Integrand size = 28, antiderivative size = 180 \[ \int (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2} \, dx=-\frac {2 \left (b^2-4 a c\right ) d \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{21 c}+\frac {(b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}}{7 c d}-\frac {\left (b^2-4 a c\right )^{9/4} d^{3/2} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{21 c^2 \sqrt {a+b x+c x^2}} \] Output:
-2/21*(-4*a*c+b^2)*d*(2*c*d*x+b*d)^(1/2)*(c*x^2+b*x+a)^(1/2)/c+1/7*(2*c*d* x+b*d)^(5/2)*(c*x^2+b*x+a)^(1/2)/c/d-1/21*(-4*a*c+b^2)^(9/4)*d^(3/2)*(-c*( c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)*EllipticF((2*c*d*x+b*d)^(1/2)/(-4*a*c+b^2 )^(1/4)/d^(1/2),I)/c^2/(c*x^2+b*x+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.13 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.61 \[ \int (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2} \, dx=\frac {1}{14} d \sqrt {d (b+2 c x)} \sqrt {a+x (b+c x)} \left (8 (a+x (b+c x))+\frac {\left (b^2-4 a c\right ) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {5}{4},\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{c \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}}}\right ) \] Input:
Integrate[(b*d + 2*c*d*x)^(3/2)*Sqrt[a + b*x + c*x^2],x]
Output:
(d*Sqrt[d*(b + 2*c*x)]*Sqrt[a + x*(b + c*x)]*(8*(a + x*(b + c*x)) + ((b^2 - 4*a*c)*Hypergeometric2F1[-1/2, 1/4, 5/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/( c*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])))/14
Time = 0.38 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {1109, 1116, 1115, 1113, 762}
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.
| \(\displaystyle \int \sqrt {a+b x+c x^2} (b d+2 c d x)^{3/2} \, dx\) |
| \(\Big \downarrow \) 1109 |
| \(\displaystyle \frac {\sqrt {a+b x+c x^2} (b d+2 c d x)^{5/2}}{7 c d}-\frac {\left (b^2-4 a c\right ) \int \frac {(b d+2 c x d)^{3/2}}{\sqrt {c x^2+b x+a}}dx}{14 c}\) |
| \(\Big \downarrow \) 1116 |
| \(\displaystyle \frac {\sqrt {a+b x+c x^2} (b d+2 c d x)^{5/2}}{7 c d}-\frac {\left (b^2-4 a c\right ) \left (\frac {1}{3} d^2 \left (b^2-4 a c\right ) \int \frac {1}{\sqrt {b d+2 c x d} \sqrt {c x^2+b x+a}}dx+\frac {4}{3} d \sqrt {a+b x+c x^2} \sqrt {b d+2 c d x}\right )}{14 c}\) |
| \(\Big \downarrow \) 1115 |
| \(\displaystyle \frac {\sqrt {a+b x+c x^2} (b d+2 c d x)^{5/2}}{7 c d}-\frac {\left (b^2-4 a c\right ) \left (\frac {d^2 \left (b^2-4 a c\right ) \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \int \frac {1}{\sqrt {b d+2 c x d} \sqrt {-\frac {c^2 x^2}{b^2-4 a c}-\frac {b c x}{b^2-4 a c}-\frac {a c}{b^2-4 a c}}}dx}{3 \sqrt {a+b x+c x^2}}+\frac {4}{3} d \sqrt {a+b x+c x^2} \sqrt {b d+2 c d x}\right )}{14 c}\) |
