\(\int \frac {(d+e x)^{3/2} (A+B x+C x^2)}{\sqrt {a+b x+c x^2}} \, dx\) [101]

Optimal result

Integrand size = 34, antiderivative size = 693 \[ \int \frac {(d+e x)^{3/2} \left (A+B x+C x^2\right )}{\sqrt {a+b x+c x^2}} \, dx=-\frac {2 \left (15 b C d+\frac {6 c C d^2}{e}-35 A c e+25 a C e-\frac {24 b^2 C e}{c}-7 B (3 c d-4 b e)\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}}{105 c^2}+\frac {2 \left (7 B c-6 b C-\frac {2 c C d}{e}\right ) (d+e x)^{3/2} \sqrt {a+b x+c x^2}}{35 c^2}+\frac {2 C (d+e x)^{5/2} \sqrt {a+b x+c x^2}}{7 c e}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (48 b^3 C e^3-8 b c e^2 (9 b C d+7 b B e+13 a C e)+c^3 \left (6 C d^3-7 d e (3 B d+20 A e)\right )+c^2 e \left (a e (82 C d+63 B e)+b \left (12 C d^2+91 B d e+70 A e^2\right )\right )\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{105 c^4 e^2 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right ) \left (24 b^2 C e^2-c e (15 b C d+28 b B e+25 a C e)-c^2 \left (6 C d^2-7 e (3 B d+5 A e)\right )\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{105 c^4 e^2 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \] Output:

-2/105*(15*C*b*d+6*c*C*d^2/e-35*A*c*e+25*C*a*e-24*b^2*C*e/c-7*B*(-4*b*e+3* 
c*d))*(e*x+d)^(1/2)*(c*x^2+b*x+a)^(1/2)/c^2+2/35*(7*B*c-6*b*C-2*c*C*d/e)*( 
e*x+d)^(3/2)*(c*x^2+b*x+a)^(1/2)/c^2+2/7*C*(e*x+d)^(5/2)*(c*x^2+b*x+a)^(1/ 
2)/c/e-1/105*2^(1/2)*(-4*a*c+b^2)^(1/2)*(48*b^3*C*e^3-8*b*c*e^2*(7*B*b*e+1 
3*C*a*e+9*C*b*d)+c^3*(6*C*d^3-7*d*e*(20*A*e+3*B*d))+c^2*e*(a*e*(63*B*e+82* 
C*d)+b*(70*A*e^2+91*B*d*e+12*C*d^2)))*(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/e 
^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)- 
2/105*2^(1/2)*(-4*a*c+b^2)^(1/2)*(a*e^2-b*d*e+c*d^2)*(24*b^2*C*e^2-c*e*(28 
*B*b*e+25*C*a*e+15*C*b*d)-c^2*(6*C*d^2-7*e*(5*A*e+3*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^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 = 36.22 (sec) , antiderivative size = 9972, normalized size of antiderivative = 14.39 \[ \int \frac {(d+e x)^{3/2} \left (A+B x+C x^2\right )}{\sqrt {a+b x+c x^2}} \, dx=\text {Result too large to show} \] Input:

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

Output:

Result too large to show
 

Rubi [A] (warning: unable to verify)

Time = 2.84 (sec) , antiderivative size = 740, normalized size of antiderivative = 1.07, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {2184, 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.

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

\(\Big \downarrow \) 2184

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1236

\(\displaystyle \frac {2 C (d+e x)^{5/2} \sqrt {a+b x+c x^2}}{7 c e}-\frac {\frac {2 \int -\frac {\sqrt {d+e x} \left (6 C d e b^2+18 a C e^2 b-c d (3 C d+7 B e) b+c e (35 A c d-19 a C d-21 a B e)+\left (-\left (\left (6 C d^2-7 e (3 B d+5 A e)\right ) c^2\right )-e (15 b C d+28 b B e+25 a C e) c+24 b^2 C e^2\right ) x\right )}{2 \sqrt {c x^2+b x+a}}dx}{5 c}+\frac {2 (d+e x)^{3/2} \sqrt {a+b x+c x^2} (6 b C e-7 B c e+2 c C d)}{5 c}}{7 c e}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 C (d+e x)^{5/2} \sqrt {a+b x+c x^2}}{7 c e}-\frac {\frac {2 (d+e x)^{3/2} \sqrt {a+b x+c x^2} (6 b C e-7 B c e+2 c C d)}{5 c}-\frac {\int \frac {\sqrt {d+e x} \left (6 C d e b^2+18 a C e^2 b-c d (3 C d+7 B e) b+c e (35 A c d-19 a C d-21 a B e)+\left (-\left (\left (6 C d^2-7 e (3 B d+5 A e)\right ) c^2\right )-e (15 b C d+28 b B e+25 a C e) c+24 b^2 C e^2\right ) x\right )}{\sqrt {c x^2+b x+a}}dx}{5 c}}{7 c e}\)

