\(\int \frac {(b+2 c x) \sqrt {a+b x+c x^2}}{(d+e x)^{7/2}} \, dx\) [654]

Optimal result

Integrand size = 30, antiderivative size = 674 \[ \int \frac {(b+2 c x) \sqrt {a+b x+c x^2}}{(d+e x)^{7/2}} \, dx=\frac {2 \left (16 c^2 d^2-b^2 e^2-4 c e (4 b d-5 a e)\right ) \sqrt {a+b x+c x^2}}{15 e^2 \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2}}+\frac {4 (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) \sqrt {a+b x+c x^2}}{15 e^2 \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x}}-\frac {2 (8 c d+b e+10 c e x) \sqrt {a+b x+c x^2}}{5 e^2 (d+e x)^{5/2}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{15 e^3 \left (c d^2-b d e+a e^2\right )^2 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (16 c^2 d^2-b^2 e^2-4 c e (4 b d-5 a e)\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{15 e^3 \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}} \] Output:

2/15*(16*c^2*d^2-b^2*e^2-4*c*e*(-5*a*e+4*b*d))*(c*x^2+b*x+a)^(1/2)/e^2/(a* 
e^2-b*d*e+c*d^2)/(e*x+d)^(3/2)+4/15*(-b*e+2*c*d)*(4*c^2*d^2-b^2*e^2-4*c*e* 
(-2*a*e+b*d))*(c*x^2+b*x+a)^(1/2)/e^2/(a*e^2-b*d*e+c*d^2)^2/(e*x+d)^(1/2)- 
2/5*(10*c*e*x+b*e+8*c*d)*(c*x^2+b*x+a)^(1/2)/e^2/(e*x+d)^(5/2)-2/15*2^(1/2 
)*(-4*a*c+b^2)^(1/2)*(-b*e+2*c*d)*(4*c^2*d^2-b^2*e^2-4*c*e*(-2*a*e+b*d))*( 
e*x+d)^(1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)*EllipticE(1/2*(1+(2*c*x 
+b)/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),(-2*(-4*a*c+b^2)^(1/2)*e/(2*c*d-(b+( 
-4*a*c+b^2)^(1/2))*e))^(1/2))/e^3/(a*e^2-b*d*e+c*d^2)^2/(c*(e*x+d)/(2*c*d- 
(b+(-4*a*c+b^2)^(1/2))*e))^(1/2)/(c*x^2+b*x+a)^(1/2)+2/15*2^(1/2)*(-4*a*c+ 
b^2)^(1/2)*(16*c^2*d^2-b^2*e^2-4*c*e*(-5*a*e+4*b*d))*(c*(e*x+d)/(2*c*d-(b+ 
(-4*a*c+b^2)^(1/2))*e))^(1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)*Ellipt 
icF(1/2*(1+(2*c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),(-2*(-4*a*c+b^2)^(1 
/2)*e/(2*c*d-(b+(-4*a*c+b^2)^(1/2))*e))^(1/2))/e^3/(a*e^2-b*d*e+c*d^2)/(e* 
x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 35.14 (sec) , antiderivative size = 5427, normalized size of antiderivative = 8.05 \[ \int \frac {(b+2 c x) \sqrt {a+b x+c x^2}}{(d+e x)^{7/2}} \, dx=\text {Result too large to show} \] Input:

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

Output:

Result too large to show
 

Rubi [A] (warning: unable to verify)

Time = 1.93 (sec) , antiderivative size = 715, normalized size of antiderivative = 1.06, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {1229, 27, 1237, 27, 1269, 1172, 321, 327}

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.

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

\(\Big \downarrow \) 1229

\(\displaystyle -\frac {2 \int \frac {2 e^2 b^3+5 c d e b^2-8 c \left (c d^2+2 a e^2\right ) b+12 a c^2 d e-c \left (16 c^2 d^2-b^2 e^2-4 c e (4 b d-5 a e)\right ) x}{2 (d+e x)^{3/2} \sqrt {c x^2+b x+a}}dx}{15 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {2 \sqrt {a+b x+c x^2} \left (e x \left (-2 c e (7 b d-5 a e)+b^2 e^2+14 c^2 d^2\right )-c d e (5 b d-4 a e)-b e^2 (2 b d-3 a e)+8 c^2 d^3\right )}{15 e^2 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {2 e^2 b^3+5 c d e b^2-8 c \left (c d^2+2 a e^2\right ) b+12 a c^2 d e-c \left (16 c^2 d^2-b^2 e^2-4 c e (4 b d-5 a e)\right ) x}{(d+e x)^{3/2} \sqrt {c x^2+b x+a}}dx}{15 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {2 \sqrt {a+b x+c x^2} \left (e x \left (-2 c e (7 b d-5 a e)+b^2 e^2+14 c^2 d^2\right )-c d e (5 b d-4 a e)-b e^2 (2 b d-3 a e)+8 c^2 d^3\right )}{15 e^2 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1237

