\(\int \frac {(d+e x)^{7/2}}{(a+b x+c x^2)^3} \, dx\) [545]

Optimal result

Integrand size = 22, antiderivative size = 683 \[ \int \frac {(d+e x)^{7/2}}{\left (a+b x+c x^2\right )^3} \, dx=-\frac {e \left (24 c^2 d^2+b^2 e^2-4 c e (6 b d-5 a e)\right ) \sqrt {d+e x}}{4 c \left (b^2-4 a c\right )^2}-\frac {(d+e x)^{5/2} (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {(d+e x)^{3/2} \left (11 b^2 d e+4 a c d e-12 b \left (c d^2+a e^2\right )-\left (24 c^2 d^2+b^2 e^2-4 c e (6 b d-5 a e)\right ) x\right )}{4 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {\left (e (2 c d-b e) \left (12 c^2 d^2-b^2 e^2-4 c e (3 b d-4 a e)\right )+\frac {96 c^4 d^4-b^4 e^4-8 c^3 d^2 e (24 b d-19 a e)-2 b^2 c e^3 (5 b d-9 a e)+2 c^2 e^2 \left (53 b^2 d^2-76 a b d e+20 a^2 e^2\right )}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{4 \sqrt {2} c^{3/2} \left (b^2-4 a c\right )^2 \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {\left (e (2 c d-b e) \left (12 c^2 d^2-b^2 e^2-4 c e (3 b d-4 a e)\right )-\frac {96 c^4 d^4-b^4 e^4-8 c^3 d^2 e (24 b d-19 a e)-2 b^2 c e^3 (5 b d-9 a e)+2 c^2 e^2 \left (53 b^2 d^2-76 a b d e+20 a^2 e^2\right )}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{4 \sqrt {2} c^{3/2} \left (b^2-4 a c\right )^2 \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \] Output:

-1/4*e*(24*c^2*d^2+b^2*e^2-4*c*e*(-5*a*e+6*b*d))*(e*x+d)^(1/2)/c/(-4*a*c+b 
^2)^2-1/2*(e*x+d)^(5/2)*(b*d-2*a*e+(-b*e+2*c*d)*x)/(-4*a*c+b^2)/(c*x^2+b*x 
+a)^2-1/4*(e*x+d)^(3/2)*(11*b^2*d*e+4*a*c*d*e-12*b*(a*e^2+c*d^2)-(24*c^2*d 
^2+b^2*e^2-4*c*e*(-5*a*e+6*b*d))*x)/(-4*a*c+b^2)^2/(c*x^2+b*x+a)-1/8*(e*(- 
b*e+2*c*d)*(12*c^2*d^2-b^2*e^2-4*c*e*(-4*a*e+3*b*d))+(96*c^4*d^4-b^4*e^4-8 
*c^3*d^2*e*(-19*a*e+24*b*d)-2*b^2*c*e^3*(-9*a*e+5*b*d)+2*c^2*e^2*(20*a^2*e 
^2-76*a*b*d*e+53*b^2*d^2))/(-4*a*c+b^2)^(1/2))*arctanh(2^(1/2)*c^(1/2)*(e* 
x+d)^(1/2)/(2*c*d-(b-(-4*a*c+b^2)^(1/2))*e)^(1/2))*2^(1/2)/c^(3/2)/(-4*a*c 
+b^2)^2/(2*c*d-(b-(-4*a*c+b^2)^(1/2))*e)^(1/2)-1/8*(e*(-b*e+2*c*d)*(12*c^2 
*d^2-b^2*e^2-4*c*e*(-4*a*e+3*b*d))-(96*c^4*d^4-b^4*e^4-8*c^3*d^2*e*(-19*a* 
e+24*b*d)-2*b^2*c*e^3*(-9*a*e+5*b*d)+2*c^2*e^2*(20*a^2*e^2-76*a*b*d*e+53*b 
^2*d^2))/(-4*a*c+b^2)^(1/2))*arctanh(2^(1/2)*c^(1/2)*(e*x+d)^(1/2)/(2*c*d- 
(b+(-4*a*c+b^2)^(1/2))*e)^(1/2))*2^(1/2)/c^(3/2)/(-4*a*c+b^2)^2/(2*c*d-(b+ 
(-4*a*c+b^2)^(1/2))*e)^(1/2)
 

Mathematica [A] (verified)

