\(\int \sqrt {d+e x} (b x+c x^2)^{3/2} \, dx\) [192]

Optimal result

Integrand size = 23, antiderivative size = 552 \[ \int \sqrt {d+e x} \left (b x+c x^2\right )^{3/2} \, dx=-\frac {2 \left (16 c^4 d^4-32 b c^3 d^3 e+9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4\right ) x \sqrt {d+e x}}{315 c^2 e^4 \sqrt {b x+c x^2}}+\frac {2 \left (8 c^3 d^3-15 b c^2 d^2 e+3 b^2 c d e^2-4 b^3 e^3\right ) \sqrt {d+e x} \sqrt {b x+c x^2}}{315 c^2 e^3}+\frac {2 \left (11 b d-\frac {6 c d^2}{e}+\frac {3 b^2 e}{c}\right ) x \sqrt {d+e x} \sqrt {b x+c x^2}}{315 e}+\frac {2 (c d+3 b e) x^2 \sqrt {d+e x} \sqrt {b x+c x^2}}{63 e}+\frac {2}{9} x \sqrt {d+e x} \left (b x+c x^2\right )^{3/2}+\frac {2 \sqrt {b} \left (16 c^4 d^4-32 b c^3 d^3 e+9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4\right ) \sqrt {x} \sqrt {d+e x} E\left (\arctan \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )|1-\frac {b e}{c d}\right )}{315 c^{5/2} e^4 \sqrt {\frac {b (d+e x)}{d (b+c x)}} \sqrt {b x+c x^2}}-\frac {2 b^{3/2} \left (8 c^3 d^3-15 b c^2 d^2 e+3 b^2 c d e^2-4 b^3 e^3\right ) \sqrt {x} \sqrt {d+e x} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right ),1-\frac {b e}{c d}\right )}{315 c^{5/2} e^3 \sqrt {\frac {b (d+e x)}{d (b+c x)}} \sqrt {b x+c x^2}} \] Output:

-2/315*(-8*b^4*e^4+7*b^3*c*d*e^3+9*b^2*c^2*d^2*e^2-32*b*c^3*d^3*e+16*c^4*d 
^4)*x*(e*x+d)^(1/2)/c^2/e^4/(c*x^2+b*x)^(1/2)+2/315*(-4*b^3*e^3+3*b^2*c*d* 
e^2-15*b*c^2*d^2*e+8*c^3*d^3)*(e*x+d)^(1/2)*(c*x^2+b*x)^(1/2)/c^2/e^3+2/31 
5*(11*b*d-6*c*d^2/e+3*b^2*e/c)*x*(e*x+d)^(1/2)*(c*x^2+b*x)^(1/2)/e+2/63*(3 
*b*e+c*d)*x^2*(e*x+d)^(1/2)*(c*x^2+b*x)^(1/2)/e+2/9*x*(e*x+d)^(1/2)*(c*x^2 
+b*x)^(3/2)+2/315*b^(1/2)*(-8*b^4*e^4+7*b^3*c*d*e^3+9*b^2*c^2*d^2*e^2-32*b 
*c^3*d^3*e+16*c^4*d^4)*x^(1/2)*(e*x+d)^(1/2)*EllipticE(c^(1/2)*x^(1/2)/b^( 
1/2)/(1+c*x/b)^(1/2),(1-b*e/c/d)^(1/2))/c^(5/2)/e^4/(b*(e*x+d)/d/(c*x+b))^ 
(1/2)/(c*x^2+b*x)^(1/2)-2/315*b^(3/2)*(-4*b^3*e^3+3*b^2*c*d*e^2-15*b*c^2*d 
^2*e+8*c^3*d^3)*x^(1/2)*(e*x+d)^(1/2)*InverseJacobiAM(arctan(c^(1/2)*x^(1/ 
2)/b^(1/2)),(1-b*e/c/d)^(1/2))/c^(5/2)/e^3/(b*(e*x+d)/d/(c*x+b))^(1/2)/(c* 
x^2+b*x)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 18.24 (sec) , antiderivative size = 463, normalized size of antiderivative = 0.84 \[ \int \sqrt {d+e x} \left (b x+c x^2\right )^{3/2} \, dx=\frac {2 (x (b+c x))^{3/2} \left (b e x (b+c x) (d+e x) \left (-4 b^3 e^3+3 b^2 c e^2 (d+e x)+b c^2 e \left (-15 d^2+11 d e x+50 e^2 x^2\right )+c^3 \left (8 d^3-6 d^2 e x+5 d e^2 x^2+35 e^3 x^3\right )\right )-\sqrt {\frac {b}{c}} \left (\sqrt {\frac {b}{c}} \left (16 c^4 d^4-32 b c^3 d^3 e+9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4\right ) (b+c x) (d+e x)+i b e \left (16 c^4 d^4-32 b c^3 d^3 e+9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )-i b e \left (8 c^4 d^4-17 b c^3 d^3 e+6 b^2 c^2 d^2 e^2+11 b^3 c d e^3-8 b^4 e^4\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )\right )\right )}{315 b c^2 e^4 x^2 (b+c x)^2 \sqrt {d+e x}} \] Input:

