\(\int \frac {(a+b x+c x^2)^3}{\sqrt {b d+2 c d x}} \, dx\) [1278]

Optimal result

Integrand size = 26, antiderivative size = 121 \[ \int \frac {\left (a+b x+c x^2\right )^3}{\sqrt {b d+2 c d x}} \, dx=-\frac {\left (b^2-4 a c\right )^3 \sqrt {b d+2 c d x}}{64 c^4 d}+\frac {3 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{5/2}}{320 c^4 d^3}-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{9/2}}{192 c^4 d^5}+\frac {(b d+2 c d x)^{13/2}}{832 c^4 d^7} \]

[Out]

3/320*(-4*a*c+b^2)^2*(2*c*d*x+b*d)^(5/2)/c^4/d^3-1/192*(-4*a*c+b^2)*(2*c*d*x+b*d)^(9/2)/c^4/d^5+1/832*(2*c*d*x
+b*d)^(13/2)/c^4/d^7-1/64*(-4*a*c+b^2)^3*(2*c*d*x+b*d)^(1/2)/c^4/d

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {697} \[ \int \frac {\left (a+b x+c x^2\right )^3}{\sqrt {b d+2 c d x}} \, dx=-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{9/2}}{192 c^4 d^5}+\frac {3 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{5/2}}{320 c^4 d^3}-\frac {\left (b^2-4 a c\right )^3 \sqrt {b d+2 c d x}}{64 c^4 d}+\frac {(b d+2 c d x)^{13/2}}{832 c^4 d^7} \]

[In]

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

[Out]

-1/64*((b^2 - 4*a*c)^3*Sqrt[b*d + 2*c*d*x])/(c^4*d) + (3*(b^2 - 4*a*c)^2*(b*d + 2*c*d*x)^(5/2))/(320*c^4*d^3)
- ((b^2 - 4*a*c)*(b*d + 2*c*d*x)^(9/2))/(192*c^4*d^5) + (b*d + 2*c*d*x)^(13/2)/(832*c^4*d^7)

Rule 697

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e,
0] && IGtQ[p, 0] &&  !(EqQ[m, 3] && NeQ[p, 1])

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (-b^2+4 a c\right )^3}{64 c^3 \sqrt {b d+2 c d x}}+\frac {3 \left (-b^2+4 a c\right )^2 (b d+2 c d x)^{3/2}}{64 c^3 d^2}+\frac {3 \left (-b^2+4 a c\right ) (b d+2 c d x)^{7/2}}{64 c^3 d^4}+\frac {(b d+2 c d x)^{11/2}}{64 c^3 d^6}\right ) \, dx \\ & = -\frac {\left (b^2-4 a c\right )^3 \sqrt {b d+2 c d x}}{64 c^4 d}+\frac {3 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{5/2}}{320 c^4 d^3}-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{9/2}}{192 c^4 d^5}+\frac {(b d+2 c d x)^{13/2}}{832 c^4 d^7} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.10 \[ \int \frac {\left (a+b x+c x^2\right )^3}{\sqrt {b d+2 c d x}} \, dx=\frac {\sqrt {d (b+2 c x)} \left (-195 b^6+2340 a b^4 c-9360 a^2 b^2 c^2+12480 a^3 c^3+117 b^4 (b+2 c x)^2-936 a b^2 c (b+2 c x)^2+1872 a^2 c^2 (b+2 c x)^2-65 b^2 (b+2 c x)^4+260 a c (b+2 c x)^4+15 (b+2 c x)^6\right )}{12480 c^4 d} \]

[In]

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

[Out]

(Sqrt[d*(b + 2*c*x)]*(-195*b^6 + 2340*a*b^4*c - 9360*a^2*b^2*c^2 + 12480*a^3*c^3 + 117*b^4*(b + 2*c*x)^2 - 936
*a*b^2*c*(b + 2*c*x)^2 + 1872*a^2*c^2*(b + 2*c*x)^2 - 65*b^2*(b + 2*c*x)^4 + 260*a*c*(b + 2*c*x)^4 + 15*(b + 2
*c*x)^6))/(12480*c^4*d)