\(\displaystyle -\frac {-\frac {2 \int -\frac {c \left (d e^2 b^3-\left (11 c d^2 e-a e^3\right ) b^2+4 c d \left (2 c d^2+5 a e^2\right ) b-4 a c e \left (c d^2+5 a e^2\right )+2 (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) x\right )}{2 \sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{a e^2-b d e+c d^2}-\frac {4 \sqrt {a+b x+c x^2} (2 c d-b e) \left (-4 c e (b d-2 a e)-b^2 e^2+4 c^2 d^2\right )}{\sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}}{15 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {2 \sqrt {a+b x+c x^2} \left (e x \left (-2 c e (7 b d-5 a e)+b^2 e^2+14 c^2 d^2\right )-c d e (5 b d-4 a e)-b e^2 (2 b d-3 a e)+8 c^2 d^3\right )}{15 e^2 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {c \int \frac {d e^2 b^3-\left (11 c d^2 e-a e^3\right ) b^2+4 c d \left (2 c d^2+5 a e^2\right ) b-4 a c e \left (c d^2+5 a e^2\right )+2 (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{a e^2-b d e+c d^2}-\frac {4 \sqrt {a+b x+c x^2} (2 c d-b e) \left (-4 c e (b d-2 a e)-b^2 e^2+4 c^2 d^2\right )}{\sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}}{15 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {2 \sqrt {a+b x+c x^2} \left (e x \left (-2 c e (7 b d-5 a e)+b^2 e^2+14 c^2 d^2\right )-c d e (5 b d-4 a e)-b e^2 (2 b d-3 a e)+8 c^2 d^3\right )}{15 e^2 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1269

\(\displaystyle -\frac {\frac {c \left (\frac {2 (2 c d-b e) \left (-4 c e (b d-2 a e)-b^2 e^2+4 c^2 d^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x+a}}dx}{e}-\frac {\left (a e^2-b d e+c d^2\right ) \left (20 a c e^2-b^2 e^2-16 b c d e+16 c^2 d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{e}\right )}{a e^2-b d e+c d^2}-\frac {4 \sqrt {a+b x+c x^2} (2 c d-b e) \left (-4 c e (b d-2 a e)-b^2 e^2+4 c^2 d^2\right )}{\sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}}{15 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {2 \sqrt {a+b x+c x^2} \left (e x \left (-2 c e (7 b d-5 a e)+b^2 e^2+14 c^2 d^2\right )-c d e (5 b d-4 a e)-b e^2 (2 b d-3 a e)+8 c^2 d^3\right )}{15 e^2 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1172

\(\displaystyle -\frac {2 \sqrt {c x^2+b x+a} \left (8 c^2 d^3-c e (5 b d-4 a e) d-b e^2 (2 b d-3 a e)+e \left (14 c^2 d^2+b^2 e^2-2 c e (7 b d-5 a e)\right ) x\right )}{15 e^2 \left (c d^2-b e d+a e^2\right ) (d+e x)^{5/2}}-\frac {\frac {c \left (\frac {2 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {c x^2+b x+a}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (c d^2-b e d+a e^2\right ) \left (16 c^2 d^2-16 b c e d-b^2 e^2+20 a c e^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \int \frac {1}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}} \sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {d+e x} \sqrt {c x^2+b x+a}}\right )}{c d^2-b e d+a e^2}-\frac {4 (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) \sqrt {c x^2+b x+a}}{\left (c d^2-b e d+a e^2\right ) \sqrt {d+e x}}}{15 e^2 \left (c d^2-b e d+a e^2\right )}\)