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

Output:

(2*(x*(b + c*x))^(3/2)*(b*e*x*(b + c*x)*(d + e*x)*(-4*b^3*e^3 + 3*b^2*c*e^ 
2*(d + e*x) + b*c^2*e*(-15*d^2 + 11*d*e*x + 50*e^2*x^2) + c^3*(8*d^3 - 6*d 
^2*e*x + 5*d*e^2*x^2 + 35*e^3*x^3)) - Sqrt[b/c]*(Sqrt[b/c]*(16*c^4*d^4 - 3 
2*b*c^3*d^3*e + 9*b^2*c^2*d^2*e^2 + 7*b^3*c*d*e^3 - 8*b^4*e^4)*(b + c*x)*( 
d + e*x) + I*b*e*(16*c^4*d^4 - 32*b*c^3*d^3*e + 9*b^2*c^2*d^2*e^2 + 7*b^3* 
c*d*e^3 - 8*b^4*e^4)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE 
[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - I*b*e*(8*c^4*d^4 - 17*b*c^3* 
d^3*e + 6*b^2*c^2*d^2*e^2 + 11*b^3*c*d*e^3 - 8*b^4*e^4)*Sqrt[1 + b/(c*x)]* 
Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b 
*e)])))/(315*b*c^2*e^4*x^2*(b + c*x)^2*Sqrt[d + e*x])
 

Rubi [A] (verified)

Time = 1.25 (sec) , antiderivative size = 477, normalized size of antiderivative = 0.86, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {1162, 1236, 27, 1231, 27, 1269, 1169, 122, 120, 127, 126}

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.

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

\(\Big \downarrow \) 1162

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

\(\Big \downarrow \) 1236

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1231

\(\displaystyle \frac {2 \left (b x+c x^2\right )^{3/2} (d+e x)^{3/2}}{9 e}-\frac {\frac {-\frac {2 \int -\frac {b d \left (8 c^3 d^3-15 b c^2 e d^2+3 b^2 c e^2 d-4 b^3 e^3\right )+\left (16 c^4 d^4-32 b c^3 e d^3+9 b^2 c^2 e^2 d^2+7 b^3 c e^3 d-8 b^4 e^4\right ) x}{2 \sqrt {d+e x} \sqrt {c x^2+b x}}dx}{15 c e^2}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} \left (-4 b^3 e^3-6 c e x \left (2 b^2 e^2-b c d e+c^2 d^2\right )+3 b^2 c d e^2-15 b c^2 d^2 e+8 c^3 d^3\right )}{15 c e^2}}{7 c}+\frac {2 \left (b x+c x^2\right )^{3/2} \sqrt {d+e x} (2 c d-b e)}{7 c}}{3 e}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 \left (b x+c x^2\right )^{3/2} (d+e x)^{3/2}}{9 e}-\frac {\frac {\frac {\int \frac {b d \left (8 c^3 d^3-15 b c^2 e d^2+3 b^2 c e^2 d-4 b^3 e^3\right )+\left (16 c^4 d^4-32 b c^3 e d^3+9 b^2 c^2 e^2 d^2+7 b^3 c e^3 d-8 b^4 e^4\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{15 c e^2}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} \left (-4 b^3 e^3-6 c e x \left (2 b^2 e^2-b c d e+c^2 d^2\right )+3 b^2 c d e^2-15 b c^2 d^2 e+8 c^3 d^3\right )}{15 c e^2}}{7 c}+\frac {2 \left (b x+c x^2\right )^{3/2} \sqrt {d+e x} (2 c d-b e)}{7 c}}{3 e}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {2 \left (b x+c x^2\right )^{3/2} (d+e x)^{3/2}}{9 e}-\frac {\frac {\frac {\frac {\left (-8 b^4 e^4+7 b^3 c d e^3+9 b^2 c^2 d^2 e^2-32 b c^3 d^3 e+16 c^4 d^4\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x}}dx}{e}-\frac {4 d (c d-b e) (2 c d-b e) \left (-b^2 e^2-2 b c d e+2 c^2 d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{e}}{15 c e^2}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} \left (-4 b^3 e^3-6 c e x \left (2 b^2 e^2-b c d e+c^2 d^2\right )+3 b^2 c d e^2-15 b c^2 d^2 e+8 c^3 d^3\right )}{15 c e^2}}{7 c}+\frac {2 \left (b x+c x^2\right )^{3/2} \sqrt {d+e x} (2 c d-b e)}{7 c}}{3 e}\)