Maple [A] (verified)

Time = 2.26 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.21

method	result	size
derivativedivides	\(\frac {\frac {\left (2 c d x +b d \right )^{\frac {13}{2}}}{13}+\frac {\left (12 a c \,d^{2}-3 b^{2} d^{2}\right ) \left (2 c d x +b d \right )^{\frac {9}{2}}}{9}+\frac {\left (\left (4 a c \,d^{2}-b^{2} d^{2}\right ) \left (8 a c \,d^{2}-2 b^{2} d^{2}\right )+\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{2}\right ) \left (2 c d x +b d \right )^{\frac {5}{2}}}{5}+\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{3} \sqrt {2 c d x +b d}}{64 d^{7} c^{4}}\)	\(147\)
default	\(\frac {\frac {\left (2 c d x +b d \right )^{\frac {13}{2}}}{13}+\frac {\left (12 a c \,d^{2}-3 b^{2} d^{2}\right ) \left (2 c d x +b d \right )^{\frac {9}{2}}}{9}+\frac {\left (\left (4 a c \,d^{2}-b^{2} d^{2}\right ) \left (8 a c \,d^{2}-2 b^{2} d^{2}\right )+\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{2}\right ) \left (2 c d x +b d \right )^{\frac {5}{2}}}{5}+\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{3} \sqrt {2 c d x +b d}}{64 d^{7} c^{4}}\)	\(147\)
pseudoelliptic	\(\frac {\left (15 c^{6} x^{6}+45 b \,c^{5} x^{5}+65 a \,c^{5} x^{4}+40 b^{2} c^{4} x^{4}+130 a b \,c^{4} x^{3}+5 x^{3} b^{3} c^{3}+117 a^{2} c^{4} x^{2}+39 a \,b^{2} c^{3} x^{2}-3 x^{2} b^{4} c^{2}+117 a^{2} b \,c^{3} x -26 x a \,b^{3} c^{2}+2 x \,b^{5} c +195 c^{3} a^{3}-117 a^{2} b^{2} c^{2}+26 a \,b^{4} c -2 b^{6}\right ) \sqrt {d \left (2 c x +b \right )}}{195 d \,c^{4}}\)	\(170\)
trager	\(\frac {\left (15 c^{6} x^{6}+45 b \,c^{5} x^{5}+65 a \,c^{5} x^{4}+40 b^{2} c^{4} x^{4}+130 a b \,c^{4} x^{3}+5 x^{3} b^{3} c^{3}+117 a^{2} c^{4} x^{2}+39 a \,b^{2} c^{3} x^{2}-3 x^{2} b^{4} c^{2}+117 a^{2} b \,c^{3} x -26 x a \,b^{3} c^{2}+2 x \,b^{5} c +195 c^{3} a^{3}-117 a^{2} b^{2} c^{2}+26 a \,b^{4} c -2 b^{6}\right ) \sqrt {2 c d x +b d}}{195 d \,c^{4}}\)	\(171\)
gosper	\(\frac {\left (2 c x +b \right ) \left (15 c^{6} x^{6}+45 b \,c^{5} x^{5}+65 a \,c^{5} x^{4}+40 b^{2} c^{4} x^{4}+130 a b \,c^{4} x^{3}+5 x^{3} b^{3} c^{3}+117 a^{2} c^{4} x^{2}+39 a \,b^{2} c^{3} x^{2}-3 x^{2} b^{4} c^{2}+117 a^{2} b \,c^{3} x -26 x a \,b^{3} c^{2}+2 x \,b^{5} c +195 c^{3} a^{3}-117 a^{2} b^{2} c^{2}+26 a \,b^{4} c -2 b^{6}\right )}{195 c^{4} \sqrt {2 c d x +b d}}\)	\(174\)

[In]

