\(\int \frac {(d+e x)^{7/2}}{(a+b x+c x^2)^2} \, dx\) [539]

Optimal result

Integrand size = 22, antiderivative size = 602 \[ \int \frac {(d+e x)^{7/2}}{\left (a+b x+c x^2\right )^2} \, dx=\frac {e \left (2 c^2 d^2+3 b^2 e^2-2 c e (b d+5 a e)\right ) \sqrt {d+e x}}{c^2 \left (b^2-4 a c\right )}+\frac {e (2 c d-b e) (d+e x)^{3/2}}{c \left (b^2-4 a c\right )}-\frac {(d+e x)^{5/2} (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\left (e (2 c d-b e) \left (c^2 d^2-3 b^2 e^2-c e (b d-13 a e)\right )+\frac {8 c^4 d^4-3 b^4 e^4-4 c^3 d^2 e (4 b d-9 a e)+b^2 c e^3 (5 b d+19 a e)+c^2 e^2 \left (3 b^2 d^2-36 a b d e-20 a^2 e^2\right )}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} c^{5/2} \left (b^2-4 a c\right ) \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {\left (e (2 c d-b e) \left (c^2 d^2-3 b^2 e^2-c e (b d-13 a e)\right )-\frac {8 c^4 d^4-3 b^4 e^4-4 c^3 d^2 e (4 b d-9 a e)+b^2 c e^3 (5 b d+19 a e)+c^2 e^2 \left (3 b^2 d^2-36 a b d e-20 a^2 e^2\right )}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} c^{5/2} \left (b^2-4 a c\right ) \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \] Output:

e*(2*c^2*d^2+3*b^2*e^2-2*c*e*(5*a*e+b*d))*(e*x+d)^(1/2)/c^2/(-4*a*c+b^2)+e 
*(-b*e+2*c*d)*(e*x+d)^(3/2)/c/(-4*a*c+b^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)+1/2*(e*(-b*e+2*c*d)*(c^2*d^2-3*b^2*e^ 
2-c*e*(-13*a*e+b*d))+(8*c^4*d^4-3*b^4*e^4-4*c^3*d^2*e*(-9*a*e+4*b*d)+b^2*c 
*e^3*(19*a*e+5*b*d)+c^2*e^2*(-20*a^2*e^2-36*a*b*d*e+3*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^(5/2)/(-4*a*c+b^2)/(2*c*d-(b-(-4*a*c+b^2)^(1/2))*e 
)^(1/2)+1/2*(e*(-b*e+2*c*d)*(c^2*d^2-3*b^2*e^2-c*e*(-13*a*e+b*d))-(8*c^4*d 
^4-3*b^4*e^4-4*c^3*d^2*e*(-9*a*e+4*b*d)+b^2*c*e^3*(19*a*e+5*b*d)+c^2*e^2*( 
-20*a^2*e^2-36*a*b*d*e+3*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^(5/2) 
/(-4*a*c+b^2)/(2*c*d-(b+(-4*a*c+b^2)^(1/2))*e)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 11.15 (sec) , antiderivative size = 771, normalized size of antiderivative = 1.28 \[ \int \frac {(d+e x)^{7/2}}{\left (a+b x+c x^2\right )^2} \, dx=\frac {-\frac {2 \sqrt {c} \sqrt {d+e x} \left (-3 b^3 e^3 x+b^2 e^2 (-3 a e+c x (3 d-2 e x))+b c \left (c d^2 (d-3 e x)+a e^2 (3 d+11 e x)\right )+2 c \left (5 a^2 e^3+c^2 d^3 x+a c e \left (-3 d^2-3 d e x+4 e^2 x^2\right )\right )\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}-\frac {\left (8 i c^4 d^4-3 b^3 \left (i b+\sqrt {-b^2+4 a c}\right ) e^4-2 c^3 d^2 e \left (8 i b d+\sqrt {-b^2+4 a c} d-18 i a e\right )+b c e^3 \left (5 i b^2 d+5 b \sqrt {-b^2+4 a c} d+19 i a b e+13 a \sqrt {-b^2+4 a c} e\right )+c^2 e^2 \left (3 i b^2 d^2+2 a e \left (-13 \sqrt {-b^2+4 a c} d-10 i a e\right )+3 b d \left (\sqrt {-b^2+4 a c} d-12 i a e\right )\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e-i \sqrt {-b^2+4 a c} e}}\right )}{\left (-b^2+4 a c\right )^{3/2} \sqrt {-c d+\frac {1}{2} \left (b-i \sqrt {-b^2+4 a c}\right ) e}}-\frac {\left (-8 i c^4 d^4-3 b^3 \left (-i b+\sqrt {-b^2+4 a c}\right ) e^4-2 c^3 d^2 e \left (-8 i b d+\sqrt {-b^2+4 a c} d+18 i a e\right )+b c e^3 \left (-5 i b^2 d+5 b \sqrt {-b^2+4 a c} d-19 i a b e+13 a \sqrt {-b^2+4 a c} e\right )+c^2 e^2 \left (-3 i b^2 d^2+2 a e \left (-13 \sqrt {-b^2+4 a c} d+10 i a e\right )+3 b d \left (\sqrt {-b^2+4 a c} d+12 i a e\right )\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e+i \sqrt {-b^2+4 a c} e}}\right )}{\left (-b^2+4 a c\right )^{3/2} \sqrt {-c d+\frac {1}{2} \left (b+i \sqrt {-b^2+4 a c}\right ) e}}}{2 c^{5/2}} \] Input:

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

Output:

((-2*Sqrt[c]*Sqrt[d + e*x]*(-3*b^3*e^3*x + b^2*e^2*(-3*a*e + c*x*(3*d - 2* 
e*x)) + b*c*(c*d^2*(d - 3*e*x) + a*e^2*(3*d + 11*e*x)) + 2*c*(5*a^2*e^3 + 
c^2*d^3*x + a*c*e*(-3*d^2 - 3*d*e*x + 4*e^2*x^2))))/((b^2 - 4*a*c)*(a + x* 
(b + c*x))) - (((8*I)*c^4*d^4 - 3*b^3*(I*b + Sqrt[-b^2 + 4*a*c])*e^4 - 2*c 
^3*d^2*e*((8*I)*b*d + Sqrt[-b^2 + 4*a*c]*d - (18*I)*a*e) + b*c*e^3*((5*I)* 
b^2*d + 5*b*Sqrt[-b^2 + 4*a*c]*d + (19*I)*a*b*e + 13*a*Sqrt[-b^2 + 4*a*c]* 
e) + c^2*e^2*((3*I)*b^2*d^2 + 2*a*e*(-13*Sqrt[-b^2 + 4*a*c]*d - (10*I)*a*e 
) + 3*b*d*(Sqrt[-b^2 + 4*a*c]*d - (12*I)*a*e)))*ArcTan[(Sqrt[2]*Sqrt[c]*Sq 
rt[d + e*x])/Sqrt[-2*c*d + b*e - I*Sqrt[-b^2 + 4*a*c]*e]])/((-b^2 + 4*a*c) 
^(3/2)*Sqrt[-(c*d) + ((b - I*Sqrt[-b^2 + 4*a*c])*e)/2]) - (((-8*I)*c^4*d^4 
 - 3*b^3*((-I)*b + Sqrt[-b^2 + 4*a*c])*e^4 - 2*c^3*d^2*e*((-8*I)*b*d + Sqr 
t[-b^2 + 4*a*c]*d + (18*I)*a*e) + b*c*e^3*((-5*I)*b^2*d + 5*b*Sqrt[-b^2 + 
4*a*c]*d - (19*I)*a*b*e + 13*a*Sqrt[-b^2 + 4*a*c]*e) + c^2*e^2*((-3*I)*b^2 
*d^2 + 2*a*e*(-13*Sqrt[-b^2 + 4*a*c]*d + (10*I)*a*e) + 3*b*d*(Sqrt[-b^2 + 
4*a*c]*d + (12*I)*a*e)))*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c* 
d + b*e + I*Sqrt[-b^2 + 4*a*c]*e]])/((-b^2 + 4*a*c)^(3/2)*Sqrt[-(c*d) + (( 
b + I*Sqrt[-b^2 + 4*a*c])*e)/2]))/(2*c^(5/2))
 

Rubi [A] (verified)

Time = 1.41 (sec) , antiderivative size = 597, 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, 1196, 1196, 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.