Time = 15.05 (sec) , antiderivative size = 720, normalized size of antiderivative = 1.05 \[ \int \frac {(d+e x)^{7/2}}{\left (a+b x+c x^2\right )^3} \, dx=\frac {-4 \sqrt {c} \sqrt {b^2-4 a c} \sqrt {d+e x} \left (b^4 e^3 x^2+b^2 \left (a^2 e^3+c^2 d x \left (-8 d^2+55 d e x-10 e^2 x^2\right )+a c e \left (7 d^2-58 d e x+5 e^2 x^2\right )\right )+4 c \left (5 a^3 e^3-6 c^3 d^3 x^3-a c^2 d x \left (10 d^2+d e x+8 e^2 x^2\right )+a^2 c e \left (11 d^2+4 d e x+9 e^2 x^2\right )\right )-4 b c \left (a^2 e^2 (9 d-7 e x)+9 c^2 d^2 x^2 (d-e x)+a c \left (5 d^3-14 d^2 e x+11 d e^2 x^2-4 e^3 x^3\right )\right )+b^3 \left (2 a e^3 x+c \left (2 d^3+13 d^2 e x-16 d e^2 x^2-e^3 x^3\right )\right )\right )-\sqrt {4 c d+2 \left (-b+\sqrt {b^2-4 a c}\right ) e} \left (96 c^3 d^3+b^2 \left (b-\sqrt {b^2-4 a c}\right ) e^3-8 c^2 d e \left (18 b d+3 \sqrt {b^2-4 a c} d-13 a e\right )+2 c e^2 \left (23 b^2 d+12 b \sqrt {b^2-4 a c} d-26 a b e-10 a \sqrt {b^2-4 a c} e\right )\right ) (a+x (b+c x))^2 \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-b e+\sqrt {b^2-4 a c} e}}\right )+\sqrt {4 c d-2 \left (b+\sqrt {b^2-4 a c}\right ) e} \left (96 c^3 d^3+b^2 \left (b+\sqrt {b^2-4 a c}\right ) e^3+8 c^2 d e \left (-18 b d+3 \sqrt {b^2-4 a c} d+13 a e\right )+2 c e^2 \left (23 b^2 d+10 a \sqrt {b^2-4 a c} e-2 b \left (6 \sqrt {b^2-4 a c} d+13 a e\right )\right )\right ) (a+x (b+c x))^2 \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{16 c^{3/2} \left (b^2-4 a c\right )^{5/2} (a+x (b+c x))^2} \] Input:

Integrate[(d + e*x)^(7/2)/(a + b*x + c*x^2)^3,x]
 

Output:

(-4*Sqrt[c]*Sqrt[b^2 - 4*a*c]*Sqrt[d + e*x]*(b^4*e^3*x^2 + b^2*(a^2*e^3 + 
c^2*d*x*(-8*d^2 + 55*d*e*x - 10*e^2*x^2) + a*c*e*(7*d^2 - 58*d*e*x + 5*e^2 
*x^2)) + 4*c*(5*a^3*e^3 - 6*c^3*d^3*x^3 - a*c^2*d*x*(10*d^2 + d*e*x + 8*e^ 
2*x^2) + a^2*c*e*(11*d^2 + 4*d*e*x + 9*e^2*x^2)) - 4*b*c*(a^2*e^2*(9*d - 7 
*e*x) + 9*c^2*d^2*x^2*(d - e*x) + a*c*(5*d^3 - 14*d^2*e*x + 11*d*e^2*x^2 - 
 4*e^3*x^3)) + b^3*(2*a*e^3*x + c*(2*d^3 + 13*d^2*e*x - 16*d*e^2*x^2 - e^3 
*x^3))) - Sqrt[4*c*d + 2*(-b + Sqrt[b^2 - 4*a*c])*e]*(96*c^3*d^3 + b^2*(b 
- Sqrt[b^2 - 4*a*c])*e^3 - 8*c^2*d*e*(18*b*d + 3*Sqrt[b^2 - 4*a*c]*d - 13* 
a*e) + 2*c*e^2*(23*b^2*d + 12*b*Sqrt[b^2 - 4*a*c]*d - 26*a*b*e - 10*a*Sqrt 
[b^2 - 4*a*c]*e))*(a + x*(b + c*x))^2*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e* 
x])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]] + Sqrt[4*c*d - 2*(b + Sqrt[b^ 
2 - 4*a*c])*e]*(96*c^3*d^3 + b^2*(b + Sqrt[b^2 - 4*a*c])*e^3 + 8*c^2*d*e*( 
-18*b*d + 3*Sqrt[b^2 - 4*a*c]*d + 13*a*e) + 2*c*e^2*(23*b^2*d + 10*a*Sqrt[ 
b^2 - 4*a*c]*e - 2*b*(6*Sqrt[b^2 - 4*a*c]*d + 13*a*e)))*(a + x*(b + c*x))^ 
2*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a 
*c])*e]])/(16*c^(3/2)*(b^2 - 4*a*c)^(5/2)*(a + x*(b + c*x))^2)
 

Rubi [A] (verified)

Time = 1.43 (sec) , antiderivative size = 674, normalized size of antiderivative = 0.99, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {1164, 27, 1233, 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.