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

[Out]

1/64/d^7/c^4*(1/13*(2*c*d*x+b*d)^(13/2)+1/9*(12*a*c*d^2-3*b^2*d^2)*(2*c*d*x+b*d)^(9/2)+1/5*((4*a*c*d^2-b^2*d^2
)*(8*a*c*d^2-2*b^2*d^2)+(4*a*c*d^2-b^2*d^2)^2)*(2*c*d*x+b*d)^(5/2)+(4*a*c*d^2-b^2*d^2)^3*(2*c*d*x+b*d)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.36 \[ \int \frac {\left (a+b x+c x^2\right )^3}{\sqrt {b d+2 c d x}} \, dx=\frac {{\left (15 \, c^{6} x^{6} + 45 \, b c^{5} x^{5} - 2 \, b^{6} + 26 \, a b^{4} c - 117 \, a^{2} b^{2} c^{2} + 195 \, a^{3} c^{3} + 5 \, {\left (8 \, b^{2} c^{4} + 13 \, a c^{5}\right )} x^{4} + 5 \, {\left (b^{3} c^{3} + 26 \, a b c^{4}\right )} x^{3} - 3 \, {\left (b^{4} c^{2} - 13 \, a b^{2} c^{3} - 39 \, a^{2} c^{4}\right )} x^{2} + {\left (2 \, b^{5} c - 26 \, a b^{3} c^{2} + 117 \, a^{2} b c^{3}\right )} x\right )} \sqrt {2 \, c d x + b d}}{195 \, c^{4} d} \]

[In]

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

[Out]

1/195*(15*c^6*x^6 + 45*b*c^5*x^5 - 2*b^6 + 26*a*b^4*c - 117*a^2*b^2*c^2 + 195*a^3*c^3 + 5*(8*b^2*c^4 + 13*a*c^
5)*x^4 + 5*(b^3*c^3 + 26*a*b*c^4)*x^3 - 3*(b^4*c^2 - 13*a*b^2*c^3 - 39*a^2*c^4)*x^2 + (2*b^5*c - 26*a*b^3*c^2
+ 117*a^2*b*c^3)*x)*sqrt(2*c*d*x + b*d)/(c^4*d)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 248 vs. \(2 (116) = 232\).

Time = 1.07 (sec) , antiderivative size = 248, normalized size of antiderivative = 2.05 \[ \int \frac {\left (a+b x+c x^2\right )^3}{\sqrt {b d+2 c d x}} \, dx=\begin {cases} \frac {\frac {\sqrt {b d + 2 c d x} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )}{64 c^{3}} + \frac {\left (b d + 2 c d x\right )^{\frac {5}{2}} \cdot \left (48 a^{2} c^{2} - 24 a b^{2} c + 3 b^{4}\right )}{320 c^{3} d^{2}} + \frac {\left (12 a c - 3 b^{2}\right ) \left (b d + 2 c d x\right )^{\frac {9}{2}}}{576 c^{3} d^{4}} + \frac {\left (b d + 2 c d x\right )^{\frac {13}{2}}}{832 c^{3} d^{6}}}{c d} & \text {for}\: c d \neq 0 \\\frac {a^{3} x + \frac {3 a^{2} b x^{2}}{2} + \frac {b c^{2} x^{6}}{2} + \frac {c^{3} x^{7}}{7} + \frac {x^{5} \cdot \left (3 a c^{2} + 3 b^{2} c\right )}{5} + \frac {x^{4} \cdot \left (6 a b c + b^{3}\right )}{4} + \frac {x^{3} \cdot \left (3 a^{2} c + 3 a b^{2}\right )}{3}}{\sqrt {b d}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise(((sqrt(b*d + 2*c*d*x)*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)/(64*c**3) + (b*d + 2*c
*d*x)**(5/2)*(48*a**2*c**2 - 24*a*b**2*c + 3*b**4)/(320*c**3*d**2) + (12*a*c - 3*b**2)*(b*d + 2*c*d*x)**(9/2)/
(576*c**3*d**4) + (b*d + 2*c*d*x)**(13/2)/(832*c**3*d**6))/(c*d), Ne(c*d, 0)), ((a**3*x + 3*a**2*b*x**2/2 + b*
c**2*x**6/2 + c**3*x**7/7 + x**5*(3*a*c**2 + 3*b**2*c)/5 + x**4*(6*a*b*c + b**3)/4 + x**3*(3*a**2*c + 3*a*b**2
)/3)/sqrt(b*d), True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 778 vs. \(2 (105) = 210\).