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

\(\Big \downarrow \) 1164

\(\displaystyle -\frac {\int \frac {(d+e x)^{3/2} \left (4 c d^2-7 b e d+10 a e^2-3 e (2 c d-b e) x\right )}{2 \left (c x^2+b x+a\right )}dx}{b^2-4 a c}-\frac {(d+e x)^{5/2} (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {(d+e x)^{3/2} \left (4 c d^2-e (7 b d-10 a e)-3 e (2 c d-b e) x\right )}{c x^2+b x+a}dx}{2 \left (b^2-4 a c\right )}-\frac {(d+e x)^{5/2} (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\)

\(\Big \downarrow \) 1196

\(\displaystyle -\frac {\frac {\int \frac {\sqrt {d+e x} \left (4 c^2 d^3-c e (7 b d-16 a e) d-3 a b e^3-e \left (2 c^2 d^2+3 b^2 e^2-2 c e (b d+5 a e)\right ) x\right )}{c x^2+b x+a}dx}{c}-\frac {2 e (d+e x)^{3/2} (2 c d-b e)}{c}}{2 \left (b^2-4 a c\right )}-\frac {(d+e x)^{5/2} (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\)

\(\Big \downarrow \) 1196

\(\displaystyle -\frac {\frac {\frac {\int \frac {4 c^3 d^4-c^2 e (7 b d-18 a e) d^2+3 a b^2 e^4-5 a c e^3 (b d+2 a e)+e (2 c d-b e) \left (c^2 d^2-3 b^2 e^2-c e (b d-13 a e)\right ) x}{\sqrt {d+e x} \left (c x^2+b x+a\right )}dx}{c}-\frac {2 e \sqrt {d+e x} \left (-2 c e (5 a e+b d)+3 b^2 e^2+2 c^2 d^2\right )}{c}}{c}-\frac {2 e (d+e x)^{3/2} (2 c d-b e)}{c}}{2 \left (b^2-4 a c\right )}-\frac {(d+e x)^{5/2} (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\)

\(\Big \downarrow \) 1197

\(\displaystyle -\frac {\frac {\frac {2 \int \frac {e \left (\left (c d^2-b e d+a e^2\right ) \left (2 c^2 d^2-2 b c e d+3 b^2 e^2-10 a c e^2\right )+(2 c d-b e) \left (c^2 d^2-3 b^2 e^2-c e (b d-13 a e)\right ) (d+e x)\right )}{c d^2-b e d+a e^2+c (d+e x)^2-(2 c d-b e) (d+e x)}d\sqrt {d+e x}}{c}-\frac {2 e \sqrt {d+e x} \left (-2 c e (5 a e+b d)+3 b^2 e^2+2 c^2 d^2\right )}{c}}{c}-\frac {2 e (d+e x)^{3/2} (2 c d-b e)}{c}}{2 \left (b^2-4 a c\right )}-\frac {(d+e x)^{5/2} (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {\frac {2 e \int \frac {\left (c d^2-b e d+a e^2\right ) \left (2 c^2 d^2+3 b^2 e^2-2 c e (b d+5 a e)\right )+(2 c d-b e) \left (c^2 d^2-3 b^2 e^2-c e (b d-13 a e)\right ) (d+e x)}{c d^2-b e d+a e^2+c (d+e x)^2-(2 c d-b e) (d+e x)}d\sqrt {d+e x}}{c}-\frac {2 e \sqrt {d+e x} \left (-2 c e (5 a e+b d)+3 b^2 e^2+2 c^2 d^2\right )}{c}}{c}-\frac {2 e (d+e x)^{3/2} (2 c d-b e)}{c}}{2 \left (b^2-4 a c\right )}-\frac {(d+e x)^{5/2} (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\)

