\(\int \frac {\sqrt {a+b x+c x^2} (A+B x+C x^2)}{(d+e x)^{3/2}} \, dx\) [97]

Optimal result

Integrand size = 34, antiderivative size = 726 \[ \int \frac {\sqrt {a+b x+c x^2} \left (A+B x+C x^2\right )}{(d+e x)^{3/2}} \, dx=-\frac {2 \sqrt {d+e x} \left (b C e^2 (b d-a e)+c^2 \left (24 C d^3-5 d e (4 B d-3 A e)\right )+c e \left (a e (9 C d-5 B e)-5 b \left (5 C d^2-4 B d e+3 A e^2\right )\right )+3 c e^2 \left (5 B c d+b C d-\frac {6 c C d^2}{e}-5 A c e-a C e\right ) x\right ) \sqrt {a+b x+c x^2}}{15 c e^3 \left (c d^2-b d e+a e^2\right )}-\frac {2 \left (C d^2-e (B d-A e)\right ) \left (a+b x+c x^2\right )^{3/2}}{e \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (2 b^2 C e^2+c e (8 b C d-5 b B e-6 a C e)-c^2 \left (48 C d^2-10 e (4 B d-3 A e)\right )\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{15 c^2 e^4 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (b C e^2 (b d-a e)-2 c^2 d \left (24 C d^2-5 e (4 B d-3 A e)\right )+c e \left (32 b C d^2-5 b e (5 B d-3 A e)-2 a e (9 C d-5 B e)\right )\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{15 c^2 e^4 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \] Output:

-2/15*(e*x+d)^(1/2)*(b*C*e^2*(-a*e+b*d)+c^2*(24*C*d^3-5*d*e*(-3*A*e+4*B*d) 
)+c*e*(a*e*(-5*B*e+9*C*d)-5*b*(3*A*e^2-4*B*d*e+5*C*d^2))+3*c*e^2*(5*B*c*d+ 
C*b*d-6*c*C*d^2/e-5*A*c*e-C*a*e)*x)*(c*x^2+b*x+a)^(1/2)/c/e^3/(a*e^2-b*d*e 
+c*d^2)-2*(C*d^2-e*(-A*e+B*d))*(c*x^2+b*x+a)^(3/2)/e/(a*e^2-b*d*e+c*d^2)/( 
e*x+d)^(1/2)-1/15*2^(1/2)*(-4*a*c+b^2)^(1/2)*(2*b^2*C*e^2+c*e*(-5*B*b*e-6* 
C*a*e+8*C*b*d)-c^2*(48*C*d^2-10*e*(-3*A*e+4*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^2/e^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)+2/15*2^(1/2)*(-4*a*c+b^2)^(1/2)*(b*C*e^2*(-a*e+b*d)-2*c^2*d*( 
24*C*d^2-5*e*(-3*A*e+4*B*d))+c*e*(32*b*C*d^2-5*b*e*(-3*A*e+5*B*d)-2*a*e*(- 
5*B*e+9*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))/c^2/e^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 = 35.69 (sec) , antiderivative size = 13240, normalized size of antiderivative = 18.24 \[ \int \frac {\sqrt {a+b x+c x^2} \left (A+B x+C x^2\right )}{(d+e x)^{3/2}} \, dx=\text {Result too large to show} \] Input:

Integrate[(Sqrt[a + b*x + c*x^2]*(A + B*x + C*x^2))/(d + e*x)^(3/2),x]
 

Output:

Result too large to show
 

Rubi [A] (verified)

Time = 2.51 (sec) , antiderivative size = 774, normalized size of antiderivative = 1.07, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {2181, 27, 1231, 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.

