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

Optimal result

Integrand size = 24, antiderivative size = 580 \[ \int \frac {(d+e x)^{7/2}}{\sqrt {a+b x+c x^2}} \, dx=\frac {2 e \left (71 c^2 d^2+24 b^2 e^2-c e (71 b d+25 a e)\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}}{105 c^3}+\frac {12 e (2 c d-b e) (d+e x)^{3/2} \sqrt {a+b x+c x^2}}{35 c^2}+\frac {2 e (d+e x)^{5/2} \sqrt {a+b x+c x^2}}{7 c}+\frac {8 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (11 c^2 d^2+6 b^2 e^2-c e (11 b d+13 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{105 c^4 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right ) \left (71 c^2 d^2+24 b^2 e^2-c e (71 b d+25 a e)\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{105 c^4 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \] Output:

2/105*e*(71*c^2*d^2+24*b^2*e^2-c*e*(25*a*e+71*b*d))*(e*x+d)^(1/2)*(c*x^2+b 
*x+a)^(1/2)/c^3+12/35*e*(-b*e+2*c*d)*(e*x+d)^(3/2)*(c*x^2+b*x+a)^(1/2)/c^2 
+2/7*e*(e*x+d)^(5/2)*(c*x^2+b*x+a)^(1/2)/c+8/105*2^(1/2)*(-4*a*c+b^2)^(1/2 
)*(-b*e+2*c*d)*(11*c^2*d^2+6*b^2*e^2-c*e*(13*a*e+11*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))/c^4/(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)-2/105*2^(1/2)*(-4*a*c+b^2)^(1/2)*(a*e^2-b*d*e+c*d^2)*(71*c 
^2*d^2+24*b^2*e^2-c*e*(25*a*e+71*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)*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))/c^4/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 34.55 (sec) , antiderivative size = 1318, normalized size of antiderivative = 2.27 \[ \int \frac {(d+e x)^{7/2}}{\sqrt {a+b x+c x^2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(Sqrt[d + e*x]*(a + b*x + c*x^2)*((-2*e*(-122*c^2*d^2 + 89*b*c*d*e - 24*b^ 
2*e^2 + 25*a*c*e^2))/(105*c^3) - (4*e^2*(-11*c*d + 3*b*e)*x)/(35*c^2) + (2 
*e^3*x^2)/(7*c)))/Sqrt[a + x*(b + c*x)] + ((d + e*x)^(3/2)*Sqrt[a + b*x + 
c*x^2]*(-16*(-2*c*d + b*e)*Sqrt[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + 
 Sqrt[(b^2 - 4*a*c)*e^2])]*(11*c^2*d^2 + 6*b^2*e^2 - c*e*(11*b*d + 13*a*e) 
)*(c*(-1 + d/(d + e*x))^2 + (e*(b - (b*d)/(d + e*x) + (a*e)/(d + e*x)))/(d 
 + e*x)) + ((4*I)*Sqrt[2]*(-2*c*d + b*e)*(2*c*d - b*e + Sqrt[(b^2 - 4*a*c) 
*e^2])*(11*c^2*d^2 + 6*b^2*e^2 - c*e*(11*b*d + 13*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*Ar 
cSinh[(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]*(-105* 
c^4*d^4 + 24*b^3*e^3*(-(b*e) + Sqrt[(b^2 - 4*a*c)*e^2]) - 2*c^3*d^2*(-105* 
b*d*e - 127*a*e^2 + 44*d*Sqrt[(b^2 - 4*a*c)*e^2]) + 4*b*c*e^2*(29*b^2*d*e 
+ 19*a*b*e^2 - 23*b*d*Sqrt[(b^2 - 4*a*c)*e^2] - 13*a*e*Sqrt[(b^2 - 4*a*c)* 
e^2]) + c^2*e*(-221*b^2*d^2*e + 2*b*d*(-127*a*e^2 + 66*d*Sqrt[(b^2 - 4*a*c 
)*e^2]) + a*e*(-25*a*e^2 + 104*d*Sqrt[(b^2 - 4*a*c)*e^2])))*Sqrt[(Sqrt[...
 