\(\displaystyle \int \frac {(d+e x)^{7/2}}{\left (a+b x+c x^2\right )^3} \, dx\)

\(\Big \downarrow \) 1164

\(\displaystyle -\frac {\int \frac {(d+e x)^{3/2} \left (12 c d^2-11 b e d+10 a e^2+e (2 c d-b e) x\right )}{2 \left (c x^2+b x+a\right )^2}dx}{2 \left (b^2-4 a c\right )}-\frac {(d+e x)^{5/2} (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {(d+e x)^{3/2} \left (12 c d^2-e (11 b d-10 a e)+e (2 c d-b e) x\right )}{\left (c x^2+b x+a\right )^2}dx}{4 \left (b^2-4 a c\right )}-\frac {(d+e x)^{5/2} (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 1233

\(\displaystyle -\frac {\frac {\int -\frac {48 c^3 d^4-4 c^2 e (21 b d-19 a e) d^2+a b^2 e^4+5 c e^2 \left (7 b^2 d^2-12 a b e d+4 a^2 e^2\right )+e (2 c d-b e) \left (12 c^2 d^2-b^2 e^2-4 c e (3 b d-4 a e)\right ) x}{2 \sqrt {d+e x} \left (c x^2+b x+a\right )}dx}{c \left (b^2-4 a c\right )}-\frac {\sqrt {d+e x} \left (x (2 c d-b e) \left (-4 c e (3 b d-2 a e)+b^2 e^2+12 c^2 d^2\right )-\left (b^2 \left (a e^3+11 c d^2 e\right )\right )+12 b c d \left (3 a e^2+c d^2\right )-4 a c e \left (5 a e^2+7 c d^2\right )\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}}{4 \left (b^2-4 a c\right )}-\frac {(d+e x)^{5/2} (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {\int \frac {48 c^3 d^4-4 c^2 e (21 b d-19 a e) d^2+a b^2 e^4+5 c e^2 \left (7 b^2 d^2-12 a b e d+4 a^2 e^2\right )+e (2 c d-b e) \left (12 c^2 d^2-b^2 e^2-4 c e (3 b d-4 a e)\right ) x}{\sqrt {d+e x} \left (c x^2+b x+a\right )}dx}{2 c \left (b^2-4 a c\right )}-\frac {\sqrt {d+e x} \left (x (2 c d-b e) \left (-4 c e (3 b d-2 a e)+b^2 e^2+12 c^2 d^2\right )-\left (b^2 \left (a e^3+11 c d^2 e\right )\right )+12 b c d \left (3 a e^2+c d^2\right )-4 a c e \left (5 a e^2+7 c d^2\right )\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}}{4 \left (b^2-4 a c\right )}-\frac {(d+e x)^{5/2} (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 1197

\(\displaystyle -\frac {-\frac {\int \frac {e \left (\left (c d^2-b e d+a e^2\right ) \left (24 c^2 d^2-24 b c e d+b^2 e^2+20 a c e^2\right )+(2 c d-b e) \left (12 c^2 d^2-b^2 e^2-4 c e (3 b d-4 a e)\right ) (d+e x)\right )}{c d^2-b e d+a e^2+c (d+e x)^2-(2 c d-b e) (d+e x)}d\sqrt {d+e x}}{c \left (b^2-4 a c\right )}-\frac {\sqrt {d+e x} \left (x (2 c d-b e) \left (-4 c e (3 b d-2 a e)+b^2 e^2+12 c^2 d^2\right )-\left (b^2 \left (a e^3+11 c d^2 e\right )\right )+12 b c d \left (3 a e^2+c d^2\right )-4 a c e \left (5 a e^2+7 c d^2\right )\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}}{4 \left (b^2-4 a c\right )}-\frac {(d+e x)^{5/2} (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {e \int \frac {\left (c d^2-b e d+a e^2\right ) \left (24 c^2 d^2+b^2 e^2-4 c e (6 b d-5 a e)\right )+(2 c d-b e) \left (12 c^2 d^2-b^2 e^2-4 c e (3 b d-4 a e)\right ) (d+e x)}{c d^2-b e d+a e^2+c (d+e x)^2-(2 c d-b e) (d+e x)}d\sqrt {d+e x}}{c \left (b^2-4 a c\right )}-\frac {\sqrt {d+e x} \left (x (2 c d-b e) \left (-4 c e (3 b d-2 a e)+b^2 e^2+12 c^2 d^2\right )-\left (b^2 \left (a e^3+11 c d^2 e\right )\right )+12 b c d \left (3 a e^2+c d^2\right )-4 a c e \left (5 a e^2+7 c d^2\right )\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}}{4 \left (b^2-4 a c\right )}-\frac {(d+e x)^{5/2} (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 1480