Time = 0.20 (sec) , antiderivative size = 778, normalized size of antiderivative = 6.43 \[ \int \frac {\left (a+b x+c x^2\right )^3}{\sqrt {b d+2 c d x}} \, dx=\frac {960960 \, \sqrt {2 \, c d x + b d} a^{3} - 48048 \, a^{2} {\left (\frac {10 \, {\left (3 \, \sqrt {2 \, c d x + b d} b d - {\left (2 \, c d x + b d\right )}^{\frac {3}{2}}\right )} b}{c d} - \frac {15 \, \sqrt {2 \, c d x + b d} b^{2} d^{2} - 10 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b d + 3 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}}}{c d^{2}}\right )} + 572 \, a {\left (\frac {84 \, {\left (15 \, \sqrt {2 \, c d x + b d} b^{2} d^{2} - 10 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b d + 3 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}}\right )} b^{2}}{c^{2} d^{2}} - \frac {36 \, {\left (35 \, \sqrt {2 \, c d x + b d} b^{3} d^{3} - 35 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b^{2} d^{2} + 21 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} b d - 5 \, {\left (2 \, c d x + b d\right )}^{\frac {7}{2}}\right )} b}{c^{2} d^{3}} + \frac {315 \, \sqrt {2 \, c d x + b d} b^{4} d^{4} - 420 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b^{3} d^{3} + 378 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} b^{2} d^{2} - 180 \, {\left (2 \, c d x + b d\right )}^{\frac {7}{2}} b d + 35 \, {\left (2 \, c d x + b d\right )}^{\frac {9}{2}}}{c^{2} d^{4}}\right )} - \frac {3432 \, {\left (35 \, \sqrt {2 \, c d x + b d} b^{3} d^{3} - 35 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b^{2} d^{2} + 21 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} b d - 5 \, {\left (2 \, c d x + b d\right )}^{\frac {7}{2}}\right )} b^{3}}{c^{3} d^{3}} + \frac {572 \, {\left (315 \, \sqrt {2 \, c d x + b d} b^{4} d^{4} - 420 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b^{3} d^{3} + 378 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} b^{2} d^{2} - 180 \, {\left (2 \, c d x + b d\right )}^{\frac {7}{2}} b d + 35 \, {\left (2 \, c d x + b d\right )}^{\frac {9}{2}}\right )} b^{2}}{c^{3} d^{4}} - \frac {130 \, {\left (693 \, \sqrt {2 \, c d x + b d} b^{5} d^{5} - 1155 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b^{4} d^{4} + 1386 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} b^{3} d^{3} - 990 \, {\left (2 \, c d x + b d\right )}^{\frac {7}{2}} b^{2} d^{2} + 385 \, {\left (2 \, c d x + b d\right )}^{\frac {9}{2}} b d - 63 \, {\left (2 \, c d x + b d\right )}^{\frac {11}{2}}\right )} b}{c^{3} d^{5}} + \frac {5 \, {\left (3003 \, \sqrt {2 \, c d x + b d} b^{6} d^{6} - 6006 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b^{5} d^{5} + 9009 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} b^{4} d^{4} - 8580 \, {\left (2 \, c d x + b d\right )}^{\frac {7}{2}} b^{3} d^{3} + 5005 \, {\left (2 \, c d x + b d\right )}^{\frac {9}{2}} b^{2} d^{2} - 1638 \, {\left (2 \, c d x + b d\right )}^{\frac {11}{2}} b d + 231 \, {\left (2 \, c d x + b d\right )}^{\frac {13}{2}}\right )}}{c^{3} d^{6}}}{960960 \, c d} \]

