\(\int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^{7/2}} \, dx\) [641]

Optimal result

Integrand size = 24, antiderivative size = 597 \[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^{7/2}} \, dx=-\frac {2 \sqrt {a+b x+c x^2}}{5 e (d+e x)^{5/2}}+\frac {2 (2 c d-b e) \sqrt {a+b x+c x^2}}{15 e \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2}}+\frac {4 \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \sqrt {a+b x+c x^2}}{15 e \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{15 e^2 \left (c d^2-b d e+a e^2\right )^2 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{15 e^2 \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}} \] Output:

-2/5*(c*x^2+b*x+a)^(1/2)/e/(e*x+d)^(5/2)+2/15*(-b*e+2*c*d)*(c*x^2+b*x+a)^( 
1/2)/e/(a*e^2-b*d*e+c*d^2)/(e*x+d)^(3/2)+4/15*(c^2*d^2+b^2*e^2-c*e*(3*a*e+ 
b*d))*(c*x^2+b*x+a)^(1/2)/e/(a*e^2-b*d*e+c*d^2)^2/(e*x+d)^(1/2)-2/15*2^(1/ 
2)*(-4*a*c+b^2)^(1/2)*(c^2*d^2+b^2*e^2-c*e*(3*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^2/(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)*(-b*e+ 
2*c*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)*EllipticF(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^2/(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 = 34.00 (sec) , antiderivative size = 1252, normalized size of antiderivative = 2.10 \[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^{7/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

Sqrt[d + e*x]*Sqrt[a + x*(b + c*x)]*(-2/(5*e*(d + e*x)^3) - (2*(-2*c*d + b 
*e))/(15*e*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^2) - (4*(-(c^2*d^2) + b*c*d*e 
 - b^2*e^2 + 3*a*c*e^2))/(15*e*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x))) - ((d 
 + e*x)^(3/2)*Sqrt[a + x*(b + c*x)]*(4*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c* 
d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e) 
)*(c*(-1 + d/(d + e*x))^2 + (e*(b - (b*d)/(d + e*x) + (a*e)/(d + e*x)))/(d 
 + e*x)) - (I*Sqrt[2]*(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(c^2*d^2 + b 
^2*e^2 - c*e*(b*d + 3*a*e))*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] - (2*a*e^2)/(d + 
 e*x) - 2*c*d*(-1 + d/(d + e*x)) + b*e*(-1 + (2*d)/(d + e*x)))/(2*c*d - b* 
e + Sqrt[(b^2 - 4*a*c)*e^2])]*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] + (2*a*e^2)/(d 
 + e*x) + 2*c*d*(-1 + d/(d + e*x)) + b*(e - (2*d*e)/(d + e*x)))/(-2*c*d + 
b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - 
 b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], 
 -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a 
*c)*e^2]))])/Sqrt[d + e*x] + (I*Sqrt[2]*(-(b^3*e^3) + b^2*e^2*(2*c*d + Sqr 
t[(b^2 - 4*a*c)*e^2]) + b*c*e*(4*a*e^2 - d*Sqrt[(b^2 - 4*a*c)*e^2]) + c*(c 
*d^2*Sqrt[(b^2 - 4*a*c)*e^2] - a*e^2*(8*c*d + 3*Sqrt[(b^2 - 4*a*c)*e^2]))) 
*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] - (2*a*e^2)/(d + e*x) - 2*c*d*(-1 + d/(d + 
e*x)) + b*e*(-1 + (2*d)/(d + e*x)))/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2] 
)]*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] + (2*a*e^2)/(d + e*x) + 2*c*d*(-1 + d/...
 

Rubi [A] (verified)

Time = 0.91 (sec) , antiderivative size = 644, normalized size of antiderivative = 1.08, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {1161, 1237, 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.