\(\displaystyle -\frac {-\frac {e \left (\frac {1}{2} \left (\frac {2 c^2 e^2 \left (20 a^2 e^2-76 a b d e+53 b^2 d^2\right )-2 b^2 c e^3 (5 b d-9 a e)-8 c^3 d^2 e (24 b d-19 a e)-b^4 e^4+96 c^4 d^4}{e \sqrt {b^2-4 a c}}+(2 c d-b e) \left (-4 c e (3 b d-4 a e)-b^2 e^2+12 c^2 d^2\right )\right ) \int \frac {1}{\frac {1}{2} \left (\left (b-\sqrt {b^2-4 a c}\right ) e-2 c d\right )+c (d+e x)}d\sqrt {d+e x}+\frac {1}{2} \left ((2 c d-b e) \left (-4 c e (3 b d-4 a e)-b^2 e^2+12 c^2 d^2\right )-\frac {2 c^2 e^2 \left (20 a^2 e^2-76 a b d e+53 b^2 d^2\right )-2 b^2 c e^3 (5 b d-9 a e)-8 c^3 d^2 e (24 b d-19 a e)-b^4 e^4+96 c^4 d^4}{e \sqrt {b^2-4 a c}}\right ) \int \frac {1}{\frac {1}{2} \left (\left (b+\sqrt {b^2-4 a c}\right ) e-2 c d\right )+c (d+e x)}d\sqrt {d+e x}\right )}{c \left (b^2-4 a c\right )}-\frac {\sqrt {d+e x} \left (x (2 c d-b e) \left (-4 c e (3 b d-2 a e)+b^2 e^2+12 c^2 d^2\right )-\left (b^2 \left (a e^3+11 c d^2 e\right )\right )+12 b c d \left (3 a e^2+c d^2\right )-4 a c e \left (5 a e^2+7 c d^2\right )\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}}{4 \left (b^2-4 a c\right )}-\frac {(d+e x)^{5/2} (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {-\frac {e \left (-\frac {\left (\frac {2 c^2 e^2 \left (20 a^2 e^2-76 a b d e+53 b^2 d^2\right )-2 b^2 c e^3 (5 b d-9 a e)-8 c^3 d^2 e (24 b d-19 a e)-b^4 e^4+96 c^4 d^4}{e \sqrt {b^2-4 a c}}+(2 c d-b e) \left (-4 c e (3 b d-4 a e)-b^2 e^2+12 c^2 d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {2} \sqrt {c} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}-\frac {\left ((2 c d-b e) \left (-4 c e (3 b d-4 a e)-b^2 e^2+12 c^2 d^2\right )-\frac {2 c^2 e^2 \left (20 a^2 e^2-76 a b d e+53 b^2 d^2\right )-2 b^2 c e^3 (5 b d-9 a e)-8 c^3 d^2 e (24 b d-19 a e)-b^4 e^4+96 c^4 d^4}{e \sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {2} \sqrt {c} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{c \left (b^2-4 a c\right )}-\frac {\sqrt {d+e x} \left (x (2 c d-b e) \left (-4 c e (3 b d-2 a e)+b^2 e^2+12 c^2 d^2\right )-\left (b^2 \left (a e^3+11 c d^2 e\right )\right )+12 b c d \left (3 a e^2+c d^2\right )-4 a c e \left (5 a e^2+7 c d^2\right )\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}}{4 \left (b^2-4 a c\right )}-\frac {(d+e x)^{5/2} (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\)

Input:

Int[(d + e*x)^(7/2)/(a + b*x + c*x^2)^3,x]
 

Output:

-1/2*((d + e*x)^(5/2)*(b*d - 2*a*e + (2*c*d - b*e)*x))/((b^2 - 4*a*c)*(a + 
 b*x + c*x^2)^2) - (-((Sqrt[d + e*x]*(12*b*c*d*(c*d^2 + 3*a*e^2) - 4*a*c*e 
*(7*c*d^2 + 5*a*e^2) - b^2*(11*c*d^2*e + a*e^3) + (2*c*d - b*e)*(12*c^2*d^ 
2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e))*x))/(c*(b^2 - 4*a*c)*(a + b*x + c*x^2 
))) - (e*(-((((2*c*d - b*e)*(12*c^2*d^2 - b^2*e^2 - 4*c*e*(3*b*d - 4*a*e)) 
 + (96*c^4*d^4 - b^4*e^4 - 8*c^3*d^2*e*(24*b*d - 19*a*e) - 2*b^2*c*e^3*(5* 
b*d - 9*a*e) + 2*c^2*e^2*(53*b^2*d^2 - 76*a*b*d*e + 20*a^2*e^2))/(Sqrt[b^2 
 - 4*a*c]*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b - Sq 
rt[b^2 - 4*a*c])*e]])/(Sqrt[2]*Sqrt[c]*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c] 
)*e])) - (((2*c*d - b*e)*(12*c^2*d^2 - b^2*e^2 - 4*c*e*(3*b*d - 4*a*e)) - 
(96*c^4*d^4 - b^4*e^4 - 8*c^3*d^2*e*(24*b*d - 19*a*e) - 2*b^2*c*e^3*(5*b*d 
 - 9*a*e) + 2*c^2*e^2*(53*b^2*d^2 - 76*a*b*d*e + 20*a^2*e^2))/(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[c]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e 
])))/(c*(b^2 - 4*a*c)))/(4*(b^2 - 4*a*c))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 221

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]
 

rule 1164

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(d + e*x)^(m - 1)*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b*x 
+ c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a* 
c))   Int[(d + e*x)^(m - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2* 
c*d^2*(2*p + 3) + e*(b*e - 2*d*c)*(m + 2*p + 2)*x, x]*(a + b*x + c*x^2)^(p 
+ 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[p, -1] && GtQ[m, 1] && Int 
QuadraticQ[a, b, c, d, e, m, p, x]
 

rule 1197

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]
 

rule 1233

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(d + e*x)^(m - 1))*(a + b*x + c*x^2) 
^(p + 1)*((2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g - c 
*(b*e*f + b*d*g + 2*a*e*g))*x)/(c*(p + 1)*(b^2 - 4*a*c))), x] - Simp[1/(c*( 
p + 1)*(b^2 - 4*a*c))   Int[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Sim 
p[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2*a*e*(e*f 
*(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*( 
m + p + 1) + 2*c^2*d*f*(m + 2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2* 
p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && LtQ[p, -1] && 
GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, b, c, d, e, f, g]) | 
|  !ILtQ[m + 2*p + 3, 0])
 

rule 1480

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]
 

Maple [A] (verified)

Time = 5.86 (sec) , antiderivative size = 785, normalized size of antiderivative = 1.15

method	result	size
pseudoelliptic	\(-\frac {5 \left (\left (-\frac {2 \left (\frac {3 c^{2} d^{2}}{4}+e \left (a e -\frac {3 b d}{4}\right ) c -\frac {b^{2} e^{2}}{16}\right ) \left (b e -2 c d \right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}}{5}+\left (\frac {4 c^{2} d^{2}}{5}+e \left (a e -\frac {4 b d}{5}\right ) c -\frac {b^{2} e^{2}}{20}\right ) \left (3 c^{2} d^{2}+\left (a \,e^{2}-3 b d e \right ) c +\frac {b^{2} e^{2}}{2}\right )\right ) e \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {2}\, \left (c \,x^{2}+b x +a \right )^{2} \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )+\left (e \sqrt {2}\, \left (c \,x^{2}+b x +a \right )^{2} \left (\frac {2 \left (\frac {3 c^{2} d^{2}}{4}+e \left (a e -\frac {3 b d}{4}\right ) c -\frac {b^{2} e^{2}}{16}\right ) \left (b e -2 c d \right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}}{5}+\left (\frac {4 c^{2} d^{2}}{5}+e \left (a e -\frac {4 b d}{5}\right ) c -\frac {b^{2} e^{2}}{20}\right ) \left (3 c^{2} d^{2}+\left (a \,e^{2}-3 b d e \right ) c +\frac {b^{2} e^{2}}{2}\right )\right ) \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )+\sqrt {e x +d}\, \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \left (-\frac {6 d^{3} c^{4} x^{3}}{5}-2 d x \left (\frac {4 a \,e^{2} x^{2}}{5}+\frac {d x \left (-9 b x +a \right ) e}{10}+d^{2} \left (\frac {9 b x}{10}+a \right )\right ) c^{3}+\left (\frac {9 x^{2} a \left (\frac {4 b x}{9}+a \right ) e^{3}}{5}+\frac {4 d x \left (-\frac {5}{8} b^{2} x^{2}-\frac {11}{4} a b x +a^{2}\right ) e^{2}}{5}+\frac {11 d^{2} \left (\frac {5}{4} b^{2} x^{2}+\frac {14}{11} a b x +a^{2}\right ) e}{5}-\left (\frac {2 b x}{5}+a \right ) d^{3} b \right ) c^{2}+\left (\left (\frac {1}{4} a \,b^{2} x^{2}+\frac {7}{5} a^{2} b x -\frac {1}{20} b^{3} x^{3}+a^{3}\right ) e^{3}-\frac {9 d b \left (\frac {4}{9} b^{2} x^{2}+\frac {29}{18} a b x +a^{2}\right ) e^{2}}{5}+\frac {7 d^{2} \left (\frac {13 b x}{7}+a \right ) b^{2} e}{20}+\frac {b^{3} d^{3}}{10}\right ) c +\frac {b^{2} e^{3} \left (b x +a \right )^{2}}{20}\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) \sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\right )}{16 \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \left (c \,x^{2}+b x +a \right )^{2} c \left (a c -\frac {b^{2}}{4}\right )^{2}}\)	\(785\)
derivativedivides	\(\text {Expression too large to display}\)	\(1277\)
default	\(\text {Expression too large to display}\)	\(1277\)