[In]

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

[Out]

1/960960*(960960*sqrt(2*c*d*x + b*d)*a^3 - 48048*a^2*(10*(3*sqrt(2*c*d*x + b*d)*b*d - (2*c*d*x + b*d)^(3/2))*b
/(c*d) - (15*sqrt(2*c*d*x + b*d)*b^2*d^2 - 10*(2*c*d*x + b*d)^(3/2)*b*d + 3*(2*c*d*x + b*d)^(5/2))/(c*d^2)) +
572*a*(84*(15*sqrt(2*c*d*x + b*d)*b^2*d^2 - 10*(2*c*d*x + b*d)^(3/2)*b*d + 3*(2*c*d*x + b*d)^(5/2))*b^2/(c^2*d
^2) - 36*(35*sqrt(2*c*d*x + b*d)*b^3*d^3 - 35*(2*c*d*x + b*d)^(3/2)*b^2*d^2 + 21*(2*c*d*x + b*d)^(5/2)*b*d - 5
*(2*c*d*x + b*d)^(7/2))*b/(c^2*d^3) + (315*sqrt(2*c*d*x + b*d)*b^4*d^4 - 420*(2*c*d*x + b*d)^(3/2)*b^3*d^3 + 3
78*(2*c*d*x + b*d)^(5/2)*b^2*d^2 - 180*(2*c*d*x + b*d)^(7/2)*b*d + 35*(2*c*d*x + b*d)^(9/2))/(c^2*d^4)) - 3432
*(35*sqrt(2*c*d*x + b*d)*b^3*d^3 - 35*(2*c*d*x + b*d)^(3/2)*b^2*d^2 + 21*(2*c*d*x + b*d)^(5/2)*b*d - 5*(2*c*d*
x + b*d)^(7/2))*b^3/(c^3*d^3) + 572*(315*sqrt(2*c*d*x + b*d)*b^4*d^4 - 420*(2*c*d*x + b*d)^(3/2)*b^3*d^3 + 378
*(2*c*d*x + b*d)^(5/2)*b^2*d^2 - 180*(2*c*d*x + b*d)^(7/2)*b*d + 35*(2*c*d*x + b*d)^(9/2))*b^2/(c^3*d^4) - 130
*(693*sqrt(2*c*d*x + b*d)*b^5*d^5 - 1155*(2*c*d*x + b*d)^(3/2)*b^4*d^4 + 1386*(2*c*d*x + b*d)^(5/2)*b^3*d^3 -
990*(2*c*d*x + b*d)^(7/2)*b^2*d^2 + 385*(2*c*d*x + b*d)^(9/2)*b*d - 63*(2*c*d*x + b*d)^(11/2))*b/(c^3*d^5) + 5
*(3003*sqrt(2*c*d*x + b*d)*b^6*d^6 - 6006*(2*c*d*x + b*d)^(3/2)*b^5*d^5 + 9009*(2*c*d*x + b*d)^(5/2)*b^4*d^4 -
8580*(2*c*d*x + b*d)^(7/2)*b^3*d^3 + 5005*(2*c*d*x + b*d)^(9/2)*b^2*d^2 - 1638*(2*c*d*x + b*d)^(11/2)*b*d + 2
31*(2*c*d*x + b*d)^(13/2))/(c^3*d^6))/(c*d)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 778 vs. \(2 (105) = 210\).