\(\Big \downarrow \) 1480

\(\displaystyle -\frac {\frac {\frac {2 e \left (\frac {1}{2} \left (\frac {c^2 e^2 \left (-20 a^2 e^2-36 a b d e+3 b^2 d^2\right )+b^2 c e^3 (19 a e+5 b d)-4 c^3 d^2 e (4 b d-9 a e)-3 b^4 e^4+8 c^4 d^4}{e \sqrt {b^2-4 a c}}+(2 c d-b e) \left (-c e (b d-13 a e)-3 b^2 e^2+c^2 d^2\right )\right ) \int \frac {1}{\frac {1}{2} \left (\left (b-\sqrt {b^2-4 a c}\right ) e-2 c d\right )+c (d+e x)}d\sqrt {d+e x}+\frac {1}{2} \left ((2 c d-b e) \left (-c e (b d-13 a e)-3 b^2 e^2+c^2 d^2\right )-\frac {c^2 e^2 \left (-20 a^2 e^2-36 a b d e+3 b^2 d^2\right )+b^2 c e^3 (19 a e+5 b d)-4 c^3 d^2 e (4 b d-9 a e)-3 b^4 e^4+8 c^4 d^4}{e \sqrt {b^2-4 a c}}\right ) \int \frac {1}{\frac {1}{2} \left (\left (b+\sqrt {b^2-4 a c}\right ) e-2 c d\right )+c (d+e x)}d\sqrt {d+e x}\right )}{c}-\frac {2 e \sqrt {d+e x} \left (-2 c e (5 a e+b d)+3 b^2 e^2+2 c^2 d^2\right )}{c}}{c}-\frac {2 e (d+e x)^{3/2} (2 c d-b e)}{c}}{2 \left (b^2-4 a c\right )}-\frac {(d+e x)^{5/2} (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {\frac {\frac {2 e \left (-\frac {\left (\frac {c^2 e^2 \left (-20 a^2 e^2-36 a b d e+3 b^2 d^2\right )+b^2 c e^3 (19 a e+5 b d)-4 c^3 d^2 e (4 b d-9 a e)-3 b^4 e^4+8 c^4 d^4}{e \sqrt {b^2-4 a c}}+(2 c d-b e) \left (-c e (b d-13 a e)-3 b^2 e^2+c^2 d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {2} \sqrt {c} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}-\frac {\left ((2 c d-b e) \left (-c e (b d-13 a e)-3 b^2 e^2+c^2 d^2\right )-\frac {c^2 e^2 \left (-20 a^2 e^2-36 a b d e+3 b^2 d^2\right )+b^2 c e^3 (19 a e+5 b d)-4 c^3 d^2 e (4 b d-9 a e)-3 b^4 e^4+8 c^4 d^4}{e \sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {2} \sqrt {c} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{c}-\frac {2 e \sqrt {d+e x} \left (-2 c e (5 a e+b d)+3 b^2 e^2+2 c^2 d^2\right )}{c}}{c}-\frac {2 e (d+e x)^{3/2} (2 c d-b e)}{c}}{2 \left (b^2-4 a c\right )}-\frac {(d+e x)^{5/2} (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\)

Input:

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

Output:

-(((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*e*(2*c*d - b*e)*(d + e*x)^(3/2))/c + ((-2*e*(2*c^2*d^2 
 + 3*b^2*e^2 - 2*c*e*(b*d + 5*a*e))*Sqrt[d + e*x])/c + (2*e*(-((((2*c*d - 
b*e)*(c^2*d^2 - 3*b^2*e^2 - c*e*(b*d - 13*a*e)) + (8*c^4*d^4 - 3*b^4*e^4 - 
 4*c^3*d^2*e*(4*b*d - 9*a*e) + b^2*c*e^3*(5*b*d + 19*a*e) + c^2*e^2*(3*b^2 
*d^2 - 36*a*b*d*e - 20*a^2*e^2))/(Sqrt[b^2 - 4*a*c]*e))*ArcTanh[(Sqrt[2]*S 
qrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[2]*S 
qrt[c]*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e])) - (((2*c*d - b*e)*(c^2*d^ 
2 - 3*b^2*e^2 - c*e*(b*d - 13*a*e)) - (8*c^4*d^4 - 3*b^4*e^4 - 4*c^3*d^2*e 
*(4*b*d - 9*a*e) + b^2*c*e^3*(5*b*d + 19*a*e) + c^2*e^2*(3*b^2*d^2 - 36*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)/c)/(2*(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 1196

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + 
(c_.)*(x_)^2), x_Symbol] :> Simp[g*((d + e*x)^m/(c*m)), x] + Simp[1/c   Int 
[(d + e*x)^(m - 1)*(Simp[c*d*f - a*e*g + (g*c*d - b*e*g + c*e*f)*x, x]/(a + 
 b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && FractionQ[m] & 
& GtQ[m, 0]
 

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 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 = 1.93 (sec) , antiderivative size = 644, normalized size of antiderivative = 1.07