| \(\Big \downarrow \) 1113 |
| \(\displaystyle \frac {\sqrt {a+b x+c x^2} (b d+2 c d x)^{5/2}}{7 c d}-\frac {\left (b^2-4 a c\right ) \left (\frac {2 d \left (b^2-4 a c\right ) \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \int \frac {1}{\sqrt {1-\frac {(b d+2 c x d)^2}{\left (b^2-4 a c\right ) d^2}}}d\sqrt {b d+2 c x d}}{3 c \sqrt {a+b x+c x^2}}+\frac {4}{3} d \sqrt {a+b x+c x^2} \sqrt {b d+2 c d x}\right )}{14 c}\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle \frac {\sqrt {a+b x+c x^2} (b d+2 c d x)^{5/2}}{7 c d}-\frac {\left (b^2-4 a c\right ) \left (\frac {2 d^{3/2} \left (b^2-4 a c\right )^{5/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{3 c \sqrt {a+b x+c x^2}}+\frac {4}{3} d \sqrt {a+b x+c x^2} \sqrt {b d+2 c d x}\right )}{14 c}\) |
Input:
Int[(b*d + 2*c*d*x)^(3/2)*Sqrt[a + b*x + c*x^2],x]
Output:
((b*d + 2*c*d*x)^(5/2)*Sqrt[a + b*x + c*x^2])/(7*c*d) - ((b^2 - 4*a*c)*((4 *d*Sqrt[b*d + 2*c*d*x]*Sqrt[a + b*x + c*x^2])/3 + (2*(b^2 - 4*a*c)^(5/4)*d ^(3/2)*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[ b*d + 2*c*d*x]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])], -1])/(3*c*Sqrt[a + b*x + c* x^2])))/(14*c)
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) )*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]
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[d*p*((b^2 - 4*a*c)/(b*e*(m + 2*p + 1))) Int[(d + e*x)^m*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[2*c*d - b* e, 0] && NeQ[m + 2*p + 3, 0] && GtQ[p, 0] && !LtQ[m, -1] && !(IGtQ[(m - 1 )/2, 0] && ( !IntegerQ[p] || LtQ[m, 2*p])) && RationalQ[m] && IntegerQ[2*p]
Int[1/(Sqrt[(d_) + (e_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_ Symbol] :> Simp[(4/e)*Sqrt[-c/(b^2 - 4*a*c)] Subst[Int[1/Sqrt[Simp[1 - b^ 2*(x^4/(d^2*(b^2 - 4*a*c))), x]], x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0] && LtQ[c/(b^2 - 4*a*c), 0]
Int[((d_) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Sym bol] :> Simp[Sqrt[(-c)*((a + b*x + c*x^2)/(b^2 - 4*a*c))]/Sqrt[a + b*x + c* x^2] Int[(d + e*x)^m/Sqrt[(-a)*(c/(b^2 - 4*a*c)) - b*c*(x/(b^2 - 4*a*c)) - c^2*(x^2/(b^2 - 4*a*c))], x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c* d - b*e, 0] && EqQ[m^2, 1/4]
Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_S ymbol] :> Simp[2*d*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] + Simp[d^2*(m - 1)*((b^2 - 4*a*c)/(b^2*(m + 2*p + 1))) Int[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && (IntegerQ[2*p] || (IntegerQ[m] && RationalQ[p]) || OddQ[m])
Leaf count of result is larger than twice the leaf count of optimal. \(489\) vs. \(2(152)=304\).