Time = 0.27 (sec) , antiderivative size = 778, normalized size of antiderivative = 6.43 \[ \int \frac {\left (a+b x+c x^2\right )^3}{\sqrt {b d+2 c d x}} \, dx=\frac {960960 \, \sqrt {2 \, c d x + b d} a^{3} - \frac {480480 \, {\left (3 \, \sqrt {2 \, c d x + b d} b d - {\left (2 \, c d x + b d\right )}^{\frac {3}{2}}\right )} a^{2} b}{c d} + \frac {48048 \, {\left (15 \, \sqrt {2 \, c d x + b d} b^{2} d^{2} - 10 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b d + 3 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}}\right )} a b^{2}}{c^{2} d^{2}} + \frac {48048 \, {\left (15 \, \sqrt {2 \, c d x + b d} b^{2} d^{2} - 10 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b d + 3 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}}\right )} a^{2}}{c d^{2}} - \frac {3432 \, {\left (35 \, \sqrt {2 \, c d x + b d} b^{3} d^{3} - 35 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b^{2} d^{2} + 21 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} b d - 5 \, {\left (2 \, c d x + b d\right )}^{\frac {7}{2}}\right )} b^{3}}{c^{3} d^{3}} - \frac {20592 \, {\left (35 \, \sqrt {2 \, c d x + b d} b^{3} d^{3} - 35 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b^{2} d^{2} + 21 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} b d - 5 \, {\left (2 \, c d x + b d\right )}^{\frac {7}{2}}\right )} a b}{c^{2} d^{3}} + \frac {572 \, {\left (315 \, \sqrt {2 \, c d x + b d} b^{4} d^{4} - 420 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b^{3} d^{3} + 378 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} b^{2} d^{2} - 180 \, {\left (2 \, c d x + b d\right )}^{\frac {7}{2}} b d + 35 \, {\left (2 \, c d x + b d\right )}^{\frac {9}{2}}\right )} b^{2}}{c^{3} d^{4}} + \frac {572 \, {\left (315 \, \sqrt {2 \, c d x + b d} b^{4} d^{4} - 420 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b^{3} d^{3} + 378 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} b^{2} d^{2} - 180 \, {\left (2 \, c d x + b d\right )}^{\frac {7}{2}} b d + 35 \, {\left (2 \, c d x + b d\right )}^{\frac {9}{2}}\right )} a}{c^{2} d^{4}} - \frac {130 \, {\left (693 \, \sqrt {2 \, c d x + b d} b^{5} d^{5} - 1155 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b^{4} d^{4} + 1386 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} b^{3} d^{3} - 990 \, {\left (2 \, c d x + b d\right )}^{\frac {7}{2}} b^{2} d^{2} + 385 \, {\left (2 \, c d x + b d\right )}^{\frac {9}{2}} b d - 63 \, {\left (2 \, c d x + b d\right )}^{\frac {11}{2}}\right )} b}{c^{3} d^{5}} + \frac {5 \, {\left (3003 \, \sqrt {2 \, c d x + b d} b^{6} d^{6} - 6006 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b^{5} d^{5} + 9009 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} b^{4} d^{4} - 8580 \, {\left (2 \, c d x + b d\right )}^{\frac {7}{2}} b^{3} d^{3} + 5005 \, {\left (2 \, c d x + b d\right )}^{\frac {9}{2}} b^{2} d^{2} - 1638 \, {\left (2 \, c d x + b d\right )}^{\frac {11}{2}} b d + 231 \, {\left (2 \, c d x + b d\right )}^{\frac {13}{2}}\right )}}{c^{3} d^{6}}}{960960 \, c d} \]

[In]

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

[Out]