method	result	size
pseudoelliptic	\(\frac {\frac {5 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 ) \left (\frac {13 \left (b e -2 c d \right ) \left (\left (a c -\frac {3 b^{2}}{13}\right ) e^{2}-\frac {b c d e}{13}+\frac {c^{2} d^{2}}{13}\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}}{20}+\left (\left (a c -\frac {b^{2}}{5}\right ) e^{2}-\frac {b c d e}{5}+\frac {c^{2} d^{2}}{5}\right ) \left (\left (a c -\frac {3 b^{2}}{4}\right ) e^{2}+2 b c d e -2 c^{2} d^{2}\right )\right ) \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 )}{2}+\frac {5 \sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \left (\left (-\frac {13 \left (b e -2 c d \right ) \left (\left (a c -\frac {3 b^{2}}{13}\right ) e^{2}-\frac {b c d e}{13}+\frac {c^{2} d^{2}}{13}\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}}{20}+\left (\left (a c -\frac {b^{2}}{5}\right ) e^{2}-\frac {b c d e}{5}+\frac {c^{2} d^{2}}{5}\right ) \left (\left (a c -\frac {3 b^{2}}{4}\right ) e^{2}+2 b c d e -2 c^{2} d^{2}\right )\right ) e \sqrt {2}\, \left (c \,x^{2}+b x +a \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}\, \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \left (\left (\frac {4 a \,c^{2} x^{2}}{5}+\left (-\frac {1}{5} b^{2} x^{2}+a^{2}+\frac {11}{10} a b x \right ) c -\frac {3 \left (b x +a \right ) b^{2}}{10}\right ) e^{3}+\frac {3 d \left (-2 a c x +\left (b x +a \right ) b \right ) c \,e^{2}}{10}-\frac {3 d^{2} c^{2} \left (\frac {b x}{2}+a \right ) e}{5}+\frac {c^{2} \left (2 c x +b \right ) d^{3}}{10}\right )\right )}{2}}{\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 ) c^{2} \left (a c -\frac {b^{2}}{4}\right )}\)	\(644\)
derivativedivides	\(2 e^{3} \left (\frac {\sqrt {e x +d}}{c^{2}}-\frac {\frac {-\frac {\left (3 a b c \,e^{3}-6 d \,e^{2} a \,c^{2}-b^{3} e^{3}+3 d \,e^{2} b^{2} c -3 d^{2} e b \,c^{2}+2 d^{3} c^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}}{2 e^{2} \left (4 a c -b^{2}\right )}-\frac {\left (2 e^{4} a^{2} c -a \,b^{2} e^{4}+b^{3} d \,e^{3}-3 b^{2} c \,d^{2} e^{2}+4 b \,c^{2} d^{3} e -2 d^{4} c^{3}\right ) \sqrt {e x +d}}{2 e^{2} \left (4 a c -b^{2}\right )}}{c \left (e x +d \right )^{2}+b e \left (e x +d \right )-2 c d \left (e x +d \right )+a \,e^{2}-b d e +c \,d^{2}}+\frac {2 c \left (-\frac {\left (20 e^{4} a^{2} c^{2}-19 a \,b^{2} c \,e^{4}+36 a b \,c^{2} d \,e^{3}-36 d^{2} e^{2} a \,c^{3}+3 b^{4} e^{4}-5 d \,e^{3} b^{3} c -3 d^{2} e^{2} b^{2} c^{2}+16 d^{3} e b \,c^{3}-8 d^{4} c^{4}+13 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b c \,e^{3}-26 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,c^{2} d \,e^{2}-3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} e^{3}+5 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} c d \,e^{2}+3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b \,c^{2} d^{2} e -2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{3} d^{3}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}+\frac {\left (-20 e^{4} a^{2} c^{2}+19 a \,b^{2} c \,e^{4}-36 a b \,c^{2} d \,e^{3}+36 d^{2} e^{2} a \,c^{3}-3 b^{4} e^{4}+5 d \,e^{3} b^{3} c +3 d^{2} e^{2} b^{2} c^{2}-16 d^{3} e b \,c^{3}+8 d^{4} c^{4}+13 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b c \,e^{3}-26 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,c^{2} d \,e^{2}-3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} e^{3}+5 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} c d \,e^{2}+3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b \,c^{2} d^{2} e -2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{3} d^{3}\right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{e^{2} \left (4 a c -b^{2}\right )}}{c^{2}}\right )\)	\(945\)
