\(\int \sqrt {d+e x} (a+b x+c x^2)^{3/2} \, dx\) [642]

Optimal result

Integrand size = 24, antiderivative size = 683 \[ \int \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2} \, dx=-\frac {8 (2 c d-b e) \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}}{315 c^2 e^3}+\frac {2 (d+e x)^{3/2} \left (8 c^2 d^2+b^2 e^2-c e (13 b d-14 a e)-5 c e (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{105 c e^3}+\frac {2 (d+e x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{9 e}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (16 c^4 d^4-8 b^4 e^4-4 c^3 d^2 e (8 b d-15 a e)+b^2 c e^3 (7 b d+57 a e)+3 c^2 e^2 \left (3 b^2 d^2-20 a b d e-28 a^2 e^2\right )\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{315 c^3 e^4 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}+\frac {8 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{315 c^3 e^4 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \] Output:

-8/315*(-b*e+2*c*d)*(2*c^2*d^2-b^2*e^2-2*c*e*(-3*a*e+b*d))*(e*x+d)^(1/2)*( 
c*x^2+b*x+a)^(1/2)/c^2/e^3+2/105*(e*x+d)^(3/2)*(8*c^2*d^2+b^2*e^2-c*e*(-14 
*a*e+13*b*d)-5*c*e*(-b*e+2*c*d)*x)*(c*x^2+b*x+a)^(1/2)/c/e^3+2/9*(e*x+d)^( 
3/2)*(c*x^2+b*x+a)^(3/2)/e-1/315*2^(1/2)*(-4*a*c+b^2)^(1/2)*(16*c^4*d^4-8* 
b^4*e^4-4*c^3*d^2*e*(-15*a*e+8*b*d)+b^2*c*e^3*(57*a*e+7*b*d)+3*c^2*e^2*(-2 
8*a^2*e^2-20*a*b*d*e+3*b^2*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^3/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)+8/315 
*2^(1/2)*(-4*a*c+b^2)^(1/2)*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)*(2*c^2*d^2-b^ 
2*e^2-2*c*e*(-3*a*e+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^3/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 = 34.97 (sec) , antiderivative size = 7541, normalized size of antiderivative = 11.04 \[ \int \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2} \, dx=\text {Result too large to show} \] Input:

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

Output:

Result too large to show
 

Rubi [A] (verified)

Time = 1.18 (sec) , antiderivative size = 734, normalized size of antiderivative = 1.07, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {1162, 1236, 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.

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

\(\Big \downarrow \) 1162

\(\displaystyle \frac {2 (d+e x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{9 e}-\frac {\int \sqrt {d+e x} (b d-2 a e+(2 c d-b e) x) \sqrt {c x^2+b x+a}dx}{3 e}\)

\(\Big \downarrow \) 1236

\(\displaystyle \frac {2 (d+e x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{9 e}-\frac {\frac {2 \int \frac {\left (3 d e b^2+c d^2 b+a e^2 b-16 a c d e+2 \left (c^2 d^2+2 b^2 e^2-c e (b d+7 a e)\right ) x\right ) \sqrt {c x^2+b x+a}}{2 \sqrt {d+e x}}dx}{7 c}+\frac {2 \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{7 c}}{3 e}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 (d+e x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{9 e}-\frac {\frac {\int \frac {\left (3 d e b^2+c d^2 b+a e^2 b-16 a c d e+2 \left (c^2 d^2+2 b^2 e^2-c e (b d+7 a e)\right ) x\right ) \sqrt {c x^2+b x+a}}{\sqrt {d+e x}}dx}{7 c}+\frac {2 \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{7 c}}{3 e}\)