Rubi [A] (verified)

Time = 1.10 (sec) , antiderivative size = 620, normalized size of antiderivative = 1.07, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {1166, 27, 1236, 27, 1236, 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.

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

\(\Big \downarrow \) 1166

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1236

\(\displaystyle \frac {\frac {2 \int \frac {\sqrt {d+e x} \left (35 c^2 d^3-c e (17 b d+61 a e) d+6 b e^2 (b d+3 a e)+e \left (71 c^2 d^2+24 b^2 e^2-c e (71 b d+25 a e)\right ) x\right )}{2 \sqrt {c x^2+b x+a}}dx}{5 c}+\frac {12 e (d+e x)^{3/2} \sqrt {a+b x+c x^2} (2 c d-b e)}{5 c}}{7 c}+\frac {2 e (d+e x)^{5/2} \sqrt {a+b x+c x^2}}{7 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {\sqrt {d+e x} \left (35 c^2 d^3-c e (17 b d+61 a e) d+6 b e^2 (b d+3 a e)+e \left (71 c^2 d^2+24 b^2 e^2-c e (71 b d+25 a e)\right ) x\right )}{\sqrt {c x^2+b x+a}}dx}{5 c}+\frac {12 e (d+e x)^{3/2} \sqrt {a+b x+c x^2} (2 c d-b e)}{5 c}}{7 c}+\frac {2 e (d+e x)^{5/2} \sqrt {a+b x+c x^2}}{7 c}\)

\(\Big \downarrow \) 1236

\(\displaystyle \frac {\frac {\frac {2 \int \frac {105 c^3 d^4-2 c^2 e (61 b d+127 a e) d^2-24 b^2 e^3 (b d+a e)+c e^2 \left (89 b^2 d^2+150 a b e d+25 a^2 e^2\right )+8 e (2 c d-b e) \left (11 c^2 d^2+6 b^2 e^2-c e (11 b d+13 a e)\right ) x}{2 \sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{3 c}+\frac {2 e \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (-c e (25 a e+71 b d)+24 b^2 e^2+71 c^2 d^2\right )}{3 c}}{5 c}+\frac {12 e (d+e x)^{3/2} \sqrt {a+b x+c x^2} (2 c d-b e)}{5 c}}{7 c}+\frac {2 e (d+e x)^{5/2} \sqrt {a+b x+c x^2}}{7 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\int \frac {105 c^3 d^4-2 c^2 e (61 b d+127 a e) d^2-24 b^2 e^3 (b d+a e)+c e^2 \left (89 b^2 d^2+150 a b e d+25 a^2 e^2\right )+8 e (2 c d-b e) \left (11 c^2 d^2+6 b^2 e^2-c e (11 b d+13 a e)\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{3 c}+\frac {2 e \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (-c e (25 a e+71 b d)+24 b^2 e^2+71 c^2 d^2\right )}{3 c}}{5 c}+\frac {12 e (d+e x)^{3/2} \sqrt {a+b x+c x^2} (2 c d-b e)}{5 c}}{7 c}+\frac {2 e (d+e x)^{5/2} \sqrt {a+b x+c x^2}}{7 c}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {\frac {\frac {8 (2 c d-b e) \left (-c e (13 a e+11 b d)+6 b^2 e^2+11 c^2 d^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x+a}}dx-\left (a e^2-b d e+c d^2\right ) \left (-25 a c e^2+24 b^2 e^2-71 b c d e+71 c^2 d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{3 c}+\frac {2 e \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (-c e (25 a e+71 b d)+24 b^2 e^2+71 c^2 d^2\right )}{3 c}}{5 c}+\frac {12 e (d+e x)^{3/2} \sqrt {a+b x+c x^2} (2 c d-b e)}{5 c}}{7 c}+\frac {2 e (d+e x)^{5/2} \sqrt {a+b x+c x^2}}{7 c}\)