Time = 1.36 (sec) , antiderivative size = 490, normalized size of antiderivative = 2.72
| method | result | size |
| risch | \(\frac {\left (12 c^{2} x^{2}+12 c b x +8 a c +b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, \left (2 c x +b \right ) d^{2}}{21 c \sqrt {d \left (2 c x +b \right )}}-\frac {\left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right ) \left (\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \sqrt {\frac {-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\right ) d^{2} \sqrt {d \left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )}}{21 c \sqrt {2 c^{2} d \,x^{3}+3 b c d \,x^{2}+2 a d x c +b^{2} d x +d a b}\, \sqrt {d \left (2 c x +b \right )}\, \sqrt {c \,x^{2}+b x +a}}\) | \(490\) |
| default | \(-\frac {\sqrt {d \left (2 c x +b \right )}\, \sqrt {c \,x^{2}+b x +a}\, d \left (-48 c^{5} x^{5}+16 \sqrt {-4 a c +b^{2}}\, \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-2 c x +\sqrt {-4 a c +b^{2}}-b}{\sqrt {-4 a c +b^{2}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}}}{2}, \sqrt {2}\right ) a^{2} c^{2}-8 \sqrt {-4 a c +b^{2}}\, \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-2 c x +\sqrt {-4 a c +b^{2}}-b}{\sqrt {-4 a c +b^{2}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}}}{2}, \sqrt {2}\right ) a \,b^{2} c +\sqrt {-4 a c +b^{2}}\, \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-2 c x +\sqrt {-4 a c +b^{2}}-b}{\sqrt {-4 a c +b^{2}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}}}{2}, \sqrt {2}\right ) b^{4}-120 b \,x^{4} c^{4}-80 a \,c^{4} x^{3}-100 b^{2} c^{3} x^{3}-120 a b \,c^{3} x^{2}-30 c^{2} x^{2} b^{3}-32 a^{2} c^{3} x -44 a \,b^{2} c^{2} x -2 b^{4} c x -16 a^{2} b \,c^{2}-2 a \,b^{3} c \right )}{42 c^{2} \left (2 x^{3} c^{2}+3 b c \,x^{2}+2 a c x +b^{2} x +a b \right )}\) | \(564\) |
| elliptic | \(\text {Expression too large to display}\) | \(1268\) |
Input:
int((2*c*d*x+b*d)^(3/2)*(c*x^2+b*x+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/21/c*(12*c^2*x^2+12*b*c*x+8*a*c+b^2)*(c*x^2+b*x+a)^(1/2)*(2*c*x+b)*d^2/( d*(2*c*x+b))^(1/2)-1/21/c*(16*a^2*c^2-8*a*b^2*c+b^4)*(1/2/c*(-b+(-4*a*c+b^ 2)^(1/2))+1/2*(b+(-4*a*c+b^2)^(1/2))/c)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/ (1/2/c*(-b+(-4*a*c+b^2)^(1/2))+1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)*((x+1/ 2*b/c)/(-1/2*(b+(-4*a*c+b^2)^(1/2))/c+1/2*b/c))^(1/2)*((x-1/2/c*(-b+(-4*a* c+b^2)^(1/2)))/(-1/2*(b+(-4*a*c+b^2)^(1/2))/c-1/2/c*(-b+(-4*a*c+b^2)^(1/2) )))^(1/2)/(2*c^2*d*x^3+3*b*c*d*x^2+2*a*c*d*x+b^2*d*x+a*b*d)^(1/2)*Elliptic F(((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(1/2/c*(-b+(-4*a*c+b^2)^(1/2))+1/2*(b+ (-4*a*c+b^2)^(1/2))/c))^(1/2),((-1/2*(b+(-4*a*c+b^2)^(1/2))/c-1/2/c*(-b+(- 4*a*c+b^2)^(1/2)))/(-1/2*(b+(-4*a*c+b^2)^(1/2))/c+1/2*b/c))^(1/2))*d^2*(d* (2*c*x+b)*(c*x^2+b*x+a))^(1/2)/(d*(2*c*x+b))^(1/2)/(c*x^2+b*x+a)^(1/2)