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

\(\Big \downarrow \) 2181

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1231

\(\displaystyle \frac {-\frac {2 \int -\frac {\left (c d^2-b e d+a e^2\right ) \left (-C d e b^2+24 c C d^2 b-a C e^2 b-5 c e (4 B d-3 A e) b-2 a c e (6 C d-5 B e)-\left (-\left (\left (48 C d^2-10 e (4 B d-3 A e)\right ) c^2\right )+e (8 b C d-5 b B e-6 a C e) c+2 b^2 C e^2\right ) x\right )}{2 \sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{15 c e^2}-\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (3 c e^2 x \left (-a C e-5 A c e+b C d+5 B c d-\frac {6 c C d^2}{e}\right )+c e \left (a e (9 C d-5 B e)-5 b \left (3 A e^2-4 B d e+5 C d^2\right )\right )+b C e^2 (b d-a e)+c^2 \left (24 C d^3-5 d e (4 B d-3 A e)\right )\right )}{15 c e^2}}{e \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (a+b x+c x^2\right )^{3/2} \left (C d^2-e (B d-A e)\right )}{e \sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\left (a e^2-b d e+c d^2\right ) \int \frac {-C d e b^2+24 c C d^2 b-a C e^2 b-5 c e (4 B d-3 A e) b-2 a c e (6 C d-5 B e)-\left (-\left (\left (48 C d^2-10 e (4 B d-3 A e)\right ) c^2\right )+e (8 b C d-5 b B e-6 a C e) c+2 b^2 C e^2\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{15 c e^2}-\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (3 c e^2 x \left (-a C e-5 A c e+b C d+5 B c d-\frac {6 c C d^2}{e}\right )+c e \left (a e (9 C d-5 B e)-5 b \left (3 A e^2-4 B d e+5 C d^2\right )\right )+b C e^2 (b d-a e)+c^2 \left (24 C d^3-5 d e (4 B d-3 A e)\right )\right )}{15 c e^2}}{e \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (a+b x+c x^2\right )^{3/2} \left (C d^2-e (B d-A e)\right )}{e \sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {\frac {\left (a e^2-b d e+c d^2\right ) \left (\frac {\left (c e \left (-2 a e (9 C d-5 B e)-5 b e (5 B d-3 A e)+32 b C d^2\right )+b C e^2 (b d-a e)-2 c^2 d \left (24 C d^2-5 e (4 B d-3 A e)\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{e}-\frac {\left (c e (-6 a C e-5 b B e+8 b C d)-\left (c^2 \left (48 C d^2-10 e (4 B d-3 A e)\right )\right )+2 b^2 C e^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x+a}}dx}{e}\right )}{15 c e^2}-\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (3 c e^2 x \left (-a C e-5 A c e+b C d+5 B c d-\frac {6 c C d^2}{e}\right )+c e \left (a e (9 C d-5 B e)-5 b \left (3 A e^2-4 B d e+5 C d^2\right )\right )+b C e^2 (b d-a e)+c^2 \left (24 C d^3-5 d e (4 B d-3 A e)\right )\right )}{15 c e^2}}{e \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (a+b x+c x^2\right )^{3/2} \left (C d^2-e (B d-A e)\right )}{e \sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1172

\(\displaystyle \frac {\frac {\left (c d^2-b e d+a e^2\right ) \left (\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (-2 d \left (24 C d^2-5 e (4 B d-3 A e)\right ) c^2+e \left (32 b C d^2-5 b e (5 B d-3 A e)-2 a e (9 C d-5 B e)\right ) c+b C e^2 (b d-a e)\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \int \frac {1}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}} \sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {d+e x} \sqrt {c x^2+b x+a}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (-\left (\left (48 C d^2-10 e (4 B d-3 A e)\right ) c^2\right )+e (8 b C d-5 b B e-6 a C e) c+2 b^2 C e^2\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {c x^2+b x+a}}\right )}{15 c e^2}-\frac {2 \sqrt {d+e x} \left (\left (24 C d^3-5 d e (4 B d-3 A e)\right ) c^2+e \left (a e (9 C d-5 B e)-5 b \left (5 C d^2-4 B e d+3 A e^2\right )\right ) c+3 e^2 \left (-\frac {6 c C d^2}{e}+5 B c d+b C d-5 A c e-a C e\right ) x c+b C e^2 (b d-a e)\right ) \sqrt {c x^2+b x+a}}{15 c e^2}}{e \left (c d^2-b e d+a e^2\right )}-\frac {2 \left (C d^2-e (B d-A e)\right ) \left (c x^2+b x+a\right )^{3/2}}{e \left (c d^2-b e d+a e^2\right ) \sqrt {d+e x}}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {\frac {\left (c d^2-b e d+a e^2\right ) \left (\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (-2 d \left (24 C d^2-5 e (4 B d-3 A e)\right ) c^2+e \left (32 b C d^2-5 b e (5 B d-3 A e)-2 a e (9 C d-5 B e)\right ) c+b C e^2 (b d-a e)\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {c x^2+b x+a}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (-\left (\left (48 C d^2-10 e (4 B d-3 A e)\right ) c^2\right )+e (8 b C d-5 b B e-6 a C e) c+2 b^2 C e^2\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {c x^2+b x+a}}\right )}{15 c e^2}-\frac {2 \sqrt {d+e x} \left (\left (24 C d^3-5 d e (4 B d-3 A e)\right ) c^2+e \left (a e (9 C d-5 B e)-5 b \left (5 C d^2-4 B e d+3 A e^2\right )\right ) c+3 e^2 \left (-\frac {6 c C d^2}{e}+5 B c d+b C d-5 A c e-a C e\right ) x c+b C e^2 (b d-a e)\right ) \sqrt {c x^2+b x+a}}{15 c e^2}}{e \left (c d^2-b e d+a e^2\right )}-\frac {2 \left (C d^2-e (B d-A e)\right ) \left (c x^2+b x+a\right )^{3/2}}{e \left (c d^2-b e d+a e^2\right ) \sqrt {d+e x}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {\frac {\left (a e^2-b d e+c d^2\right ) \left (\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \left (c e \left (-2 a e (9 C d-5 B e)-5 b e (5 B d-3 A e)+32 b C d^2\right )+b C e^2 (b d-a e)-2 c^2 d \left (24 C d^2-5 e (4 B d-3 A e)\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (c e (-6 a C e-5 b B e+8 b C d)-\left (c^2 \left (48 C d^2-10 e (4 B d-3 A e)\right )\right )+2 b^2 C e^2\right ) E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}\right )}{15 c e^2}-\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (3 c e^2 x \left (-a C e-5 A c e+b C d+5 B c d-\frac {6 c C d^2}{e}\right )+c e \left (a e (9 C d-5 B e)-5 b \left (3 A e^2-4 B d e+5 C d^2\right )\right )+b C e^2 (b d-a e)+c^2 \left (24 C d^3-5 d e (4 B d-3 A e)\right )\right )}{15 c e^2}}{e \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (a+b x+c x^2\right )^{3/2} \left (C d^2-e (B d-A e)\right )}{e \sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}\)