\(\Big \downarrow \) 1169

\(\displaystyle \frac {2 \left (b x+c x^2\right )^{3/2} (d+e x)^{3/2}}{9 e}-\frac {\frac {\frac {\frac {\sqrt {x} \sqrt {b+c x} \left (-8 b^4 e^4+7 b^3 c d e^3+9 b^2 c^2 d^2 e^2-32 b c^3 d^3 e+16 c^4 d^4\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}}dx}{e \sqrt {b x+c x^2}}-\frac {4 d \sqrt {x} \sqrt {b+c x} (c d-b e) (2 c d-b e) \left (-b^2 e^2-2 b c d e+2 c^2 d^2\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}}{15 c e^2}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} \left (-4 b^3 e^3-6 c e x \left (2 b^2 e^2-b c d e+c^2 d^2\right )+3 b^2 c d e^2-15 b c^2 d^2 e+8 c^3 d^3\right )}{15 c e^2}}{7 c}+\frac {2 \left (b x+c x^2\right )^{3/2} \sqrt {d+e x} (2 c d-b e)}{7 c}}{3 e}\)

\(\Big \downarrow \) 122

\(\displaystyle \frac {2 \left (b x+c x^2\right )^{3/2} (d+e x)^{3/2}}{9 e}-\frac {\frac {\frac {\frac {\sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (-8 b^4 e^4+7 b^3 c d e^3+9 b^2 c^2 d^2 e^2-32 b c^3 d^3 e+16 c^4 d^4\right ) \int \frac {\sqrt {\frac {e x}{d}+1}}{\sqrt {x} \sqrt {\frac {c x}{b}+1}}dx}{e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {4 d \sqrt {x} \sqrt {b+c x} (c d-b e) (2 c d-b e) \left (-b^2 e^2-2 b c d e+2 c^2 d^2\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}}{15 c e^2}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} \left (-4 b^3 e^3-6 c e x \left (2 b^2 e^2-b c d e+c^2 d^2\right )+3 b^2 c d e^2-15 b c^2 d^2 e+8 c^3 d^3\right )}{15 c e^2}}{7 c}+\frac {2 \left (b x+c x^2\right )^{3/2} \sqrt {d+e x} (2 c d-b e)}{7 c}}{3 e}\)

\(\Big \downarrow \) 120

\(\displaystyle \frac {2 \left (b x+c x^2\right )^{3/2} (d+e x)^{3/2}}{9 e}-\frac {\frac {\frac {\frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (-8 b^4 e^4+7 b^3 c d e^3+9 b^2 c^2 d^2 e^2-32 b c^3 d^3 e+16 c^4 d^4\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {4 d \sqrt {x} \sqrt {b+c x} (c d-b e) (2 c d-b e) \left (-b^2 e^2-2 b c d e+2 c^2 d^2\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}}{15 c e^2}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} \left (-4 b^3 e^3-6 c e x \left (2 b^2 e^2-b c d e+c^2 d^2\right )+3 b^2 c d e^2-15 b c^2 d^2 e+8 c^3 d^3\right )}{15 c e^2}}{7 c}+\frac {2 \left (b x+c x^2\right )^{3/2} \sqrt {d+e x} (2 c d-b e)}{7 c}}{3 e}\)