\(\Big \downarrow \) 1236

\(\displaystyle \frac {2 C (d+e x)^{5/2} \sqrt {a+b x+c x^2}}{7 c e}-\frac {\frac {2 (d+e x)^{3/2} \sqrt {a+b x+c x^2} (6 b C e-7 B c e+2 c C d)}{5 c}-\frac {\frac {2 \int -\frac {24 C d e^2 b^3+\left (24 a C e^3-c d e (33 C d+28 B e)\right ) b^2+c \left (3 c C d^3+7 c e (6 B d+5 A e) d-2 a e^2 (47 C d+14 B e)\right ) b-c e \left (35 A c \left (3 c d^2-a e^2\right )+a \left (25 a C e^2-3 c d (17 C d+28 B e)\right )\right )+\left (\left (6 C d^3-7 d e (3 B d+20 A e)\right ) c^3+e \left (a e (82 C d+63 B e)+b \left (12 C d^2+91 B e d+70 A e^2\right )\right ) c^2-8 b e^2 (9 b C d+7 b B e+13 a C e) c+48 b^3 C e^3\right ) x}{2 \sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{3 c}+\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (-c e (25 a C e+28 b B e+15 b C d)-\left (c^2 \left (6 C d^2-7 e (5 A e+3 B d)\right )\right )+24 b^2 C e^2\right )}{3 c}}{5 c}}{7 c e}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 C (d+e x)^{5/2} \sqrt {a+b x+c x^2}}{7 c e}-\frac {\frac {2 (d+e x)^{3/2} \sqrt {a+b x+c x^2} (6 b C e-7 B c e+2 c C d)}{5 c}-\frac {\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (-c e (25 a C e+28 b B e+15 b C d)-\left (c^2 \left (6 C d^2-7 e (5 A e+3 B d)\right )\right )+24 b^2 C e^2\right )}{3 c}-\frac {\int \frac {24 C d e^2 b^3+\left (24 a C e^3-c d e (33 C d+28 B e)\right ) b^2+c \left (3 c C d^3+7 c e (6 B d+5 A e) d-2 a e^2 (47 C d+14 B e)\right ) b-c e \left (35 A c \left (3 c d^2-a e^2\right )-a \left (51 c C d^2+84 B c e d-25 a C e^2\right )\right )+\left (\left (6 C d^3-7 d e (3 B d+20 A e)\right ) c^3+e \left (a e (82 C d+63 B e)+b \left (12 C d^2+91 B e d+70 A e^2\right )\right ) c^2-8 b e^2 (9 b C d+7 b B e+13 a C e) c+48 b^3 C e^3\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{3 c}}{5 c}}{7 c e}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {2 C (d+e x)^{5/2} \sqrt {a+b x+c x^2}}{7 c e}-\frac {\frac {2 (d+e x)^{3/2} \sqrt {a+b x+c x^2} (6 b C e-7 B c e+2 c C d)}{5 c}-\frac {\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (-c e (25 a C e+28 b B e+15 b C d)-\left (c^2 \left (6 C d^2-7 e (5 A e+3 B d)\right )\right )+24 b^2 C e^2\right )}{3 c}-\frac {\frac {\left (c^2 e \left (a e (63 B e+82 C d)+b \left (70 A e^2+91 B d e+12 C d^2\right )\right )-8 b c e^2 (13 a C e+7 b B e+9 b C d)+c^3 \left (6 C d^3-7 d e (20 A e+3 B d)\right )+48 b^3 C e^3\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x+a}}dx}{e}+\frac {\left (a e^2-b d e+c d^2\right ) \left (-c e (25 a C e+28 b B e+15 b C d)-\left (c^2 \left (6 C d^2-7 e (5 A e+3 B d)\right )\right )+24 b^2 C e^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{e}}{3 c}}{5 c}}{7 c e}\)