Input:

Int[(Sqrt[a + b*x + c*x^2]*(A + B*x + C*x^2))/(d + e*x)^(3/2),x]
 

Output:

(-2*(C*d^2 - e*(B*d - A*e))*(a + b*x + c*x^2)^(3/2))/(e*(c*d^2 - b*d*e + a 
*e^2)*Sqrt[d + e*x]) + ((-2*Sqrt[d + e*x]*(b*C*e^2*(b*d - a*e) + c^2*(24*C 
*d^3 - 5*d*e*(4*B*d - 3*A*e)) + c*e*(a*e*(9*C*d - 5*B*e) - 5*b*(5*C*d^2 - 
4*B*d*e + 3*A*e^2)) + 3*c*e^2*(5*B*c*d + b*C*d - (6*c*C*d^2)/e - 5*A*c*e - 
 a*C*e)*x)*Sqrt[a + b*x + c*x^2])/(15*c*e^2) + ((c*d^2 - b*d*e + a*e^2)*(- 
((Sqrt[2]*Sqrt[b^2 - 4*a*c]*(2*b^2*C*e^2 + c*e*(8*b*C*d - 5*b*B*e - 6*a*C* 
e) - c^2*(48*C*d^2 - 10*e*(4*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]*(b*C* 
e^2*(b*d - a*e) - 2*c^2*d*(24*C*d^2 - 5*e*(4*B*d - 3*A*e)) + c*e*(32*b*C*d 
^2 - 5*b*e*(5*B*d - 3*A*e) - 2*a*e*(9*C*d - 5*B*e)))*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])))/(15*c*e^2))/(e*(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 1231

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

rule 2181

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_ 
), x_Symbol] :> With[{Qx = PolynomialQuotient[Pq, d + e*x, x], R = Polynomi 
alRemainder[Pq, d + e*x, x]}, Simp[(e*R*(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*ExpandToSum[(m 
+ 1)*(c*d^2 - b*d*e + a*e^2)*Qx + c*d*R*(m + 1) - b*e*R*(m + p + 2) - c*e*R 
*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, e, p}, x] && PolyQ[Pq, 
x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1]
 

Maple [A] (verified)

Time = 4.84 (sec) , antiderivative size = 1215, normalized size of antiderivative = 1.67

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1215\)
risch	\(\text {Expression too large to display}\)	\(1864\)
default	\(\text {Expression too large to display}\)	\(8221\)

Input:

int((c*x^2+b*x+a)^(1/2)*(C*x^2+B*x+A)/(e*x+d)^(3/2),x,method=_RETURNVERBOS 
E)
 