\(\Big \downarrow \) 321

\(\displaystyle -\frac {\frac {c \left (\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (-4 c e (b d-2 a e)-b^2 e^2+4 c^2 d^2\right ) \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \left (20 a c e^2-b^2 e^2-16 b c d e+16 c^2 d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}\right )}{a e^2-b d e+c d^2}-\frac {4 \sqrt {a+b x+c x^2} (2 c d-b e) \left (-4 c e (b d-2 a e)-b^2 e^2+4 c^2 d^2\right )}{\sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}}{15 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {2 \sqrt {a+b x+c x^2} \left (e x \left (-2 c e (7 b d-5 a e)+b^2 e^2+14 c^2 d^2\right )-c d e (5 b d-4 a e)-b e^2 (2 b d-3 a e)+8 c^2 d^3\right )}{15 e^2 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 327

\(\displaystyle -\frac {\frac {c \left (\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (-4 c e (b d-2 a e)-b^2 e^2+4 c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \left (20 a c e^2-b^2 e^2-16 b c d e+16 c^2 d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}\right )}{a e^2-b d e+c d^2}-\frac {4 \sqrt {a+b x+c x^2} (2 c d-b e) \left (-4 c e (b d-2 a e)-b^2 e^2+4 c^2 d^2\right )}{\sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}}{15 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {2 \sqrt {a+b x+c x^2} \left (e x \left (-2 c e (7 b d-5 a e)+b^2 e^2+14 c^2 d^2\right )-c d e (5 b d-4 a e)-b e^2 (2 b d-3 a e)+8 c^2 d^3\right )}{15 e^2 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )}\)

Input:

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

Output:

(-2*(8*c^2*d^3 - c*d*e*(5*b*d - 4*a*e) - b*e^2*(2*b*d - 3*a*e) + e*(14*c^2 
*d^2 + b^2*e^2 - 2*c*e*(7*b*d - 5*a*e))*x)*Sqrt[a + b*x + c*x^2])/(15*e^2* 
(c*d^2 - b*d*e + a*e^2)*(d + e*x)^(5/2)) - ((-4*(2*c*d - b*e)*(4*c^2*d^2 - 
 b^2*e^2 - 4*c*e*(b*d - 2*a*e))*Sqrt[a + b*x + c*x^2])/((c*d^2 - b*d*e + a 
*e^2)*Sqrt[d + e*x]) + (c*((2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(2*c*d - b*e)*(4*c 
^2*d^2 - b^2*e^2 - 4*c*e*(b*d - 2*a*e))*Sqrt[d + e*x]*Sqrt[-((c*(a + b*x + 
 c*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2* 
c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d - (b + S 
qrt[b^2 - 4*a*c])*e)])/(c*e*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4* 
a*c])*e)]*Sqrt[a + b*x + c*x^2]) - (2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(c*d^2 - b 
*d*e + a*e^2)*(16*c^2*d^2 - 16*b*c*d*e - b^2*e^2 + 20*a*c*e^2)*Sqrt[(c*(d 
+ e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[-((c*(a + b*x + c*x^2))/ 
(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt 
[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 
 4*a*c])*e)])/(c*e*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2])))/(c*d^2 - b*d*e + 
 a*e^2))/(15*e^2*(c*d^2 - b*d*e + a*e^2))
 

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 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c 
/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 
0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 1172

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Sy 
mbol] :> Simp[2*Rt[b^2 - 4*a*c, 2]*(d + e*x)^m*(Sqrt[(-c)*((a + b*x + c*x^2 
)/(b^2 - 4*a*c))]/(c*Sqrt[a + b*x + c*x^2]*(2*c*((d + e*x)/(2*c*d - b*e - e 
*Rt[b^2 - 4*a*c, 2])))^m))   Subst[Int[(1 + 2*e*Rt[b^2 - 4*a*c, 2]*(x^2/(2* 
c*d - b*e - e*Rt[b^2 - 4*a*c, 2])))^m/Sqrt[1 - x^2], x], x, Sqrt[(b + Rt[b^ 
2 - 4*a*c, 2] + 2*c*x)/(2*Rt[b^2 - 4*a*c, 2])]], x] /; FreeQ[{a, b, c, d, e 
}, x] && EqQ[m^2, 1/4]
 