Input:

int((e*x+d)^(7/2)/(c*x^2+b*x+a)^3,x,method=_RETURNVERBOSE)
 

Output:

-5/16/((b*e-2*c*d+(-4*e^2*(a*c-1/4*b^2))^(1/2))*c)^(1/2)/((-b*e+2*c*d+(-4* 
e^2*(a*c-1/4*b^2))^(1/2))*c)^(1/2)*((-2/5*(3/4*c^2*d^2+e*(a*e-3/4*b*d)*c-1 
/16*b^2*e^2)*(b*e-2*c*d)*(-4*e^2*(a*c-1/4*b^2))^(1/2)+(4/5*c^2*d^2+e*(a*e- 
4/5*b*d)*c-1/20*b^2*e^2)*(3*c^2*d^2+(a*e^2-3*b*d*e)*c+1/2*b^2*e^2))*e*((b* 
e-2*c*d+(-4*e^2*(a*c-1/4*b^2))^(1/2))*c)^(1/2)*2^(1/2)*(c*x^2+b*x+a)^2*arc 
tanh((e*x+d)^(1/2)*c*2^(1/2)/((-b*e+2*c*d+(-4*e^2*(a*c-1/4*b^2))^(1/2))*c) 
^(1/2))+(e*2^(1/2)*(c*x^2+b*x+a)^2*(2/5*(3/4*c^2*d^2+e*(a*e-3/4*b*d)*c-1/1 
6*b^2*e^2)*(b*e-2*c*d)*(-4*e^2*(a*c-1/4*b^2))^(1/2)+(4/5*c^2*d^2+e*(a*e-4/ 
5*b*d)*c-1/20*b^2*e^2)*(3*c^2*d^2+(a*e^2-3*b*d*e)*c+1/2*b^2*e^2))*arctan(( 
e*x+d)^(1/2)*c*2^(1/2)/((b*e-2*c*d+(-4*e^2*(a*c-1/4*b^2))^(1/2))*c)^(1/2)) 
+(e*x+d)^(1/2)*((b*e-2*c*d+(-4*e^2*(a*c-1/4*b^2))^(1/2))*c)^(1/2)*(-6/5*d^ 
3*c^4*x^3-2*d*x*(4/5*a*e^2*x^2+1/10*d*x*(-9*b*x+a)*e+d^2*(9/10*b*x+a))*c^3 
+(9/5*x^2*a*(4/9*b*x+a)*e^3+4/5*d*x*(-5/8*b^2*x^2-11/4*a*b*x+a^2)*e^2+11/5 
*d^2*(5/4*b^2*x^2+14/11*a*b*x+a^2)*e-(2/5*b*x+a)*d^3*b)*c^2+((1/4*a*b^2*x^ 
2+7/5*a^2*b*x-1/20*b^3*x^3+a^3)*e^3-9/5*d*b*(4/9*b^2*x^2+29/18*a*b*x+a^2)* 
e^2+7/20*d^2*(13/7*b*x+a)*b^2*e+1/10*b^3*d^3)*c+1/20*b^2*e^3*(b*x+a)^2)*(- 
4*e^2*(a*c-1/4*b^2))^(1/2))*((-b*e+2*c*d+(-4*e^2*(a*c-1/4*b^2))^(1/2))*c)^ 
(1/2))/(-4*e^2*(a*c-1/4*b^2))^(1/2)/(c*x^2+b*x+a)^2/c/(a*c-1/4*b^2)^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 8780 vs. \(2 (627) = 1254\).