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*(c*e*x 
^2+b*e*x+a*e)*(A*e^2-B*d*e+C*d^2)/e^4/((x+d/e)*(c*e*x^2+b*e*x+a*e))^(1/2)+ 
2/5*C/e^2*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*(1/e^2*(B* 
c*e+C*b*e-C*c*d)-2/5*C/e^2*(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*((A*b*e^3-A*c*d*e^2+B*a*e^3-B*b*d*e^2+B*c*d^2*e-C*a* 
d*e^2+C*b*d^2*e-C*c*d^3)/e^4-(A*e^2-B*d*e+C*d^2)/e^4*(b*e-c*d)+b/e^3*(A*e^ 
2-B*d*e+C*d^2)-2/5*C/e^2*a*d-2/3*(1/e^2*(B*c*e+C*b*e-C*c*d)-2/5*C/e^2*(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*(1/e^3*(A*c*e^2+B*b*e^2-B*c*d*e+C*a*e^2-C*b*d*e+C 
*c*d^2)+(A*e^2-B*d*e+C*d^2)/e^3*c-2/5*C/e^2*(3/2*a*e+3/2*b*d)-2/3*(1/e^2*( 
B*c*e+C*b*e-C*c*d)-2/5*C/e^2*(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-1/2/c*(-b+(...
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 814, normalized size of antiderivative = 1.12 \[ \int \frac {\sqrt {a+b x+c x^2} \left (A+B x+C x^2\right )}{(d+e x)^{3/2}} \, dx =\text {Too large to display} \] Input:

integrate((c*x^2+b*x+a)^(1/2)*(C*x^2+B*x+A)/(e*x+d)^(3/2),x, algorithm="fr 
icas")
 

Output:

-2/45*((48*C*c^3*d^4 - 8*(4*C*b*c^2 + 5*B*c^3)*d^3*e - (7*C*b^2*c - 30*A*c 
^3 - (42*C*a + 25*B*b)*c^2)*d^2*e^2 - (2*C*b^3 + 15*(2*B*a + A*b)*c^2 - (9 
*C*a*b + 5*B*b^2)*c)*d*e^3 + (48*C*c^3*d^3*e - 8*(4*C*b*c^2 + 5*B*c^3)*d^2 
*e^2 - (7*C*b^2*c - 30*A*c^3 - (42*C*a + 25*B*b)*c^2)*d*e^3 - (2*C*b^3 + 1 
5*(2*B*a + A*b)*c^2 - (9*C*a*b + 5*B*b^2)*c)*e^4)*x)*sqrt(c*e)*weierstrass 
PInverse(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*(48*C*c^3*d^3*e - 8*(C*b 
*c^2 + 5*B*c^3)*d^2*e^2 - (2*C*b^2*c - 30*A*c^3 - (6*C*a + 5*B*b)*c^2)*d*e 
^3 + (48*C*c^3*d^2*e^2 - 8*(C*b*c^2 + 5*B*c^3)*d*e^3 - (2*C*b^2*c - 30*A*c 
^3 - (6*C*a + 5*B*b)*c^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), weierstras 
sPInverse(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*(3*C*c^3*e^4*x^2 - 24* 
C*c^3*d^2*e^2 - 15*A*c^3*e^4 + (C*b*c^2 + 20*B*c^3)*d*e^3 - (6*C*c^3*d*e^3 
 - (C*b*c^2 + 5*B*c^3)*e^4)*x)*sqrt(c*x^2 + b*x + a)*sqrt(e*x + d))/(c^3*e 
^6*x + c^3*d*e^5)
 

Sympy [F]

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

integrate((c*x**2+b*x+a)**(1/2)*(C*x**2+B*x+A)/(e*x+d)**(3/2),x)
 

Output:

Integral((A + B*x + C*x**2)*sqrt(a + b*x + c*x**2)/(d + e*x)**(3/2), x)
 

Maxima [F]

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

integrate((c*x^2+b*x+a)^(1/2)*(C*x^2+B*x+A)/(e*x+d)^(3/2),x, algorithm="ma 
xima")
 

Output:

integrate((C*x^2 + B*x + A)*sqrt(c*x^2 + b*x + a)/(e*x + d)^(3/2), x)
 

Giac [F]

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

integrate((c*x^2+b*x+a)^(1/2)*(C*x^2+B*x+A)/(e*x+d)^(3/2),x, algorithm="gi 
ac")
 

Output:

integrate((C*x^2 + B*x + A)*sqrt(c*x^2 + b*x + a)/(e*x + d)^(3/2), x)
 

Mupad [F(-1)]

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

int(((A + B*x + C*x^2)*(a + b*x + c*x^2)^(1/2))/(d + e*x)^(3/2),x)
 

Output:

int(((A + B*x + C*x^2)*(a + b*x + c*x^2)^(1/2))/(d + e*x)^(3/2), x)
 

Reduce [F]

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

int((c*x^2+b*x+a)^(1/2)*(C*x^2+B*x+A)/(e*x+d)^(3/2),x)
 

Output:

int((c*x^2+b*x+a)^(1/2)*(C*x^2+B*x+A)/(e*x+d)^(3/2),x)