\(\Big \downarrow \) 1172

\(\displaystyle \frac {2 C (d+e x)^{5/2} \sqrt {c x^2+b x+a}}{7 c e}-\frac {\frac {2 (2 c C d-7 B c e+6 b C e) (d+e x)^{3/2} \sqrt {c x^2+b x+a}}{5 c}-\frac {\frac {2 \left (-\left (\left (6 C d^2-7 e (3 B d+5 A e)\right ) c^2\right )-e (15 b C d+28 b B e+25 a C e) c+24 b^2 C e^2\right ) \sqrt {d+e x} \sqrt {c x^2+b x+a}}{3 c}-\frac {\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (c d^2-b e d+a e^2\right ) \left (-\left (\left (6 C d^2-7 e (3 B d+5 A e)\right ) c^2\right )-e (15 b C d+28 b B e+25 a C e) c+24 b^2 C e^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \int \frac {1}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}} \sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {d+e x} \sqrt {c x^2+b x+a}}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (\left (6 C d^3-7 d e (3 B d+20 A e)\right ) c^3+e \left (a e (82 C d+63 B e)+b \left (12 C d^2+91 B e d+70 A e^2\right )\right ) c^2-8 b e^2 (9 b C d+7 b B e+13 a C e) c+48 b^3 C e^3\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {c x^2+b x+a}}}{3 c}}{5 c}}{7 c e}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {2 C (d+e x)^{5/2} \sqrt {a+b x+c x^2}}{7 c e}-\frac {\frac {2 (d+e x)^{3/2} \sqrt {a+b x+c x^2} (6 b C e-7 B c e+2 c C d)}{5 c}-\frac {\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (-c e (25 a C e+28 b B e+15 b C d)-\left (c^2 \left (6 C d^2-7 e (5 A e+3 B d)\right )\right )+24 b^2 C e^2\right )}{3 c}-\frac {\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (c^2 e \left (a e (63 B e+82 C d)+b \left (70 A e^2+91 B d e+12 C d^2\right )\right )-8 b c e^2 (13 a C e+7 b B e+9 b C d)+c^3 \left (6 C d^3-7 d e (20 A e+3 B d)\right )+48 b^3 C e^3\right ) \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \left (-c e (25 a C e+28 b B e+15 b C d)-\left (c^2 \left (6 C d^2-7 e (5 A e+3 B d)\right )\right )+24 b^2 C e^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}}{3 c}}{5 c}}{7 c e}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {2 C (d+e x)^{5/2} \sqrt {a+b x+c x^2}}{7 c e}-\frac {\frac {2 (d+e x)^{3/2} \sqrt {a+b x+c x^2} (6 b C e-7 B c e+2 c C d)}{5 c}-\frac {\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (-c e (25 a C e+28 b B e+15 b C d)-\left (c^2 \left (6 C d^2-7 e (5 A e+3 B d)\right )\right )+24 b^2 C e^2\right )}{3 c}-\frac {\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \left (-c e (25 a C e+28 b B e+15 b C d)-\left (c^2 \left (6 C d^2-7 e (5 A e+3 B d)\right )\right )+24 b^2 C e^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (c^2 e \left (a e (63 B e+82 C d)+b \left (70 A e^2+91 B d e+12 C d^2\right )\right )-8 b c e^2 (13 a C e+7 b B e+9 b C d)+c^3 \left (6 C d^3-7 d e (20 A e+3 B d)\right )+48 b^3 C e^3\right ) E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}}{3 c}}{5 c}}{7 c e}\)

Input:

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

Output:

(2*C*(d + e*x)^(5/2)*Sqrt[a + b*x + c*x^2])/(7*c*e) - ((2*(2*c*C*d - 7*B*c 
*e + 6*b*C*e)*(d + e*x)^(3/2)*Sqrt[a + b*x + c*x^2])/(5*c) - ((2*(24*b^2*C 
*e^2 - c*e*(15*b*C*d + 28*b*B*e + 25*a*C*e) - c^2*(6*C*d^2 - 7*e*(3*B*d + 
5*A*e)))*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2])/(3*c) - ((Sqrt[2]*Sqrt[b^2 - 
 4*a*c]*(48*b^3*C*e^3 - 8*b*c*e^2*(9*b*C*d + 7*b*B*e + 13*a*C*e) + c^3*(6* 
C*d^3 - 7*d*e*(3*B*d + 20*A*e)) + c^2*e*(a*e*(82*C*d + 63*B*e) + b*(12*C*d 
^2 + 91*B*d*e + 70*A*e^2)))*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)]*Sq 
rt[a + b*x + c*x^2]) + (2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(c*d^2 - b*d*e + a*e^2 
)*(24*b^2*C*e^2 - c*e*(15*b*C*d + 28*b*B*e + 25*a*C*e) - c^2*(6*C*d^2 - 7* 
e*(3*B*d + 5*A*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]))/(3*c))/(5*c))/(7*c*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 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]
 