Time = 0.08 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.67 \[ \int (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2} \, dx=-\frac {\sqrt {2} {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {c^{2} d} d {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right ) - 2 \, {\left (12 \, c^{4} d x^{2} + 12 \, b c^{3} d x + {\left (b^{2} c^{2} + 8 \, a c^{3}\right )} d\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}}{42 \, c^{3}} \] Input:
integrate((2*c*d*x+b*d)^(3/2)*(c*x^2+b*x+a)^(1/2),x, algorithm="fricas")
Output:
-1/42*(sqrt(2)*(b^4 - 8*a*b^2*c + 16*a^2*c^2)*sqrt(c^2*d)*d*weierstrassPIn verse((b^2 - 4*a*c)/c^2, 0, 1/2*(2*c*x + b)/c) - 2*(12*c^4*d*x^2 + 12*b*c^ 3*d*x + (b^2*c^2 + 8*a*c^3)*d)*sqrt(2*c*d*x + b*d)*sqrt(c*x^2 + b*x + a))/ c^3
\[ \int (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2} \, dx=\int \left (d \left (b + 2 c x\right )\right )^{\frac {3}{2}} \sqrt {a + b x + c x^{2}}\, dx \] Input:
integrate((2*c*d*x+b*d)**(3/2)*(c*x**2+b*x+a)**(1/2),x)
Output:
Integral((d*(b + 2*c*x))**(3/2)*sqrt(a + b*x + c*x**2), x)
\[ \int (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2} \, dx=\int { {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} \sqrt {c x^{2} + b x + a} \,d x } \] Input:
integrate((2*c*d*x+b*d)^(3/2)*(c*x^2+b*x+a)^(1/2),x, algorithm="maxima")
Output:
integrate((2*c*d*x + b*d)^(3/2)*sqrt(c*x^2 + b*x + a), x)
\[ \int (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2} \, dx=\int { {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} \sqrt {c x^{2} + b x + a} \,d x } \] Input:
integrate((2*c*d*x+b*d)^(3/2)*(c*x^2+b*x+a)^(1/2),x, algorithm="giac")
Output:
integrate((2*c*d*x + b*d)^(3/2)*sqrt(c*x^2 + b*x + a), x)
Timed out. \[ \int (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2} \, dx=\int {\left (b\,d+2\,c\,d\,x\right )}^{3/2}\,\sqrt {c\,x^2+b\,x+a} \,d x \] Input:
int((b*d + 2*c*d*x)^(3/2)*(a + b*x + c*x^2)^(1/2),x)
Output:
int((b*d + 2*c*d*x)^(3/2)*(a + b*x + c*x^2)^(1/2), x)
\[ \int (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2} \, dx=\frac {\sqrt {d}\, d \left (6 \sqrt {2 c x +b}\, \sqrt {c \,x^{2}+b x +a}\, a \,b^{2}+8 \sqrt {2 c x +b}\, \sqrt {c \,x^{2}+b x +a}\, a b c x +8 \sqrt {2 c x +b}\, \sqrt {c \,x^{2}+b x +a}\, a \,c^{2} x^{2}+4 \sqrt {2 c x +b}\, \sqrt {c \,x^{2}+b x +a}\, b^{3} x +4 \sqrt {2 c x +b}\, \sqrt {c \,x^{2}+b x +a}\, b^{2} c \,x^{2}+32 \left (\int \frac {\sqrt {2 c x +b}\, \sqrt {c \,x^{2}+b x +a}\, x^{2}}{4 a \,c^{3} x^{3}+2 b^{2} c^{2} x^{3}+6 a b \,c^{2} x^{2}+3 b^{3} c \,x^{2}+4 a^{2} c^{2} x +4 a \,b^{2} c x +b^{4} x +2 a^{2} b c +a \,b^{3}}d x \right ) a^{3} c^{4}-6 \left (\int \frac {\sqrt {2 c x +b}\, \sqrt {c \,x^{2}+b x +a}\, x^{2}}{4 a \,c^{3} x^{3}+2 