\(\Big \downarrow \) 1172

\(\displaystyle \frac {\frac {\frac {\frac {8 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (-c e (13 a e+11 b d)+6 b^2 e^2+11 c^2 d^2\right ) \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \left (-25 a c e^2+24 b^2 e^2-71 b c d e+71 c^2 d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \int \frac {1}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}} \sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c \sqrt {d+e x} \sqrt {a+b x+c x^2}}}{3 c}+\frac {2 e \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (-c e (25 a e+71 b d)+24 b^2 e^2+71 c^2 d^2\right )}{3 c}}{5 c}+\frac {12 e (d+e x)^{3/2} \sqrt {a+b x+c x^2} (2 c d-b e)}{5 c}}{7 c}+\frac {2 e (d+e x)^{5/2} \sqrt {a+b x+c x^2}}{7 c}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {\frac {\frac {\frac {8 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (-c e (13 a e+11 b d)+6 b^2 e^2+11 c^2 d^2\right ) \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \left (-25 a c e^2+24 b^2 e^2-71 b c d e+71 c^2 d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c \sqrt {d+e x} \sqrt {a+b x+c x^2}}}{3 c}+\frac {2 e \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (-c e (25 a e+71 b d)+24 b^2 e^2+71 c^2 d^2\right )}{3 c}}{5 c}+\frac {12 e (d+e x)^{3/2} \sqrt {a+b x+c x^2} (2 c d-b e)}{5 c}}{7 c}+\frac {2 e (d+e x)^{5/2} \sqrt {a+b x+c x^2}}{7 c}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {\frac {\frac {\frac {8 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (-c e (13 a e+11 b d)+6 b^2 e^2+11 c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \left (-25 a c e^2+24 b^2 e^2-71 b c d e+71 c^2 d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c \sqrt {d+e x} \sqrt {a+b x+c x^2}}}{3 c}+\frac {2 e \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (-c e (25 a e+71 b d)+24 b^2 e^2+71 c^2 d^2\right )}{3 c}}{5 c}+\frac {12 e (d+e x)^{3/2} \sqrt {a+b x+c x^2} (2 c d-b e)}{5 c}}{7 c}+\frac {2 e (d+e x)^{5/2} \sqrt {a+b x+c x^2}}{7 c}\)

Input:

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

Output:

(2*e*(d + e*x)^(5/2)*Sqrt[a + b*x + c*x^2])/(7*c) + ((12*e*(2*c*d - b*e)*( 
d + e*x)^(3/2)*Sqrt[a + b*x + c*x^2])/(5*c) + ((2*e*(71*c^2*d^2 + 24*b^2*e 
^2 - c*e*(71*b*d + 25*a*e))*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2])/(3*c) + ( 
(8*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(2*c*d - b*e)*(11*c^2*d^2 + 6*b^2*e^2 - c*e*( 
11*b*d + 13*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*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)*(71*c^2*d^ 
2 - 71*b*c*d*e + 24*b^2*e^2 - 25*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))]*Ellip 
ticF[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*Sqrt 
[d + e*x]*Sqrt[a + b*x + c*x^2]))/(3*c))/(5*c))/(7*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 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 1166

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 
 1))), x] + Simp[1/(c*(m + 2*p + 1))   Int[(d + e*x)^(m - 2)*Simp[c*d^2*(m 
+ 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]* 
(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && If[Ration 
alQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadrat 
icQ[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 1236

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

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. \(1220\) vs. \(2(514)=1028\).

