2 \, c x + b}{{\left (c x^{2} + b x + a\right )}^{3} \sqrt {e x + d}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 5956 vs. \(2 (605) = 1210\).
Time = 2.17 (sec) , antiderivative size = 5956, normalized size of antiderivative = 8.85 \[ \int \frac {b+2 c x}{\sqrt {d+e x} \left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \]
1/32*((b^2*c^2*d^4*e - 4*a*c^3*d^4*e - 2*b^3*c*d^3*e^2 + 8*a*b*c^2*d^3*e^2 + b^4*d^2*e^3 - 2*a*b^2*c*d^2*e^3 - 8*a^2*c^2*d^2*e^3 - 2*a*b^3*d*e^4 + 8 *a^2*b*c*d*e^4 + a^2*b^2*e^5 - 4*a^3*c*e^5)^2*(2*c^2*d^2*e^2 - 2*b*c*d*e^3 + (3*b^2 - 10*a*c)*e^4)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e) + 2*(2*sqrt(b^2 - 4*a*c)*c^5*d^7*e^2 - 7*sqrt(b^2 - 4*a*c)*b*c^4*d^6*e^3 + 3*(b^2*c^3 + 10*a*c^4)*sqrt(b^2 - 4*a*c)*d^5*e^4 + 5*(2*b^3*c^2 - 15*a*b* c^3)*sqrt(b^2 - 4*a*c)*d^4*e^5 - (11*b^4*c - 48*a*b^2*c^2 - 54*a^2*c^3)*sq rt(b^2 - 4*a*c)*d^3*e^6 + 3*(b^5 + a*b^3*c - 27*a^2*b*c^2)*sqrt(b^2 - 4*a* c)*d^2*e^7 - (6*a*b^4 - 21*a^2*b^2*c - 26*a^3*c^2)*sqrt(b^2 - 4*a*c)*d*e^8 + (3*a^2*b^3 - 13*a^3*b*c)*sqrt(b^2 - 4*a*c)*e^9)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*abs(b^2*c^2*d^4*e - 4*a*c^3*d^4*e - 2*b^3*c*d^3* e^2 + 8*a*b*c^2*d^3*e^2 + b^4*d^2*e^3 - 2*a*b^2*c*d^2*e^3 - 8*a^2*c^2*d^2* e^3 - 2*a*b^3*d*e^4 + 8*a^2*b*c*d*e^4 + a^2*b^2*e^5 - 4*a^3*c*e^5) - (16*( b^2*c^8 - 4*a*c^9)*d^12*e^2 - 96*(b^3*c^7 - 4*a*b*c^8)*d^11*e^3 + 8*(29*b^ 4*c^6 - 100*a*b^2*c^7 - 64*a^2*c^8)*d^10*e^4 - 40*(7*b^5*c^5 - 12*a*b^3*c^ 6 - 64*a^2*b*c^7)*d^9*e^5 + (157*b^6*c^4 + 636*a*b^4*c^5 - 4704*a^2*b^2*c^ 6 - 1408*a^3*c^7)*d^8*e^6 - 4*(b^7*c^3 + 300*a*b^5*c^4 - 864*a^2*b^3*c^5 - 1408*a^3*b*c^6)*d^7*e^7 - 14*(3*b^8*c^2 - 50*a*b^6*c^3 + 576*a^3*b^2*c^5 + 128*a^4*c^6)*d^6*e^8 + 4*(5*b^9*c - 27*a*b^7*c^2 - 336*a^2*b^5*c^3 + 112 0*a^3*b^3*c^4 + 1344*a^4*b*c^5)*d^5*e^9 - (3*b^10 + 40*a*b^8*c - 590*a^...
Time = 20.42 (sec) , antiderivative size = 84064, normalized size of antiderivative = 124.91 \[ \int \frac {b+2 c x}{\sqrt {d+e x} \left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \]
((3*(d + e*x)^(3/2)*(6*a^2*c^2*e^6 - b^4*e^6 - 2*c^4*d^4*e^2 + 20*a*c^3*d^ 2*e^4 + 4*b*c^3*d^3*e^3 - 8*b^2*c^2*d^2*e^4 + 2*a*b^2*c*e^6 + 6*b^3*c*d*e^ 5 - 20*a*b*c^2*d*e^5))/(4*(4*a*c - b^2)*(a*e^2 + c*d^2 - b*d*e)^2) - ((d + e*x)^(1/2)*(5*b^3*e^5 - 2*c^3*d^3*e^2 + 3*b*c^2*d^2*e^3 - 19*a*b*c*e^5 + 38*a*c^2*d*e^4 - 11*b^2*c*d*e^4))/(4*(4*a*c - b^2)*(a*e^2 + c*d^2 - b*d*e) ) + (3*(b*e - 2*c*d)*(d + e*x)^(5/2)*(7*a*c^2*e^4 - 2*b^2*c*e^4 - c^3*d^2* e^2 + b*c^2*d*e^3))/(4*(4*a*c - b^2)*(a*e^2 + c*d^2 - b*d*e)^2) + (c*(d + e*x)^(7/2)*(10*a*c^2*e^4 - 3*b^2*c*e^4 - 2*c^3*d^2*e^2 + 2*b*c^2*d*e^3))/( 4*(4*a*c - b^2)*(a*e^2 + c*d^2 - b*d*e)^2))/(c^2*(d + e*x)^4 - (d + e*x)*( 4*c^2*d^3 + 2*b^2*d*e^2 - 2*a*b*e^3 + 4*a*c*d*e^2 - 6*b*c*d^2*e) - (4*c^2* d - 2*b*c*e)*(d + e*x)^3 + (d + e*x)^2*(b^2*e^2 + 6*c^2*d^2 + 2*a*c*e^2 - 6*b*c*d*e) + a^2*e^4 + c^2*d^4 + b^2*d^2*e^2 - 2*a*b*d*e^3 - 2*b*c*d^3*e + 2*a*c*d^2*e^2) - atan(((((106496*a^6*c^7*d*e^10 - 53248*a^6*b*c^6*e^11 - 192*a^2*b^9*c^2*e^11 + 3136*a^3*b^7*c^3*e^11 - 19200*a^4*b^5*c^4*e^11 + 52 224*a^5*b^3*c^5*e^11 + 8192*a^3*c^10*d^7*e^4 + 122880*a^4*c^9*d^5*e^6 + 22 1184*a^5*c^8*d^3*e^8 - 128*b^6*c^7*d^7*e^4 + 448*b^7*c^6*d^6*e^5 - 192*b^8 *c^5*d^5*e^6 - 640*b^9*c^4*d^4*e^7 + 704*b^10*c^3*d^3*e^8 - 192*b^11*c^2*d ^2*e^9 - 6144*a^2*b^2*c^9*d^7*e^4 + 21504*a^2*b^3*c^8*d^6*e^5 + 13824*a^2* b^4*c^7*d^5*e^6 - 88320*a^2*b^5*c^6*d^4*e^7 + 67200*a^2*b^6*c^5*d^3*e^8 - 1728*a^2*b^7*c^4*d^2*e^9 - 79872*a^3*b^2*c^8*d^5*e^6 + 271360*a^3*b^3*c...