default	\(2 e^{3} \left (\frac {\sqrt {e x +d}}{c^{2}}-\frac {\frac {-\frac {\left (3 a b c \,e^{3}-6 d \,e^{2} a \,c^{2}-b^{3} e^{3}+3 d \,e^{2} b^{2} c -3 d^{2} e b \,c^{2}+2 d^{3} c^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}}{2 e^{2} \left (4 a c -b^{2}\right )}-\frac {\left (2 e^{4} a^{2} c -a \,b^{2} e^{4}+b^{3} d \,e^{3}-3 b^{2} c \,d^{2} e^{2}+4 b \,c^{2} d^{3} e -2 d^{4} c^{3}\right ) \sqrt {e x +d}}{2 e^{2} \left (4 a c -b^{2}\right )}}{c \left (e x +d \right )^{2}+b e \left (e x +d \right )-2 c d \left (e x +d \right )+a \,e^{2}-b d e +c \,d^{2}}+\frac {2 c \left (-\frac {\left (20 e^{4} a^{2} c^{2}-19 a \,b^{2} c \,e^{4}+36 a b \,c^{2} d \,e^{3}-36 d^{2} e^{2} a \,c^{3}+3 b^{4} e^{4}-5 d \,e^{3} b^{3} c -3 d^{2} e^{2} b^{2} c^{2}+16 d^{3} e b \,c^{3}-8 d^{4} c^{4}+13 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b c \,e^{3}-26 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,c^{2} d \,e^{2}-3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} e^{3}+5 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} c d \,e^{2}+3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b \,c^{2} d^{2} e -2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{3} d^{3}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}+\frac {\left (-20 e^{4} a^{2} c^{2}+19 a \,b^{2} c \,e^{4}-36 a b \,c^{2} d \,e^{3}+36 d^{2} e^{2} a \,c^{3}-3 b^{4} e^{4}+5 d \,e^{3} b^{3} c +3 d^{2} e^{2} b^{2} c^{2}-16 d^{3} e b \,c^{3}+8 d^{4} c^{4}+13 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b c \,e^{3}-26 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,c^{2} d \,e^{2}-3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} e^{3}+5 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} c d \,e^{2}+3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b \,c^{2} d^{2} e -2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{3} d^{3}\right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{e^{2} \left (4 a c -b^{2}\right )}}{c^{2}}\right )\)	\(945\)
risch	\(\frac {2 e^{3} \sqrt {e x +d}}{c^{2}}-\frac {2 e^{3} \left (\frac {-\frac {\left (3 a b c \,e^{3}-6 d \,e^{2} a \,c^{2}-b^{3} e^{3}+3 d \,e^{2} b^{2} c -3 d^{2} e b \,c^{2}+2 d^{3} c^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}}{2 e^{2} \left (4 a c -b^{2}\right )}-\frac {\left (2 e^{4} a^{2} c -a \,b^{2} e^{4}+b^{3} d \,e^{3}-3 b^{2} c \,d^{2} e^{2}+4 b \,c^{2} d^{3} e -2 d^{4} c^{3}\right ) \sqrt {e x +d}}{2 e^{2} \left (4 a c -b^{2}\right )}}{c \left (e x +d \right )^{2}+b e \left (e x +d \right )-2 c d \left (e x +d \right )+a \,e^{2}-b d e +c \,d^{2}}+\frac {2 c \left (-\frac {\left (20 e^{4} a^{2} c^{2}-19 a \,b^{2} c \,e^{4}+36 a b \,c^{2} d \,e^{3}-36 d^{2} e^{2} a \,c^{3}+3 b^{4} e^{4}-5 d \,e^{3} b^{3} c -3 d^{2} e^{2} b^{2} c^{2}+16 d^{3} e b \,c^{3}-8 d^{4} c^{4}+13 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b c \,e^{3}-26 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,c^{2} d \,e^{2}-3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} e^{3}+5 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} c d \,e^{2}+3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b \,c^{2} d^{2} e -2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{3} d^{3}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}+\frac {\left (-20 e^{4} a^{2} c^{2}+19 a \,b^{2} c \,e^{4}-36 a b \,c^{2} d \,e^{3}+36 d^{2} e^{2} a \,c^{3}-3 b^{4} e^{4}+5 d \,e^{3} b^{3} c +3 d^{2} e^{2} b^{2} c^{2}-16 d^{3} e b \,c^{3}+8 d^{4} c^{4}+13 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b c \,e^{3}-26 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a \,c^{2} d \,e^{2}-3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{3} e^{3}+5 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} c d \,e^{2}+3 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b \,c^{2} d^{2} e -2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c^{3} d^{3}\right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{e^{2} \left (4 a c -b^{2}\right )}\right )}{c^{2}}\)	\(947\)