\(\Big \downarrow \) 1231

\(\displaystyle \frac {2 (d+e x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{9 e}-\frac {\frac {-\frac {2 \int \frac {5 c e (b d-2 a e) \left (3 d e b^2+c d^2 b+a e^2 b-16 a c d e\right )-2 \left (-d e b^2+4 c d^2 b-a e^2 b-2 a c d e\right ) \left (c^2 d^2+2 b^2 e^2-c e (b d+7 a e)\right )-\left (16 c^4 d^4-4 c^3 e (8 b d-15 a e) d^2-8 b^4 e^4+b^2 c e^3 (7 b d+57 a e)+3 c^2 e^2 \left (3 b^2 d^2-20 a b e d-28 a^2 e^2\right )\right ) x}{2 \sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{15 c e^2}-\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (-6 c e x \left (-c e (7 a e+b d)+2 b^2 e^2+c^2 d^2\right )-3 c^2 d e (5 b d-8 a e)+3 b c e^2 (3 a e+b d)-4 b^3 e^3+8 c^3 d^3\right )}{15 c e^2}}{7 c}+\frac {2 \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{7 c}}{3 e}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 (d+e x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{9 e}-\frac {\frac {-\frac {\int \frac {5 c e (b d-2 a e) \left (3 d e b^2+c d^2 b+a e^2 b-16 a c d e\right )-2 \left (-d e b^2+4 c d^2 b-a e^2 b-2 a c d e\right ) \left (c^2 d^2+2 b^2 e^2-c e (b d+7 a e)\right )-\left (16 c^4 d^4-4 c^3 e (8 b d-15 a e) d^2-8 b^4 e^4+b^2 c e^3 (7 b d+57 a e)+3 c^2 e^2 \left (3 b^2 d^2-20 a b e d-28 a^2 e^2\right )\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{15 c e^2}-\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (-6 c e x \left (-c e (7 a e+b d)+2 b^2 e^2+c^2 d^2\right )-3 c^2 d e (5 b d-8 a e)+3 b c e^2 (3 a e+b d)-4 b^3 e^3+8 c^3 d^3\right )}{15 c e^2}}{7 c}+\frac {2 \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{7 c}}{3 e}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {2 (d+e x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{9 e}-\frac {\frac {-\frac {\frac {4 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (6 a c e^2-b^2 e^2-2 b c d e+2 c^2 d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{e}-\frac {\left (c^2 \left (-84 a^2 e^4-60 a b d e^3+9 b^2 d^2 e^2\right )+b^2 c e^3 (57 a e+7 b d)-4 c^3 d^2 e (8 b d-15 a e)-8 b^4 e^4+16 c^4 d^4\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x+a}}dx}{e}}{15 c e^2}-\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (-6 c e x \left (-c e (7 a e+b d)+2 b^2 e^2+c^2 d^2\right )-3 c^2 d e (5 b d-8 a e)+3 b c e^2 (3 a e+b d)-4 b^3 e^3+8 c^3 d^3\right )}{15 c e^2}}{7 c}+\frac {2 \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{7 c}}{3 e}\)