rule 1229

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/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)))*((d*g - e*f*(m + 2))*(c* 
d^2 - b*d*e + a*e^2) - d*p*(2*c*d - b*e)*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 
- b*d*e + a*e^2) + p*(2*c*d - b*e)*(e*f - d*g))*x), x] - Simp[p/(e^2*(m + 1 
)*(m + 2)*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2 
)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) + b^2*e*(d*g*(p + 1) - e*f*(m + 
p + 2)) + b*(a*e^2*g*(m + 1) - c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2))) - c 
*(2*c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2)) - e*(2*a*e*g*(m + 1) - b*(d*g*( 
m - 2*p) + e*f*(m + 2*p + 2))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g 
}, x] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3, 
0]
 

rule 1237

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

rule 1269

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1310\) vs. \(2(608)=1216\).

Time = 5.74 (sec) , antiderivative size = 1311, normalized size of antiderivative = 1.95

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1311\)
default	\(\text {Expression too large to display}\)	\(19265\)

Input:

int((2*c*x+b)*(c*x^2+b*x+a)^(1/2)/(e*x+d)^(7/2),x,method=_RETURNVERBOSE)
 

Output:

((e*x+d)*(c*x^2+b*x+a))^(1/2)/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)*(-2/5*(b*e 
-2*c*d)/e^5*(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)/(x+d/e)^3-2/15 
*(10*a*c*e^2+b^2*e^2-14*b*c*d*e+14*c^2*d^2)/e^4/(a*e^2-b*d*e+c*d^2)*(c*e*x 
^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)/(x+d/e)^2-4/15*(c*e*x^2+b*e*x+a* 
e)/e^3/(a*e^2-b*d*e+c*d^2)^2*(8*a*b*c*e^3-16*a*c^2*d*e^2-b^3*e^3-2*b^2*c*d 
*e^2+12*b*c^2*d^2*e-8*c^3*d^3)/((x+d/e)*(c*e*x^2+b*e*x+a*e))^(1/2)+2*(2*c^ 
2/e^3-1/15*c*(10*a*c*e^2+b^2*e^2-14*b*c*d*e+14*c^2*d^2)/e^3/(a*e^2-b*d*e+c 
*d^2)-2/15/e^3*(b*e-c*d)*(8*a*b*c*e^3-16*a*c^2*d*e^2-b^3*e^3-2*b^2*c*d*e^2 
+12*b*c^2*d^2*e-8*c^3*d^3)/(a*e^2-b*d*e+c*d^2)^2+2/15*b/e^2/(a*e^2-b*d*e+c 
*d^2)^2*(8*a*b*c*e^3-16*a*c^2*d*e^2-b^3*e^3-2*b^2*c*d*e^2+12*b*c^2*d^2*e-8 
*c^3*d^3))*(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c)*((x+d/e)/(d/e-1/2*(b+(-4*a*c 
+b^2)^(1/2))/c))^(1/2)*((x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))/(-d/e-1/2/c*(-b+ 
(-4*a*c+b^2)^(1/2))))^(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(-d/e+1/2*(b 
+(-4*a*c+b^2)^(1/2))/c))^(1/2)/(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^( 
1/2)*EllipticF(((x+d/e)/(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2),((-d/e+1 
/2*(b+(-4*a*c+b^2)^(1/2))/c)/(-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2))+ 
4/15*c/e^2*(8*a*b*c*e^3-16*a*c^2*d*e^2-b^3*e^3-2*b^2*c*d*e^2+12*b*c^2*d^2* 
e-8*c^3*d^3)/(a*e^2-b*d*e+c*d^2)^2*(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c)*((x+ 
d/e)/(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)*((x-1/2/c*(-b+(-4*a*c+b^2)^ 
(1/2)))/(-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2)*((x+1/2*(b+(-4*a*c+...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1513 vs. \(2 (616) = 1232\).