Time = 1.80 (sec) , antiderivative size = 8780, normalized size of antiderivative = 12.86 \[ \int \frac {(d+e x)^{7/2}}{\left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)^(7/2)/(c*x^2+b*x+a)^3,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(d+e x)^{7/2}}{\left (a+b x+c x^2\right )^3} \, dx=\text {Timed out} \] Input:

integrate((e*x+d)**(7/2)/(c*x**2+b*x+a)**3,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(d+e x)^{7/2}}{\left (a+b x+c x^2\right )^3} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {7}{2}}}{{\left (c x^{2} + b x + a\right )}^{3}} \,d x } \] Input:

integrate((e*x+d)^(7/2)/(c*x^2+b*x+a)^3,x, algorithm="maxima")
 

Output:

integrate((e*x + d)^(7/2)/(c*x^2 + b*x + a)^3, x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3037 vs. \(2 (627) = 1254\).

Time = 1.70 (sec) , antiderivative size = 3037, normalized size of antiderivative = 4.45 \[ \int \frac {(d+e x)^{7/2}}{\left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)^(7/2)/(c*x^2+b*x+a)^3,x, algorithm="giac")
 

Output:

1/4*(192*(b^6*c^7 - 12*a*b^4*c^8 + 48*a^2*b^2*c^9 - 64*a^3*c^10)*d^5*e - 4 
80*(b^7*c^6 - 12*a*b^5*c^7 + 48*a^2*b^3*c^8 - 64*a^3*b*c^9)*d^4*e^2 + 4*(1 
01*b^8*c^5 - 1136*a*b^6*c^6 + 3936*a^2*b^4*c^7 - 2816*a^3*b^2*c^8 - 4864*a 
^4*c^9)*d^3*e^3 - 6*(21*b^9*c^4 - 176*a*b^7*c^5 + 96*a^2*b^5*c^6 + 2304*a^ 
3*b^3*c^7 - 4864*a^4*b*c^8)*d^2*e^4 + 4*(2*b^10*c^3 + 23*a*b^8*c^4 - 448*a 
^2*b^6*c^5 + 1888*a^3*b^4*c^6 - 2048*a^4*b^2*c^7 - 1280*a^5*c^8)*d*e^5 + ( 
b^11*c^2 - 30*a*b^9*c^3 + 224*a^2*b^7*c^4 - 448*a^3*b^5*c^5 - 768*a^4*b^3* 
c^6 + 2560*a^5*b*c^7)*e^6 - (24*c^3*d^3*e - 36*b*c^2*d^2*e^2 + 2*(5*b^2*c 
+ 16*a*c^2)*d*e^3 + (b^3 - 16*a*b*c)*e^4)*(b^4*c*e - 8*a*b^2*c^2*e + 16*a^ 
2*c^3*e)^2 - 2*(24*(b^2*c^5 - 4*a*c^6)*sqrt(b^2 - 4*a*c)*d^4*e - 48*(b^3*c 
^4 - 4*a*b*c^5)*sqrt(b^2 - 4*a*c)*d^3*e^2 + (25*b^4*c^3 - 56*a*b^2*c^4 - 1 
76*a^2*c^5)*sqrt(b^2 - 4*a*c)*d^2*e^3 - (b^5*c^2 + 40*a*b^3*c^3 - 176*a^2* 
b*c^4)*sqrt(b^2 - 4*a*c)*d*e^4 + (a*b^4*c^2 + 16*a^2*b^2*c^3 - 80*a^3*c^4) 
*sqrt(b^2 - 4*a*c)*e^5)*abs(b^4*c*e - 8*a*b^2*c^2*e + 16*a^2*c^3*e))*arcta 
n(2*sqrt(1/2)*sqrt(e*x + d)/sqrt(-(2*b^4*c^2*d - 16*a*b^2*c^3*d + 32*a^2*c 
^4*d - b^5*c*e + 8*a*b^3*c^2*e - 16*a^2*b*c^3*e + sqrt((2*b^4*c^2*d - 16*a 
*b^2*c^3*d + 32*a^2*c^4*d - b^5*c*e + 8*a*b^3*c^2*e - 16*a^2*b*c^3*e)^2 - 
4*(b^4*c^2*d^2 - 8*a*b^2*c^3*d^2 + 16*a^2*c^4*d^2 - b^5*c*d*e + 8*a*b^3*c^ 
2*d*e - 16*a^2*b*c^3*d*e + a*b^4*c*e^2 - 8*a^2*b^2*c^2*e^2 + 16*a^3*c^3*e^ 
2)*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)))/(b^4*c^2 - 8*a*b^2*c^3 + 16*a...
 