Input:

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

Output:

5/2/((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)*(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)*(13/20*(b*e-2*c*d)*((a*c-3/13*b^2)*e^2- 
1/13*b*c*d*e+1/13*c^2*d^2)*(-4*e^2*(a*c-1/4*b^2))^(1/2)+((a*c-1/5*b^2)*e^2 
-1/5*b*c*d*e+1/5*c^2*d^2)*((a*c-3/4*b^2)*e^2+2*b*c*d*e-2*c^2*d^2))*arctanh 
((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))+((-b*e+2*c*d+(-4*e^2*(a*c-1/4*b^2))^(1/2))*c)^(1/2)*((-13/20*(b*e-2*c* 
d)*((a*c-3/13*b^2)*e^2-1/13*b*c*d*e+1/13*c^2*d^2)*(-4*e^2*(a*c-1/4*b^2))^( 
1/2)+((a*c-1/5*b^2)*e^2-1/5*b*c*d*e+1/5*c^2*d^2)*((a*c-3/4*b^2)*e^2+2*b*c* 
d*e-2*c^2*d^2))*e*2^(1/2)*(c*x^2+b*x+a)*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)*(-4*e^2*(a*c-1/4*b^2))^(1/2)*((4/5 
*a*c^2*x^2+(-1/5*b^2*x^2+a^2+11/10*a*b*x)*c-3/10*(b*x+a)*b^2)*e^3+3/10*d*( 
-2*a*c*x+(b*x+a)*b)*c*e^2-3/5*d^2*c^2*(1/2*b*x+a)*e+1/10*c^2*(2*c*x+b)*d^3 
)))/(-4*e^2*(a*c-1/4*b^2))^(1/2)/(c*x^2+b*x+a)/c^2/(a*c-1/4*b^2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 10077 vs. \(2 (558) = 1116\).

Time = 12.32 (sec) , antiderivative size = 10077, normalized size of antiderivative = 16.74 \[ \int \frac {(d+e x)^{7/2}}{\left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1785 vs. \(2 (558) = 1116\).

Time = 0.92 (sec) , antiderivative size = 1785, normalized size of antiderivative = 2.97 \[ \int \frac {(d+e x)^{7/2}}{\left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

2*sqrt(e*x + d)*e^3/c^2 - (2*(e*x + d)^(3/2)*c^3*d^3*e - 2*sqrt(e*x + d)*c 
^3*d^4*e - 3*(e*x + d)^(3/2)*b*c^2*d^2*e^2 + 4*sqrt(e*x + d)*b*c^2*d^3*e^2 
 + 3*(e*x + d)^(3/2)*b^2*c*d*e^3 - 6*(e*x + d)^(3/2)*a*c^2*d*e^3 - 3*sqrt( 
e*x + d)*b^2*c*d^2*e^3 - (e*x + d)^(3/2)*b^3*e^4 + 3*(e*x + d)^(3/2)*a*b*c 
*e^4 + sqrt(e*x + d)*b^3*d*e^4 - sqrt(e*x + d)*a*b^2*e^5 + 2*sqrt(e*x + d) 
*a^2*c*e^5)/((b^2*c^2 - 4*a*c^3)*((e*x + d)^2*c - 2*(e*x + d)*c*d + c*d^2 
+ (e*x + d)*b*e - b*d*e + a*e^2)) - (16*(b^2*c^9 - 4*a*c^10)*d^5*e - 40*(b 
^3*c^8 - 4*a*b*c^9)*d^4*e^2 + 2*(11*b^4*c^7 - 8*a*b^2*c^8 - 144*a^2*c^9)*d 
^3*e^3 + (7*b^5*c^6 - 136*a*b^3*c^7 + 432*a^2*b*c^8)*d^2*e^4 - (11*b^6*c^5 
 - 118*a*b^4*c^6 + 336*a^2*b^2*c^7 - 160*a^3*c^8)*d*e^5 + (3*b^7*c^4 - 31* 
a*b^5*c^5 + 96*a^2*b^3*c^6 - 80*a^3*b*c^7)*e^6 - (2*c^3*d^3*e - 3*b*c^2*d^ 
2*e^2 - (5*b^2*c - 26*a*c^2)*d*e^3 + (3*b^3 - 13*a*b*c)*e^4)*(b^2*c^2*e - 
4*a*c^3*e)^2 - 2*(2*sqrt(b^2 - 4*a*c)*c^6*d^4*e - 4*sqrt(b^2 - 4*a*c)*b*c^ 
5*d^3*e^2 + (5*b^2*c^4 - 8*a*c^5)*sqrt(b^2 - 4*a*c)*d^2*e^3 - (3*b^3*c^3 - 
 8*a*b*c^4)*sqrt(b^2 - 4*a*c)*d*e^4 + (3*a*b^2*c^3 - 10*a^2*c^4)*sqrt(b^2 
- 4*a*c)*e^5)*abs(-b^2*c^2*e + 4*a*c^3*e))*arctan(2*sqrt(1/2)*sqrt(e*x + d 
)/sqrt(-(2*b^2*c^3*d - 8*a*c^4*d - b^3*c^2*e + 4*a*b*c^3*e + sqrt((2*b^2*c 
^3*d - 8*a*c^4*d - b^3*c^2*e + 4*a*b*c^3*e)^2 - 4*(b^2*c^3*d^2 - 4*a*c^4*d 
^2 - b^3*c^2*d*e + 4*a*b*c^3*d*e + a*b^2*c^2*e^2 - 4*a^2*c^3*e^2)*(b^2*c^3 
 - 4*a*c^4)))/(b^2*c^3 - 4*a*c^4)))/(sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 ...
 