rule 2184

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p 
_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, S 
imp[f*(d + e*x)^(m + q - 1)*((a + b*x + c*x^2)^(p + 1)/(c*e^(q - 1)*(m + q 
+ 2*p + 1))), x] + Simp[1/(c*e^q*(m + q + 2*p + 1))   Int[(d + e*x)^m*(a + 
b*x + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 
1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(b*d*e*(p + 1) + a*e^2*(m + q - 1) - c 
*d^2*(m + q + 2*p + 1) - e*(2*c*d - b*e)*(m + q + p)*x), x], x], x] /; GtQ[ 
q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, d, e, m, p}, x] && Pol 
yQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] &&  !(IGt 
Q[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1260\) vs. \(2(627)=1254\).

Time = 4.90 (sec) , antiderivative size = 1261, normalized size of antiderivative = 1.82

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1261\)
risch	\(\text {Expression too large to display}\)	\(4796\)
default	\(\text {Expression too large to display}\)	\(14084\)

Input:

int((e*x+d)^(3/2)*(C*x^2+B*x+A)/(c*x^2+b*x+a)^(1/2),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)*(2/7*C*e/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*(B*e^2+2*C*d*e-2/ 
7*C*e/c*(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*(A*e^2+2*B*d*e+C*d^2-2/7*C*e/c*(5/2*a*e+5/2*b*d)-2/5*(B*e^2+2*C*d*e 
-2/7*C*e/c*(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*(A*d^2-2/5*(B*e^2+2*C*d*e-2/7*C*e/c*(3*b*e+3*c*d) 
)/c/e*a*d-2/3*(A*e^2+2*B*d*e+C*d^2-2/7*C*e/c*(5/2*a*e+5/2*b*d)-2/5*(B*e^2+ 
2*C*d*e-2/7*C*e/c*(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)*Ellip 
ticF(((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*(2*A*d*e 
+B*d^2-4/7*C*e/c*a*d-2/5*(B*e^2+2*C*d*e-2/7*C*e/c*(3*b*e+3*c*d))/c/e*(3/2* 
a*e+3/2*b*d)-2/3*(A*e^2+2*B*d*e+C*d^2-2/7*C*e/c*(5/2*a*e+5/2*b*d)-2/5*(B*e 
^2+2*C*d*e-2/7*C*e/c*(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...
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 761, normalized size of antiderivative = 1.10 \[ \int \frac {(d+e x)^{3/2} \left (A+B x+C x^2\right )}{\sqrt {a+b x+c x^2}} \, dx =\text {Too large to display} \] Input:

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

Output:

2/315*((6*C*c^4*d^4 + 3*(3*C*b*c^3 - 7*B*c^4)*d^3*e + (39*C*b^2*c^2 + 175* 
A*c^4 - (71*C*a + 56*B*b)*c^3)*d^2*e^2 - (96*C*b^3*c + 7*(27*B*a + 25*A*b) 
*c^3 - (260*C*a*b + 119*B*b^2)*c^2)*d*e^3 + (48*C*b^4 - 105*A*a*c^3 + (75* 
C*a^2 + 147*B*a*b + 70*A*b^2)*c^2 - 8*(22*C*a*b^2 + 7*B*b^3)*c)*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)) + 3*(6*C*c^4 
*d^3*e + 3*(4*C*b*c^3 - 7*B*c^4)*d^2*e^2 - (72*C*b^2*c^2 + 140*A*c^4 - (82 
*C*a + 91*B*b)*c^3)*d*e^3 + (48*C*b^3*c + 7*(9*B*a + 10*A*b)*c^3 - 8*(13*C 
*a*b + 7*B*b^2)*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), weierstrassPInvers 
e(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*c^4*e^4*x^2 + 3*C*c^4*d^ 
2*e^2 - 3*(11*C*b*c^3 - 14*B*c^4)*d*e^3 + (24*C*b^2*c^2 + 35*A*c^4 - (25*C 
*a + 28*B*b)*c^3)*e^4 + 3*(8*C*c^4*d*e^3 - (6*C*b*c^3 - 7*B*c^4)*e^4)*x)*s 
qrt(c*x^2 + b*x + a)*sqrt(e*x + d))/(c^5*e^3)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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