b^{2} c^{2} x^{3}+6 a b \,c^{2} x^{2}+3 b^{3} c \,x^{2}+4 a^{2} c^{2} x +4 a \,b^{2} c x +b^{4} x +2 a^{2} b c +a \,b^{3}}d x \right ) a \,b^{4} c^{2}+\left (\int \frac {\sqrt {2 c x +b}\, \sqrt {c \,x^{2}+b x +a}\, x^{2}}{4 a \,c^{3} x^{3}+2 b^{2} c^{2} x^{3}+6 a b \,c^{2} x^{2}+3 b^{3} c \,x^{2}+4 a^{2} c^{2} x +4 a \,b^{2} c x +b^{4} x +2 a^{2} b c +a \,b^{3}}d x \right ) b^{6} c +32 \left (\int \frac {\sqrt {2 c x +b}\, \sqrt {c \,x^{2}+b x +a}\, x}{4 a \,c^{3} x^{3}+2 b^{2} c^{2} x^{3}+6 a b \,c^{2} x^{2}+3 b^{3} c \,x^{2}+4 a^{2} c^{2} x +4 a \,b^{2} c x +b^{4} x +2 a^{2} b c +a \,b^{3}}d x \right ) a^{3} b \,c^{3}-6 \left (\int \frac {\sqrt {2 c x +b}\, \sqrt {c \,x^{2}+b x +a}\, x}{4 a \,c^{3} x^{3}+2 b^{2} c^{2} x^{3}+6 a b \,c^{2} x^{2}+3 b^{3} c \,x^{2}+4 a^{2} c^{2} x +4 a \,b^{2} c x +b^{4} x +2 a^{2} b c +a \,b^{3}}d x \right ) a \,b^{5} c +\left (\int \frac {\sqrt {2 c x +b}\, \sqrt {c \,x^{2}+b x +a}\, x}{4 a \,c^{3} x^{3}+2 b^{2} c^{2} x^{3}+6 a b \,c^{2} x^{2}+3 b^{3} c \,x^{2}+4 a^{2} c^{2} x +4 a \,b^{2} c x +b^{4} x +2 a^{2} b c +a \,b^{3}}d x \right ) b^{7}\right )}{14 a c +7 b^{2}} \] Input:
int((2*c*d*x+b*d)^(3/2)*(c*x^2+b*x+a)^(1/2),x)
Output:
(sqrt(d)*d*(6*sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*a*b**2 + 8*sqrt(b + 2 *c*x)*sqrt(a + b*x + c*x**2)*a*b*c*x + 8*sqrt(b + 2*c*x)*sqrt(a + b*x + c* x**2)*a*c**2*x**2 + 4*sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*b**3*x + 4*sq rt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*b**2*c*x**2 + 32*int((sqrt(b + 2*c*x) *sqrt(a + b*x + c*x**2)*x**2)/(2*a**2*b*c + 4*a**2*c**2*x + a*b**3 + 4*a*b **2*c*x + 6*a*b*c**2*x**2 + 4*a*c**3*x**3 + b**4*x + 3*b**3*c*x**2 + 2*b** 2*c**2*x**3),x)*a**3*c**4 - 6*int((sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)* x**2)/(2*a**2*b*c + 4*a**2*c**2*x + a*b**3 + 4*a*b**2*c*x + 6*a*b*c**2*x** 2 + 4*a*c**3*x**3 + b**4*x + 3*b**3*c*x**2 + 2*b**2*c**2*x**3),x)*a*b**4*c **2 + int((sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*x**2)/(2*a**2*b*c + 4*a* *2*c**2*x + a*b**3 + 4*a*b**2*c*x + 6*a*b*c**2*x**2 + 4*a*c**3*x**3 + b**4 *x + 3*b**3*c*x**2 + 2*b**2*c**2*x**3),x)*b**6*c + 32*int((sqrt(b + 2*c*x) *sqrt(a + b*x + c*x**2)*x)/(2*a**2*b*c + 4*a**2*c**2*x + a*b**3 + 4*a*b**2 *c*x + 6*a*b*c**2*x**2 + 4*a*c**3*x**3 + b**4*x + 3*b**3*c*x**2 + 2*b**2*c **2*x**3),x)*a**3*b*c**3 - 6*int((sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*x )/(2*a**2*b*c + 4*a**2*c**2*x + a*b**3 + 4*a*b**2*c*x + 6*a*b*c**2*x**2 + 4*a*c**3*x**3 + b**4*x + 3*b**3*c*x**2 + 2*b**2*c**2*x**3),x)*a*b**5*c + i nt((sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*x)/(2*a**2*b*c + 4*a**2*c**2*x + a*b**3 + 4*a*b**2*c*x + 6*a*b*c**2*x**2 + 4*a*c**3*x**3 + b**4*x + 3*b** 3*c*x**2 + 2*b**2*c**2*x**3),x)*b**7))/(7*(2*a*c + b**2))