1/960960*(960960*sqrt(2*c*d*x + b*d)*a^3 - 480480*(3*sqrt(2*c*d*x + b*d)*b*d - (2*c*d*x + b*d)^(3/2))*a^2*b/(c
*d) + 48048*(15*sqrt(2*c*d*x + b*d)*b^2*d^2 - 10*(2*c*d*x + b*d)^(3/2)*b*d + 3*(2*c*d*x + b*d)^(5/2))*a*b^2/(c
^2*d^2) + 48048*(15*sqrt(2*c*d*x + b*d)*b^2*d^2 - 10*(2*c*d*x + b*d)^(3/2)*b*d + 3*(2*c*d*x + b*d)^(5/2))*a^2/
(c*d^2) - 3432*(35*sqrt(2*c*d*x + b*d)*b^3*d^3 - 35*(2*c*d*x + b*d)^(3/2)*b^2*d^2 + 21*(2*c*d*x + b*d)^(5/2)*b
*d - 5*(2*c*d*x + b*d)^(7/2))*b^3/(c^3*d^3) - 20592*(35*sqrt(2*c*d*x + b*d)*b^3*d^3 - 35*(2*c*d*x + b*d)^(3/2)
*b^2*d^2 + 21*(2*c*d*x + b*d)^(5/2)*b*d - 5*(2*c*d*x + b*d)^(7/2))*a*b/(c^2*d^3) + 572*(315*sqrt(2*c*d*x + b*d
)*b^4*d^4 - 420*(2*c*d*x + b*d)^(3/2)*b^3*d^3 + 378*(2*c*d*x + b*d)^(5/2)*b^2*d^2 - 180*(2*c*d*x + b*d)^(7/2)*
b*d + 35*(2*c*d*x + b*d)^(9/2))*b^2/(c^3*d^4) + 572*(315*sqrt(2*c*d*x + b*d)*b^4*d^4 - 420*(2*c*d*x + b*d)^(3/
2)*b^3*d^3 + 378*(2*c*d*x + b*d)^(5/2)*b^2*d^2 - 180*(2*c*d*x + b*d)^(7/2)*b*d + 35*(2*c*d*x + b*d)^(9/2))*a/(
c^2*d^4) - 130*(693*sqrt(2*c*d*x + b*d)*b^5*d^5 - 1155*(2*c*d*x + b*d)^(3/2)*b^4*d^4 + 1386*(2*c*d*x + b*d)^(5
/2)*b^3*d^3 - 990*(2*c*d*x + b*d)^(7/2)*b^2*d^2 + 385*(2*c*d*x + b*d)^(9/2)*b*d - 63*(2*c*d*x + b*d)^(11/2))*b
/(c^3*d^5) + 5*(3003*sqrt(2*c*d*x + b*d)*b^6*d^6 - 6006*(2*c*d*x + b*d)^(3/2)*b^5*d^5 + 9009*(2*c*d*x + b*d)^(
5/2)*b^4*d^4 - 8580*(2*c*d*x + b*d)^(7/2)*b^3*d^3 + 5005*(2*c*d*x + b*d)^(9/2)*b^2*d^2 - 1638*(2*c*d*x + b*d)^
(11/2)*b*d + 231*(2*c*d*x + b*d)^(13/2))/(c^3*d^6))/(c*d)

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a+b x+c x^2\right )^3}{\sqrt {b d+2 c d x}} \, dx=\frac {{\left (b\,d+2\,c\,d\,x\right )}^{13/2}}{832\,c^4\,d^7}+\frac {{\left (b\,d+2\,c\,d\,x\right )}^{9/2}\,\left (4\,a\,c-b^2\right )}{192\,c^4\,d^5}+\frac {\sqrt {b\,d+2\,c\,d\,x}\,{\left (4\,a\,c-b^2\right )}^3}{64\,c^4\,d}+\frac {3\,{\left (b\,d+2\,c\,d\,x\right )}^{5/2}\,{\left (4\,a\,c-b^2\right )}^2}{320\,c^4\,d^3} \]

[In]

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

[Out]

(b*d + 2*c*d*x)^(13/2)/(832*c^4*d^7) + ((b*d + 2*c*d*x)^(9/2)*(4*a*c - b^2))/(192*c^4*d^5) + ((b*d + 2*c*d*x)^
(1/2)*(4*a*c - b^2)^3)/(64*c^4*d) + (3*(b*d + 2*c*d*x)^(5/2)*(4*a*c - b^2)^2)/(320*c^4*d^3)