\(\Big \downarrow \) 127

\(\displaystyle \frac {2 \left (b x+c x^2\right )^{3/2} (d+e x)^{3/2}}{9 e}-\frac {\frac {\frac {\frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (-8 b^4 e^4+7 b^3 c d e^3+9 b^2 c^2 d^2 e^2-32 b c^3 d^3 e+16 c^4 d^4\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {4 d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) (2 c d-b e) \left (-b^2 e^2-2 b c d e+2 c^2 d^2\right ) \int \frac {1}{\sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1}}dx}{e \sqrt {b x+c x^2} \sqrt {d+e x}}}{15 c e^2}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} \left (-4 b^3 e^3-6 c e x \left (2 b^2 e^2-b c d e+c^2 d^2\right )+3 b^2 c d e^2-15 b c^2 d^2 e+8 c^3 d^3\right )}{15 c e^2}}{7 c}+\frac {2 \left (b x+c x^2\right )^{3/2} \sqrt {d+e x} (2 c d-b e)}{7 c}}{3 e}\)

\(\Big \downarrow \) 126

\(\displaystyle \frac {2 \left (b x+c x^2\right )^{3/2} (d+e x)^{3/2}}{9 e}-\frac {\frac {\frac {\frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (-8 b^4 e^4+7 b^3 c d e^3+9 b^2 c^2 d^2 e^2-32 b c^3 d^3 e+16 c^4 d^4\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {8 \sqrt {-b} d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) (2 c d-b e) \left (-b^2 e^2-2 b c d e+2 c^2 d^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {d+e x}}}{15 c e^2}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} \left (-4 b^3 e^3-6 c e x \left (2 b^2 e^2-b c d e+c^2 d^2\right )+3 b^2 c d e^2-15 b c^2 d^2 e+8 c^3 d^3\right )}{15 c e^2}}{7 c}+\frac {2 \left (b x+c x^2\right )^{3/2} \sqrt {d+e x} (2 c d-b e)}{7 c}}{3 e}\)

Input:

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

Output:

(2*(d + e*x)^(3/2)*(b*x + c*x^2)^(3/2))/(9*e) - ((2*(2*c*d - b*e)*Sqrt[d + 
 e*x]*(b*x + c*x^2)^(3/2))/(7*c) + ((-2*Sqrt[d + e*x]*(8*c^3*d^3 - 15*b*c^ 
2*d^2*e + 3*b^2*c*d*e^2 - 4*b^3*e^3 - 6*c*e*(c^2*d^2 - b*c*d*e + 2*b^2*e^2 
)*x)*Sqrt[b*x + c*x^2])/(15*c*e^2) + ((2*Sqrt[-b]*(16*c^4*d^4 - 32*b*c^3*d 
^3*e + 9*b^2*c^2*d^2*e^2 + 7*b^3*c*d*e^3 - 8*b^4*e^4)*Sqrt[x]*Sqrt[1 + (c* 
x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c 
*d)])/(Sqrt[c]*e*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (8*Sqrt[-b]*d*(c*d 
 - b*e)*(2*c*d - b*e)*(2*c^2*d^2 - 2*b*c*d*e - b^2*e^2)*Sqrt[x]*Sqrt[1 + ( 
c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b 
*e)/(c*d)])/(Sqrt[c]*e*Sqrt[d + e*x]*Sqrt[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 120

Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_] 
 :> Simp[2*(Sqrt[e]/b)*Rt[-b/d, 2]*EllipticE[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[- 
b/d, 2])], c*(f/(d*e))], x] /; FreeQ[{b, c, d, e, f}, x] && GtQ[c, 0] && Gt 
Q[e, 0] &&  !LtQ[-b/d, 0]
 

rule 122

Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_] 
 :> Simp[Sqrt[e + f*x]*(Sqrt[1 + d*(x/c)]/(Sqrt[c + d*x]*Sqrt[1 + f*(x/e)]) 
)   Int[Sqrt[1 + f*(x/e)]/(Sqrt[b*x]*Sqrt[1 + d*(x/c)]), x], x] /; FreeQ[{b 
, c, d, e, f}, x] &&  !(GtQ[c, 0] && GtQ[e, 0])
 

rule 126

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x 
_] :> Simp[(2/(b*Sqrt[e]))*Rt[-b/d, 2]*EllipticF[ArcSin[Sqrt[b*x]/(Sqrt[c]* 
Rt[-b/d, 2])], c*(f/(d*e))], x] /; FreeQ[{b, c, d, e, f}, x] && GtQ[c, 0] & 
& GtQ[e, 0] && (PosQ[-b/d] || NegQ[-b/f])
 

rule 127

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x 
_] :> Simp[Sqrt[1 + d*(x/c)]*(Sqrt[1 + f*(x/e)]/(Sqrt[c + d*x]*Sqrt[e + f*x 
]))   Int[1/(Sqrt[b*x]*Sqrt[1 + d*(x/c)]*Sqrt[1 + f*(x/e)]), x], x] /; Free 
Q[{b, c, d, e, f}, x] &&  !(GtQ[c, 0] && GtQ[e, 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 1169

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> 
 Simp[Sqrt[x]*(Sqrt[b + c*x]/Sqrt[b*x + c*x^2])   Int[(d + e*x)^m/(Sqrt[x]* 
Sqrt[b + c*x]), x], x] /; FreeQ[{b, c, d, e}, x] && NeQ[c*d - b*e, 0] && Eq 
Q[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 [A] (verified)

Time = 1.11 (sec) , antiderivative size = 942, normalized size of antiderivative = 1.71

method	result	size
elliptic	\(\frac {\sqrt {x \left (c x +b \right )}\, \sqrt {\left (c x +b \right ) x \left (e x +d \right )}\, \left (\frac {2 c \,x^{3} \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{9}+\frac {2 \left (2 b c e +c^{2} d -\frac {2 c \left (4 b e +4 c d \right )}{9}\right ) x^{2} \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{7 c e}+\frac {2 \left (e \,b^{2}+\frac {11 d b c}{9}-\frac {2 \left (2 b c e +c^{2} d -\frac {2 c \left (4 b e +4 c d \right )}{9}\right ) \left (3 b e +3 c d \right )}{7 c e}\right ) x \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{5 c e}+\frac {2 \left (b^{2} d -\frac {5 \left (2 b c e +c^{2} d -\frac {2 c \left (4 b e +4 c d \right )}{9}\right ) b d}{7 c e}-\frac {2 \left (e \,b^{2}+\frac {11 d b c}{9}-\frac {2 \left (2 b c e +c^{2} d -\frac {2 c \left (4 b e +4 c d \right )}{9}\right ) \left (3 b e +3 c d \right )}{7 c e}\right ) \left (2 b e +2 c d \right )}{5 c e}\right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{3 c e}-\frac {2 \left (b^{2} d -\frac {5 \left (2 b c e +c^{2} d -\frac {2 c \left (4 b e +4 c d \right )}{9}\right ) b d}{7 c e}-\frac {2 \left (e \,b^{2}+\frac {11 d b c}{9}-\frac {2 \left (2 b c e +c^{2} d -\frac {2 c \left (4 b e +4 c d \right )}{9}\right ) \left (3 b e +3 c d \right )}{7 c e}\right ) \left (2 b e +2 c d \right )}{5 c e}\right ) b \,d^{2} \sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}\, \sqrt {\frac {\frac {b}{c}+x}{-\frac {d}{e}+\frac {b}{c}}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}+\frac {b}{c}\right )}}\right )}{3 c \,e^{2} \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}+\frac {2 \left (-\frac {3 \left (e \,b^{2}+\frac {11 d b c}{9}-\frac {2 \left (2 b c e +c^{2} d -\frac {2 c \left (4 b e +4 c d \right )}{9}\right ) \left (3 b e +3 c d \right )}{7 c e}\right ) b d}{5 c e}-\frac {2 \left (b^{2} d -\frac {5 \left (2 b c e +c^{2} d -\frac {2 c \left (4 b e +4 c d \right )}{9}\right ) b d}{7 c e}-\frac {2 \left (e \,b^{2}+\frac {11 d b c}{9}-\frac {2 \left (2 b c e +c^{2} d -\frac {2 c \left (4 b e +4 c d \right )}{9}\right ) \left (3 b e +3 c d \right )}{7 c e}\right ) \left (2 b e +2 c d \right )}{5 c e}\right ) \left (b e +c d \right )}{3 c e}\right ) d \sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}\, \sqrt {\frac {\frac {b}{c}+x}{-\frac {d}{e}+\frac {b}{c}}}\, \sqrt {-\frac {e x}{d}}\, \left (\left (-\frac {d}{e}+\frac {b}{c}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}+\frac {b}{c}\right )}}\right )-\frac {b \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}+\frac {b}{c}\right )}}\right )}{c}\right )}{e \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}\right )}{\sqrt {e x +d}\, x \left (c x +b \right )}\)	\(942\)
default	\(\text {Expression too large to display}\)	\(1170\)