Mupad [B] (verification not implemented)

Time = 7.27 (sec) , antiderivative size = 32541, normalized size of antiderivative = 54.05 \[ \int \frac {(d+e x)^{7/2}}{\left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

(2*e^3*(d + e*x)^(1/2))/c^2 - atan(((((2560*a^5*c^7*e^7 + 12*a*b^8*c^3*e^7 
 - 12*b^9*c^3*d*e^6 - 184*a^2*b^6*c^4*e^7 + 1056*a^3*b^4*c^5*e^7 - 2688*a^ 
4*b^2*c^6*e^7 - 512*a^3*c^9*d^4*e^3 + 2048*a^4*c^8*d^2*e^5 + 8*b^6*c^6*d^4 
*e^3 - 16*b^7*c^5*d^3*e^4 + 20*b^8*c^4*d^2*e^5 + 384*a^2*b^2*c^8*d^4*e^3 - 
 768*a^2*b^3*c^7*d^3*e^4 + 1344*a^2*b^4*c^6*d^2*e^5 - 2816*a^3*b^2*c^7*d^2 
*e^5 + 176*a*b^7*c^4*d*e^6 - 2048*a^4*b*c^7*d*e^6 - 96*a*b^4*c^7*d^4*e^3 + 
 192*a*b^5*c^6*d^3*e^4 - 272*a*b^6*c^5*d^2*e^5 - 960*a^2*b^5*c^5*d*e^6 + 1 
024*a^3*b*c^8*d^3*e^4 + 2304*a^3*b^3*c^6*d*e^6)/(64*a^3*c^6 - b^6*c^3 + 12 
*a*b^4*c^4 - 48*a^2*b^2*c^5) - (2*(d + e*x)^(1/2)*((32*b^6*c^7*d^7 - 2048* 
a^3*c^10*d^7 - 9*b^13*e^7 - 9*b^4*e^7*(-(4*a*c - b^2)^9)^(1/2) - 384*a*b^4 
*c^8*d^7 - 26880*a^6*b*c^6*e^7 + 53760*a^6*c^7*d*e^6 - 112*b^7*c^6*d^6*e + 
 1536*a^2*b^2*c^9*d^7 - 2077*a^2*b^9*c^2*e^7 + 10656*a^3*b^7*c^3*e^7 - 302 
40*a^4*b^5*c^4*e^7 + 44800*a^5*b^3*c^5*e^7 - 25*a^2*c^2*e^7*(-(4*a*c - b^2 
)^9)^(1/2) - 17920*a^4*c^9*d^5*e^2 - 35840*a^5*c^8*d^3*e^4 + 98*b^8*c^5*d^ 
5*e^2 + 35*b^9*c^4*d^4*e^3 - 70*b^10*c^3*d^3*e^4 + 14*b^11*c^2*d^2*e^5 + 3 
5*c^4*d^4*e^3*(-(4*a*c - b^2)^9)^(1/2) + 213*a*b^11*c*e^7 + 21*b^12*c*d*e^ 
6 + 1344*a^2*b^4*c^7*d^5*e^2 + 10080*a^2*b^5*c^6*d^4*e^3 - 7840*a^2*b^6*c^ 
5*d^3*e^4 - 1008*a^2*b^7*c^4*d^2*e^5 + 7168*a^3*b^2*c^8*d^5*e^2 - 35840*a^ 
3*b^3*c^7*d^4*e^3 + 17920*a^3*b^4*c^6*d^3*e^4 + 12544*a^3*b^5*c^5*d^2*e^5 
- 44800*a^4*b^3*c^6*d^2*e^5 + 14*b^2*c^2*d^2*e^5*(-(4*a*c - b^2)^9)^(1/...
 

Reduce [F]

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

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

Output:

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