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

\(\Big \downarrow \) 1161

\(\displaystyle \frac {\int \frac {b+2 c x}{(d+e x)^{5/2} \sqrt {c x^2+b x+a}}dx}{5 e}-\frac {2 \sqrt {a+b x+c x^2}}{5 e (d+e x)^{5/2}}\)

\(\Big \downarrow \) 1237

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1237

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1269

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

\(\Big \downarrow \) 1172

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

\(\Big \downarrow \) 321

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

\(\Big \downarrow \) 327

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

Input:

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

Output:

(-2*Sqrt[a + b*x + c*x^2])/(5*e*(d + e*x)^(5/2)) + ((2*(2*c*d - b*e)*Sqrt[ 
a + b*x + c*x^2])/(3*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^(3/2)) + ((4*(c^2*d 
^2 + b^2*e^2 - c*e*(b*d + 3*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]*(c^2*d^2 + b^2*e 
^2 - c*e*(b*d + 3*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 + Sqrt[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]*(2*c*d - b*e)*(c*d^2 - b*d 
*e + a*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))/(3*(c*d^2 - b*d*e + a*e^2)))/(5*e)
 

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 1161

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(d + e*x)^(m + 1)*((a + b*x + c*x^2)^p/(e*(m + 1))), x] - Si 
mp[p/(e*(m + 1))   Int[(d + e*x)^(m + 1)*(b + 2*c*x)*(a + b*x + c*x^2)^(p - 
 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && GtQ[p, 0] && (IntegerQ[p] || 
 LtQ[m, -1]) && NeQ[m, -1] &&  !ILtQ[m + 2*p + 1, 0] && IntQuadraticQ[a, b, 
 c, d, e, m, p, x]
 

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 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. \(1148\) vs. \(2(531)=1062\).

Time = 1.80 (sec) , antiderivative size = 1149, normalized size of antiderivative = 1.92

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1149\)
default	\(\text {Expression too large to display}\)	\(12980\)

Input:

int((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/e^4* 
(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*(b*e-2*c*d) 
/e^3/(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^2/(a*e^2-b*d*e+c*d^2)^2*(3*a*c*e^2-b^2 
*e^2+b*c*d*e-c^2*d^2)/((x+d/e)*(c*e*x^2+b*e*x+a*e))^(1/2)+2*(-1/15*c*(b*e- 
2*c*d)/e^2/(a*e^2-b*d*e+c*d^2)-2/15/e^2*(b*e-c*d)*(3*a*c*e^2-b^2*e^2+b*c*d 
*e-c^2*d^2)/(a*e^2-b*d*e+c*d^2)^2+2/15*b/e/(a*e^2-b*d*e+c*d^2)^2*(3*a*c*e^ 
2-b^2*e^2+b*c*d*e-c^2*d^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+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*(3*a*c*e^2-b^2*e^2+b*c*d*e-c^2*d^2)/(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+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)*((-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))*EllipticE(((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...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1188 vs. \(2 (539) = 1078\).

Time = 0.14 (sec) , antiderivative size = 1188, normalized size of antiderivative = 1.99 \[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^{7/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

2/45*((2*c^3*d^6 - 3*b*c^2*d^5*e - 3*(b^2*c - 6*a*c^2)*d^4*e^2 + (2*b^3 - 
9*a*b*c)*d^3*e^3 + (2*c^3*d^3*e^3 - 3*b*c^2*d^2*e^4 - 3*(b^2*c - 6*a*c^2)* 
d*e^5 + (2*b^3 - 9*a*b*c)*e^6)*x^3 + 3*(2*c^3*d^4*e^2 - 3*b*c^2*d^3*e^3 - 
3*(b^2*c - 6*a*c^2)*d^2*e^4 + (2*b^3 - 9*a*b*c)*d*e^5)*x^2 + 3*(2*c^3*d^5* 
e - 3*b*c^2*d^4*e^2 - 3*(b^2*c - 6*a*c^2)*d^3*e^3 + (2*b^3 - 9*a*b*c)*d^2* 
e^4)*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*(c^3*d^5*e - b*c^2*d^4*e^2 + (b^2*c - 3*a*c^2)*d^3*e^3 + (c^3*d^2*e^4 
 - b*c^2*d*e^5 + (b^2*c - 3*a*c^2)*e^6)*x^3 + 3*(c^3*d^3*e^3 - b*c^2*d^2*e 
^4 + (b^2*c - 3*a*c^2)*d*e^5)*x^2 + 3*(c^3*d^4*e^2 - b*c^2*d^3*e^3 + (b^2* 
c - 3*a*c^2)*d^2*e^4)*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), 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))) + 3*(c^3*d^4*e^2 + b*c^2*d^3*e^3 - 
10*a*c^2*d^2*e^4 + 5*a*b*c*d*e^5 - 3*a^2*c*e^6 + 2*(c^3*d^2*e^4 - b*c^2*d* 
e^5 + (b^2*c - 3*a*c^2)*e^6)*x^2 + (6*c^3*d^3*e^3 - 7*b*c^2*d^2*e^4 - a*b* 
c*e^6 + 5*(b^2*c - 2*a*c^2)*d*e^5)*x)*sqrt(c*x^2 + b*x + a)*sqrt(e*x + ...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Giac [F]