Input:

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

Output:

1/(e*x+d)^(1/2)*(x*(c*x+b))^(1/2)*((c*x+b)*x*(e*x+d))^(1/2)/x/(c*x+b)*(2/9 
*c*x^3*(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(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+b*d*x)^(1/2)+2/5*(e*b^2+11/9*d 
*b*c-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+b*d*x)^(1/2)+2/3*(b^2*d-5/7*(2*b*c*e+c^2*d-2/9*c*(4*b 
*e+4*c*d))/c/e*b*d-2/5*(e*b^2+11/9*d*b*c-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+ 
b*d*x)^(1/2)-2/3*(b^2*d-5/7*(2*b*c*e+c^2*d-2/9*c*(4*b*e+4*c*d))/c/e*b*d-2/ 
5*(e*b^2+11/9*d*b*c-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^2*b*d^2*((x+d/e)/d*e)^(1/2)*((b/c+x)/(-d/e+b/c 
))^(1/2)*(-e*x/d)^(1/2)/(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)*EllipticF((( 
x+d/e)/d*e)^(1/2),(-d/e/(-d/e+b/c))^(1/2))+2*(-3/5*(e*b^2+11/9*d*b*c-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*b*d-2/3*(b^2*d-5 
/7*(2*b*c*e+c^2*d-2/9*c*(4*b*e+4*c*d))/c/e*b*d-2/5*(e*b^2+11/9*d*b*c-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*(b*e+c*d))*d/e*((x+d/e)/d*e)^(1/2)*((b/c+x)/(-d/e+b/c))^(1/2)*(-e*x/d)^ 
(1/2)/(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)*((-d/e+b/c)*EllipticE(((x+d/e) 
/d*e)^(1/2),(-d/e/(-d/e+b/c))^(1/2))-b/c*EllipticF(((x+d/e)/d*e)^(1/2),(-d 
/e/(-d/e+b/c))^(1/2))))
 

Fricas [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 546, normalized size of antiderivative = 0.99 \[ \int \sqrt {d+e x} \left (b x+c x^2\right )^{3/2} \, dx=\frac {2 \, {\left ({\left (16 \, c^{5} d^{5} - 40 \, b c^{4} d^{4} e + 22 \, b^{2} c^{3} d^{3} e^{2} + 7 \, b^{3} c^{2} d^{2} e^{3} + 11 \, b^{4} c d e^{4} - 8 \, b^{5} e^{5}\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right ) + 3 \, {\left (16 \, c^{5} d^{4} e - 32 \, b c^{4} d^{3} e^{2} + 9 \, b^{2} c^{3} d^{2} e^{3} + 7 \, b^{3} c^{2} d e^{4} - 8 \, b^{4} c e^{5}\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right )\right ) + 3 \, {\left (35 \, c^{5} e^{5} x^{3} + 8 \, c^{5} d^{3} e^{2} - 15 \, b c^{4} d^{2} e^{3} + 3 \, b^{2} c^{3} d e^{4} - 4 \, b^{3} c^{2} e^{5} + 5 \, {\left (c^{5} d e^{4} + 10 \, b c^{4} e^{5}\right )} x^{2} - {\left (6 \, c^{5} d^{2} e^{3} - 11 \, b c^{4} d e^{4} - 3 \, b^{2} c^{3} e^{5}\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {e x + d}\right )}}{945 \, c^{4} e^{5}} \] Input:

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

Output:

2/945*((16*c^5*d^5 - 40*b*c^4*d^4*e + 22*b^2*c^3*d^3*e^2 + 7*b^3*c^2*d^2*e 
^3 + 11*b^4*c*d*e^4 - 8*b^5*e^5)*sqrt(c*e)*weierstrassPInverse(4/3*(c^2*d^ 
2 - b*c*d*e + b^2*e^2)/(c^2*e^2), -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*b^2 
*c*d*e^2 + 2*b^3*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 + 9*b^2*c^3*d^2*e^3 + 7*b^3*c^2*d*e^4 - 8*b^4 
*c*e^5)*sqrt(c*e)*weierstrassZeta(4/3*(c^2*d^2 - b*c*d*e + b^2*e^2)/(c^2*e 
^2), -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + 2*b^3*e^3)/(c^3*e^ 
3), weierstrassPInverse(4/3*(c^2*d^2 - b*c*d*e + b^2*e^2)/(c^2*e^2), -4/27 
*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + 2*b^3*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*d*e^4 - 4*b^3*c^2*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*e^5)*x)*sqrt(c*x^2 + b*x 
)*sqrt(e*x + d))/(c^4*e^5)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \sqrt {d+e x} \left (b x+c x^2\right )^{3/2} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 18*sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*b**3*d*e**2 + 12*sqrt(x)*sqrt(d 
 + e*x)*sqrt(b + c*x)*b**3*e**3*x - 66*sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x) 
*b**2*c*d**2*e + 56*sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*b**2*c*d*e**2*x + 
200*sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*b**2*c*e**3*x**2 + 36*sqrt(x)*sqrt 
(d + e*x)*sqrt(b + c*x)*b*c**2*d**3 + 20*sqrt(x)*sqrt(d + e*x)*sqrt(b + c* 
x)*b*c**2*d**2*e*x + 220*sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*b*c**2*d*e**2 
*x**2 + 140*sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*b*c**2*e**3*x**3 - 24*sqrt 
(x)*sqrt(d + e*x)*sqrt(b + c*x)*c**3*d**3*x + 20*sqrt(x)*sqrt(d + e*x)*sqr 
t(b + c*x)*c**3*d**2*e*x**2 + 140*sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*c**3 
*d*e**2*x**3 - 24*int((sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*x)/(b**2*d*e + 
b**2*e**2*x + b*c*d**2 + 2*b*c*d*e*x + b*c*e**2*x**2 + c**2*d**2*x + c**2* 
d*e*x**2),x)*b**5*e**5 - 3*int((sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*x)/(b* 
*2*d*e + b**2*e**2*x + b*c*d**2 + 2*b*c*d*e*x + b*c*e**2*x**2 + c**2*d**2* 
x + c**2*d*e*x**2),x)*b**4*c*d*e**4 + 48*int((sqrt(x)*sqrt(d + e*x)*sqrt(b 
 + c*x)*x)/(b**2*d*e + b**2*e**2*x + b*c*d**2 + 2*b*c*d*e*x + b*c*e**2*x** 
2 + c**2*d**2*x + c**2*d*e*x**2),x)*b**3*c**2*d**2*e**3 - 69*int((sqrt(x)* 
sqrt(d + e*x)*sqrt(b + c*x)*x)/(b**2*d*e + b**2*e**2*x + b*c*d**2 + 2*b*c* 
d*e*x + b*c*e**2*x**2 + c**2*d**2*x + c**2*d*e*x**2),x)*b**2*c**3*d**3*e** 
2 - 48*int((sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x)*x)/(b**2*d*e + b**2*e**2*x 
 + b*c*d**2 + 2*b*c*d*e*x + b*c*e**2*x**2 + c**2*d**2*x + c**2*d*e*x**2...