\(\Big \downarrow \) 1172

\(\displaystyle \frac {2 (d+e x)^{3/2} \left (c x^2+b x+a\right )^{3/2}}{9 e}-\frac {\frac {2 (2 c d-b e) \sqrt {d+e x} \left (c x^2+b x+a\right )^{3/2}}{7 c}+\frac {-\frac {2 \sqrt {d+e x} \sqrt {c x^2+b x+a} \left (8 c^3 d^3-3 c^2 e (5 b d-8 a e) d-4 b^3 e^3+3 b c e^2 (b d+3 a e)-6 c e \left (c^2 d^2+2 b^2 e^2-c e (b d+7 a e)\right ) x\right )}{15 c e^2}-\frac {\frac {8 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (c d^2-b e d+a e^2\right ) \left (2 c^2 d^2-2 b c e d-b^2 e^2+6 a c e^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \int \frac {1}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}} \sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {d+e x} \sqrt {c x^2+b x+a}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (16 c^4 d^4-4 c^3 e (8 b d-15 a e) d^2-8 b^4 e^4+b^2 c e^3 (7 b d+57 a e)+c^2 \left (-84 a^2 e^4-60 a b d e^3+9 b^2 d^2 e^2\right )\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {c x^2+b x+a}}}{15 c e^2}}{7 c}}{3 e}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {2 (d+e x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{9 e}-\frac {\frac {-\frac {\frac {8 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (6 a c e^2-b^2 e^2-2 b c d e+2 c^2 d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (c^2 \left (-84 a^2 e^4-60 a b d e^3+9 b^2 d^2 e^2\right )+b^2 c e^3 (57 a e+7 b d)-4 c^3 d^2 e (8 b d-15 a e)-8 b^4 e^4+16 c^4 d^4\right ) \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}}{15 c e^2}-\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (-6 c e x \left (-c e (7 a e+b d)+2 b^2 e^2+c^2 d^2\right )-3 c^2 d e (5 b d-8 a e)+3 b c e^2 (3 a e+b d)-4 b^3 e^3+8 c^3 d^3\right )}{15 c e^2}}{7 c}+\frac {2 \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{7 c}}{3 e}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {2 (d+e x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{9 e}-\frac {\frac {-\frac {\frac {8 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (6 a c e^2-b^2 e^2-2 b c d e+2 c^2 d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (c^2 \left (-84 a^2 e^4-60 a b d e^3+9 b^2 d^2 e^2\right )+b^2 c e^3 (57 a e+7 b d)-4 c^3 d^2 e (8 b d-15 a e)-8 b^4 e^4+16 c^4 d^4\right ) E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}}{15 c e^2}-\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (-6 c e x \left (-c e (7 a e+b d)+2 b^2 e^2+c^2 d^2\right )-3 c^2 d e (5 b d-8 a e)+3 b c e^2 (3 a e+b d)-4 b^3 e^3+8 c^3 d^3\right )}{15 c e^2}}{7 c}+\frac {2 \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{7 c}}{3 e}\)

Input:

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

Output:

(2*(d + e*x)^(3/2)*(a + b*x + c*x^2)^(3/2))/(9*e) - ((2*(2*c*d - b*e)*Sqrt 
[d + e*x]*(a + b*x + c*x^2)^(3/2))/(7*c) + ((-2*Sqrt[d + e*x]*(8*c^3*d^3 - 
 4*b^3*e^3 - 3*c^2*d*e*(5*b*d - 8*a*e) + 3*b*c*e^2*(b*d + 3*a*e) - 6*c*e*( 
c^2*d^2 + 2*b^2*e^2 - c*e*(b*d + 7*a*e))*x)*Sqrt[a + b*x + c*x^2])/(15*c*e 
^2) - (-((Sqrt[2]*Sqrt[b^2 - 4*a*c]*(16*c^4*d^4 - 8*b^4*e^4 - 4*c^3*d^2*e* 
(8*b*d - 15*a*e) + b^2*c*e^3*(7*b*d + 57*a*e) + c^2*(9*b^2*d^2*e^2 - 60*a* 
b*d*e^3 - 84*a^2*e^4))*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])) + (8*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(2*c*d - b*e)*(c*d^2 - b*d 
*e + a*e^2)*(2*c^2*d^2 - 2*b*c*d*e - b^2*e^2 + 6*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]))/(15*c*e^2))/(7*c))/(3* 
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 1162

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 + 2*p + 1))), x 
] - Simp[p/(e*(m + 2*p + 1))   Int[(d + e*x)^m*Simp[b*d - 2*a*e + (2*c*d - 
b*e)*x, x]*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x 
] && GtQ[p, 0] && NeQ[m + 2*p + 1, 0] && ( !RationalQ[m] || LtQ[m, 1]) && 
!ILtQ[m + 2*p, 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 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 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. \(1664\) vs. \(2(617)=1234\).