Mupad [B] (verification not implemented)

Time = 21.17 (sec) , antiderivative size = 20000, normalized size of antiderivative = 29.28 \[ \int \frac {(d+e x)^{7/2}}{\left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \] Input:

int((d + e*x)^(7/2)/(a + b*x + c*x^2)^3,x)
 

Output:

log((e^3*(b*e - 2*c*d)*(a*e^2 + c*d^2 - b*d*e)^2*(35*b^6*e^6 + 27648*c^6*d 
^6 + 6400*a^3*c^3*e^6 + 76032*a*c^5*d^4*e^2 + 9456*a^2*b^2*c^2*e^6 + 57024 
*a^2*c^4*d^2*e^4 + 84672*b^2*c^4*d^4*e^2 - 31104*b^3*c^3*d^3*e^3 + 972*b^4 
*c^2*d^2*e^4 - 1176*a*b^4*c*e^6 - 82944*b*c^5*d^5*e + 756*b^5*c*d*e^5 - 15 
2064*a*b*c^4*d^3*e^3 - 9504*a*b^3*c^2*d*e^5 - 57024*a^2*b*c^3*d*e^5 + 8553 
6*a*b^2*c^3*d^2*e^4))/(64*c*(4*a*c - b^2)^6) - (2^(1/2)*((2^(1/2)*((c*e^3* 
(24*c^3*d^4 + a*b^2*e^4 + 20*a^2*c*e^4 - b^3*d*e^3 + 44*a*c^2*d^2*e^2 + 25 
*b^2*c*d^2*e^2 - 48*b*c^2*d^3*e - 44*a*b*c*d*e^3))/(4*a*c - b^2) - (2^(1/2 
)*c^2*e^2*(4*a*c - b^2)*(b*e - 2*c*d)*(d + e*x)^(1/2)*(-(b^17*e^7 + 471859 
2*a^5*c^12*d^7 - 4608*b^10*c^7*d^7 + b^2*e^7*(-(4*a*c - b^2)^15)^(1/2) + 9 
2160*a*b^8*c^8*d^7 - 1720320*a^8*b*c^8*e^7 + 3440640*a^8*c^9*d*e^6 + 16128 
*b^11*c^6*d^6*e - 737280*a^2*b^6*c^9*d^7 + 2949120*a^3*b^4*c^10*d^7 - 5898 
240*a^4*b^2*c^11*d^7 + 1140*a^2*b^13*c^2*e^7 - 10160*a^3*b^11*c^3*e^7 + 34 
880*a^4*b^9*c^4*e^7 + 43776*a^5*b^7*c^5*e^7 - 680960*a^6*b^5*c^6*e^7 + 186 
3680*a^7*b^3*c^7*e^7 + 13762560*a^6*c^11*d^5*e^2 + 12615680*a^7*c^10*d^3*e 
^4 - 20832*b^12*c^5*d^5*e^2 + 11760*b^13*c^4*d^4*e^3 - 2450*b^14*c^3*d^3*e 
^4 - 21*b^15*c^2*d^2*e^5 - 21*c^2*d^2*e^5*(-(4*a*c - b^2)^15)^(1/2) - 55*a 
*b^15*c*e^7 - 25*a*c*e^7*(-(4*a*c - b^2)^15)^(1/2) + 21*b^16*c*d*e^6 + 21* 
b*c*d*e^6*(-(4*a*c - b^2)^15)^(1/2) - 3064320*a^2*b^8*c^7*d^5*e^2 + 120960 
0*a^2*b^9*c^6*d^4*e^3 + 144480*a^2*b^10*c^5*d^3*e^4 - 136080*a^2*b^11*c...
 

Reduce [F]

\[ \int \frac {(d+e x)^{7/2}}{\left (a+b x+c x^2\right )^3} \, dx=\int \frac {\left (e x +d \right )^{\frac {7}{2}}}{\left (c \,x^{2}+b x +a \right )^{3}}d x \] Input:

int((e*x+d)^(7/2)/(c*x^2+b*x+a)^3,x)
 

Output:

int((e*x+d)^(7/2)/(c*x^2+b*x+a)^3,x)