Time = 0.14 (sec) , antiderivative size = 1513, normalized size of antiderivative = 2.24 \[ \int \frac {(b+2 c x) \sqrt {a+b x+c x^2}}{(d+e x)^{7/2}} \, dx=\text {Too large to display} \] Input:

integrate((2*c*x+b)*(c*x^2+b*x+a)^(1/2)/(e*x+d)^(7/2),x, algorithm="fricas 
")
 

Output:

2/45*((16*c^4*d^7 - 32*b*c^3*d^6*e + (13*b^2*c^2 + 44*a*c^3)*d^5*e^2 + (3* 
b^3*c - 44*a*b*c^2)*d^4*e^3 + (2*b^4 - 19*a*b^2*c + 60*a^2*c^2)*d^3*e^4 + 
(16*c^4*d^4*e^3 - 32*b*c^3*d^3*e^4 + (13*b^2*c^2 + 44*a*c^3)*d^2*e^5 + (3* 
b^3*c - 44*a*b*c^2)*d*e^6 + (2*b^4 - 19*a*b^2*c + 60*a^2*c^2)*e^7)*x^3 + 3 
*(16*c^4*d^5*e^2 - 32*b*c^3*d^4*e^3 + (13*b^2*c^2 + 44*a*c^3)*d^3*e^4 + (3 
*b^3*c - 44*a*b*c^2)*d^2*e^5 + (2*b^4 - 19*a*b^2*c + 60*a^2*c^2)*d*e^6)*x^ 
2 + 3*(16*c^4*d^6*e - 32*b*c^3*d^5*e^2 + (13*b^2*c^2 + 44*a*c^3)*d^4*e^3 + 
 (3*b^3*c - 44*a*b*c^2)*d^3*e^4 + (2*b^4 - 19*a*b^2*c + 60*a^2*c^2)*d^2*e^ 
5)*x)*sqrt(c*e)*weierstrassPInverse(4/3*(c^2*d^2 - b*c*d*e + (b^2 - 3*a*c) 
*e^2)/(c^2*e^2), -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*(b^2*c - 6*a*c^2)*d* 
e^2 + (2*b^3 - 9*a*b*c)*e^3)/(c^3*e^3), 1/3*(3*c*e*x + c*d + b*e)/(c*e)) + 
 6*(8*c^4*d^6*e - 12*b*c^3*d^5*e^2 + 2*(b^2*c^2 + 8*a*c^3)*d^4*e^3 + (b^3* 
c - 8*a*b*c^2)*d^3*e^4 + (8*c^4*d^3*e^4 - 12*b*c^3*d^2*e^5 + 2*(b^2*c^2 + 
8*a*c^3)*d*e^6 + (b^3*c - 8*a*b*c^2)*e^7)*x^3 + 3*(8*c^4*d^4*e^3 - 12*b*c^ 
3*d^3*e^4 + 2*(b^2*c^2 + 8*a*c^3)*d^2*e^5 + (b^3*c - 8*a*b*c^2)*d*e^6)*x^2 
 + 3*(8*c^4*d^5*e^2 - 12*b*c^3*d^4*e^3 + 2*(b^2*c^2 + 8*a*c^3)*d^3*e^4 + ( 
b^3*c - 8*a*b*c^2)*d^2*e^5)*x)*sqrt(c*e)*weierstrassZeta(4/3*(c^2*d^2 - b* 
c*d*e + (b^2 - 3*a*c)*e^2)/(c^2*e^2), -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3 
*(b^2*c - 6*a*c^2)*d*e^2 + (2*b^3 - 9*a*b*c)*e^3)/(c^3*e^3), weierstrassPI 
nverse(4/3*(c^2*d^2 - b*c*d*e + (b^2 - 3*a*c)*e^2)/(c^2*e^2), -4/27*(2*...
 

Sympy [F]

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

integrate((2*c*x+b)*(c*x**2+b*x+a)**(1/2)/(e*x+d)**(7/2),x)
 

Output:

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

Maxima [F]

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

integrate((2*c*x+b)*(c*x^2+b*x+a)^(1/2)/(e*x+d)^(7/2),x, algorithm="maxima 
")
 

Output:

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

Giac [F]

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

integrate((2*c*x+b)*(c*x^2+b*x+a)^(1/2)/(e*x+d)^(7/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

int(((b + 2*c*x)*(a + b*x + c*x^2)^(1/2))/(d + e*x)^(7/2),x)
                                                                                    
                                                                                    
 

Output:

int(((b + 2*c*x)*(a + b*x + c*x^2)^(1/2))/(d + e*x)^(7/2), x)
 

Reduce [F]

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

int((2*c*x+b)*(c*x^2+b*x+a)^(1/2)/(e*x+d)^(7/2),x)
 

Output:

int((2*c*x+b)*(c*x^2+b*x+a)^(1/2)/(e*x+d)^(7/2),x)