Time = 4.91 (sec) , antiderivative size = 1665, normalized size of antiderivative = 2.44

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1665\)
risch	\(\text {Expression too large to display}\)	\(3639\)
default	\(\text {Expression too large to display}\)	\(9174\)

Input:

int((e*x+d)^(1/2)*(c*x^2+b*x+a)^(3/2),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)*(2/9*c*x^3 
*(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)+2/7*(2*b*c*e+c^2*d-2/9*c* 
(4*b*e+4*c*d))/c/e*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 
*(2*a*c*e+e*b^2+2*d*b*c-2/9*c*(7/2*a*e+7/2*b*d)-2/7*(2*b*c*e+c^2*d-2/9*c*( 
4*b*e+4*c*d))/c/e*(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*(2*a*b*e+4/3*a*c*d+b^2*d-2/7*(2*b*c*e+c^2*d-2/9*c*(4*b*e+ 
4*c*d))/c/e*(5/2*a*e+5/2*b*d)-2/5*(2*a*c*e+e*b^2+2*d*b*c-2/9*c*(7/2*a*e+7/ 
2*b*d)-2/7*(2*b*c*e+c^2*d-2/9*c*(4*b*e+4*c*d))/c/e*(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^2*d-2/ 
5*(2*a*c*e+e*b^2+2*d*b*c-2/9*c*(7/2*a*e+7/2*b*d)-2/7*(2*b*c*e+c^2*d-2/9*c* 
(4*b*e+4*c*d))/c/e*(3*b*e+3*c*d))/c/e*a*d-2/3*(2*a*b*e+4/3*a*c*d+b^2*d-2/7 
*(2*b*c*e+c^2*d-2/9*c*(4*b*e+4*c*d))/c/e*(5/2*a*e+5/2*b*d)-2/5*(2*a*c*e+e* 
b^2+2*d*b*c-2/9*c*(7/2*a*e+7/2*b*d)-2/7*(2*b*c*e+c^2*d-2/9*c*(4*b*e+4*c*d) 
)/c/e*(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*(a^2*e+2*d*a*b-4/...
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 727, normalized size of antiderivative = 1.06 \[ \int \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2} \, dx =\text {Too large to display} \] Input:

integrate((e*x+d)^(1/2)*(c*x^2+b*x+a)^(3/2),x, algorithm="fricas")
 

Output:

2/945*((16*c^5*d^5 - 40*b*c^4*d^4*e + 2*(11*b^2*c^3 + 36*a*c^4)*d^3*e^2 + 
(7*b^3*c^2 - 108*a*b*c^3)*d^2*e^3 + (11*b^4*c - 102*a*b^2*c^2 + 312*a^2*c^ 
3)*d*e^4 - (8*b^5 - 69*a*b^3*c + 156*a^2*b*c^2)*e^5)*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*(16*c^5*d^4*e - 32*b*c^4 
*d^3*e^2 + 3*(3*b^2*c^3 + 20*a*c^4)*d^2*e^3 + (7*b^3*c^2 - 60*a*b*c^3)*d*e 
^4 - (8*b^4*c - 57*a*b^2*c^2 + 84*a^2*c^3)*e^5)*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*(35*c^5* 
e^5*x^3 + 8*c^5*d^3*e^2 - 15*b*c^4*d^2*e^3 + (3*b^2*c^3 + 29*a*c^4)*d*e^4 
- 4*(b^3*c^2 - 6*a*b*c^3)*e^5 + 5*(c^5*d*e^4 + 10*b*c^4*e^5)*x^2 - (6*c^5* 
d^2*e^3 - 11*b*c^4*d*e^4 - (3*b^2*c^3 + 77*a*c^4)*e^5)*x)*sqrt(c*x^2 + b*x 
 + a)*sqrt(e*x + d))/(c^4*e^5)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((e*x+d)^(1/2)*(c*x^2+b*x+a)^(3/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((e*x+d)^(1/2)*(c*x^2+b*x+a)^(3/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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