Time = 5.86 (sec) , antiderivative size = 1221, normalized size of antiderivative = 2.11

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1221\)
risch	\(\text {Expression too large to display}\)	\(2930\)
default	\(\text {Expression too large to display}\)	\(6947\)

Input:

int((e*x+d)^(7/2)/(c*x^2+b*x+a)^(1/2),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)*(2/7*e^3/c 
*x^2*(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)+2/5*(4*d*e^3-2/7/c*e^ 
3*(3*b*e+3*c*d))/c/e*x*(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)+2/3 
*(6*d^2*e^2-2/7*e^3/c*(5/2*a*e+5/2*b*d)-2/5*(4*d*e^3-2/7/c*e^3*(3*b*e+3*c* 
d))/c/e*(2*b*e+2*c*d))/c/e*(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2) 
+2*(d^4-2/5*(4*d*e^3-2/7/c*e^3*(3*b*e+3*c*d))/c/e*a*d-2/3*(6*d^2*e^2-2/7*e 
^3/c*(5/2*a*e+5/2*b*d)-2/5*(4*d*e^3-2/7/c*e^3*(3*b*e+3*c*d))/c/e*(2*b*e+2* 
c*d))/c/e*(1/2*a*e+1/2*b*d))*(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))+2*(4*d^3*e-4/7*a/c*d*e^3-2/5*(4*d*e^3-2/7/c*e^3*(3*b*e+3 
*c*d))/c/e*(3/2*a*e+3/2*b*d)-2/3*(6*d^2*e^2-2/7*e^3/c*(5/2*a*e+5/2*b*d)-2/ 
5*(4*d*e^3-2/7/c*e^3*(3*b*e+3*c*d))/c/e*(2*b*e+2*c*d))/c/e*(b*e+c*d))*(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-...
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 589, normalized size of antiderivative = 1.02 \[ \int \frac {(d+e x)^{7/2}}{\sqrt {a+b x+c x^2}} \, dx=\frac {2 \, {\left ({\left (139 \, c^{4} d^{4} - 278 \, b c^{3} d^{3} e + {\left (347 \, b^{2} c^{2} - 554 \, a c^{3}\right )} d^{2} e^{2} - 2 \, {\left (104 \, b^{3} c - 277 \, a b c^{2}\right )} d e^{3} + {\left (48 \, b^{4} - 176 \, a b^{2} c + 75 \, a^{2} c^{2}\right )} e^{4}\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right ) - 24 \, {\left (22 \, c^{4} d^{3} e - 33 \, b c^{3} d^{2} e^{2} + {\left (23 \, b^{2} c^{2} - 26 \, a c^{3}\right )} d e^{3} - {\left (6 \, b^{3} c - 13 \, a b c^{2}\right )} e^{4}\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )}}{27 \, c^{3} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right )\right ) + 3 \, {\left (15 \, c^{4} e^{4} x^{2} + 122 \, c^{4} d^{2} e^{2} - 89 \, b c^{3} d e^{3} + {\left (24 \, b^{2} c^{2} - 25 \, a c^{3}\right )} e^{4} + 6 \, {\left (11 \, c^{4} d e^{3} - 3 \, b c^{3} e^{4}\right )} x\right )} \sqrt {c x^{2} + b x + a} \sqrt {e x + d}\right )}}{315 \, c^{5} e} \] Input:

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

Output:

2/315*((139*c^4*d^4 - 278*b*c^3*d^3*e + (347*b^2*c^2 - 554*a*c^3)*d^2*e^2 
- 2*(104*b^3*c - 277*a*b*c^2)*d*e^3 + (48*b^4 - 176*a*b^2*c + 75*a^2*c^2)* 
e^4)*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)) - 
24*(22*c^4*d^3*e - 33*b*c^3*d^2*e^2 + (23*b^2*c^2 - 26*a*c^3)*d*e^3 - (6*b 
^3*c - 13*a*b*c^2)*e^4)*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*(15*c^4*e^4*x^2 + 122*c^4*d^2*e^ 
2 - 89*b*c^3*d*e^3 + (24*b^2*c^2 - 25*a*c^3)*e^4 + 6*(11*c^4*d*e^3 - 3*b*c 
^3*e^4)*x)*sqrt(c*x^2 + b*x + a)*sqrt(e*x + d))/(c^5*e)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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