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

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^{7/2}} \, dx=\text {too large to display} \] Input:

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

Output:

( - 2*sqrt(d + e*x)*sqrt(a + b*x + c*x**2)*b + 2*int((sqrt(d + e*x)*sqrt(a 
 + b*x + c*x**2)*x**2)/(2*a*b*d**4*e + 8*a*b*d**3*e**2*x + 12*a*b*d**2*e** 
3*x**2 + 8*a*b*d*e**4*x**3 + 2*a*b*e**5*x**4 - a*c*d**5 - 4*a*c*d**4*e*x - 
 6*a*c*d**3*e**2*x**2 - 4*a*c*d**2*e**3*x**3 - a*c*d*e**4*x**4 + 2*b**2*d* 
*4*e*x + 8*b**2*d**3*e**2*x**2 + 12*b**2*d**2*e**3*x**3 + 8*b**2*d*e**4*x* 
*4 + 2*b**2*e**5*x**5 - b*c*d**5*x - 2*b*c*d**4*e*x**2 + 2*b*c*d**3*e**2*x 
**3 + 8*b*c*d**2*e**3*x**4 + 7*b*c*d*e**4*x**5 + 2*b*c*e**5*x**6 - c**2*d* 
*5*x**2 - 4*c**2*d**4*e*x**3 - 6*c**2*d**3*e**2*x**4 - 4*c**2*d**2*e**3*x* 
*5 - c**2*d*e**4*x**6),x)*b**2*c*d**3*e**2 + 6*int((sqrt(d + e*x)*sqrt(a + 
 b*x + c*x**2)*x**2)/(2*a*b*d**4*e + 8*a*b*d**3*e**2*x + 12*a*b*d**2*e**3* 
x**2 + 8*a*b*d*e**4*x**3 + 2*a*b*e**5*x**4 - a*c*d**5 - 4*a*c*d**4*e*x - 6 
*a*c*d**3*e**2*x**2 - 4*a*c*d**2*e**3*x**3 - a*c*d*e**4*x**4 + 2*b**2*d**4 
*e*x + 8*b**2*d**3*e**2*x**2 + 12*b**2*d**2*e**3*x**3 + 8*b**2*d*e**4*x**4 
 + 2*b**2*e**5*x**5 - b*c*d**5*x - 2*b*c*d**4*e*x**2 + 2*b*c*d**3*e**2*x** 
3 + 8*b*c*d**2*e**3*x**4 + 7*b*c*d*e**4*x**5 + 2*b*c*e**5*x**6 - c**2*d**5 
*x**2 - 4*c**2*d**4*e*x**3 - 6*c**2*d**3*e**2*x**4 - 4*c**2*d**2*e**3*x**5 
 - c**2*d*e**4*x**6),x)*b**2*c*d**2*e**3*x + 6*int((sqrt(d + e*x)*sqrt(a + 
 b*x + c*x**2)*x**2)/(2*a*b*d**4*e + 8*a*b*d**3*e**2*x + 12*a*b*d**2*e**3* 
x**2 + 8*a*b*d*e**4*x**3 + 2*a*b*e**5*x**4 - a*c*d**5 - 4*a*c*d**4*e*x - 6 
*a*c*d**3*e**2*x**2 - 4*a*c*d**2*e**3*x**3 - a*c*d*e**4*x**4 + 2*b**2*d...