\(\int \frac {(d+e x)^3 (a+c x^2)}{\sqrt {f+g x}} \, dx\) [589]

Optimal result

Integrand size = 24, antiderivative size = 240 \[ \int \frac {(d+e x)^3 \left (a+c x^2\right )}{\sqrt {f+g x}} \, dx=-\frac {2 (e f-d g)^3 \left (c f^2+a g^2\right ) \sqrt {f+g x}}{g^6}+\frac {2 (e f-d g)^2 \left (3 a e g^2+c f (5 e f-2 d g)\right ) (f+g x)^{3/2}}{3 g^6}-\frac {2 (e f-d g) \left (3 a e^2 g^2+c \left (10 e^2 f^2-8 d e f g+d^2 g^2\right )\right ) (f+g x)^{5/2}}{5 g^6}+\frac {2 e \left (a e^2 g^2+c \left (10 e^2 f^2-12 d e f g+3 d^2 g^2\right )\right ) (f+g x)^{7/2}}{7 g^6}-\frac {2 c e^2 (5 e f-3 d g) (f+g x)^{9/2}}{9 g^6}+\frac {2 c e^3 (f+g x)^{11/2}}{11 g^6} \]

[Out]

2/3*(-d*g+e*f)^2*(3*a*e*g^2+c*f*(-2*d*g+5*e*f))*(g*x+f)^(3/2)/g^6-2/5*(-d*g+e*f)*(3*a*e^2*g^2+c*(d^2*g^2-8*d*e
*f*g+10*e^2*f^2))*(g*x+f)^(5/2)/g^6+2/7*e*(a*e^2*g^2+c*(3*d^2*g^2-12*d*e*f*g+10*e^2*f^2))*(g*x+f)^(7/2)/g^6-2/
9*c*e^2*(-3*d*g+5*e*f)*(g*x+f)^(9/2)/g^6+2/11*c*e^3*(g*x+f)^(11/2)/g^6-2*(-d*g+e*f)^3*(a*g^2+c*f^2)*(g*x+f)^(1
/2)/g^6

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {912, 1167} \[ \int \frac {(d+e x)^3 \left (a+c x^2\right )}{\sqrt {f+g x}} \, dx=\frac {2 e (f+g x)^{7/2} \left (a e^2 g^2+c \left (3 d^2 g^2-12 d e f g+10 e^2 f^2\right )\right )}{7 g^6}-\frac {2 (f+g x)^{5/2} (e f-d g) \left (3 a e^2 g^2+c \left (d^2 g^2-8 d e f g+10 e^2 f^2\right )\right )}{5 g^6}-\frac {2 \sqrt {f+g x} \left (a g^2+c f^2\right ) (e f-d g)^3}{g^6}+\frac {2 (f+g x)^{3/2} (e f-d g)^2 \left (3 a e g^2+c f (5 e f-2 d g)\right )}{3 g^6}-\frac {2 c e^2 (f+g x)^{9/2} (5 e f-3 d g)}{9 g^6}+\frac {2 c e^3 (f+g x)^{11/2}}{11 g^6} \]

[In]

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

[Out]

(-2*(e*f - d*g)^3*(c*f^2 + a*g^2)*Sqrt[f + g*x])/g^6 + (2*(e*f - d*g)^2*(3*a*e*g^2 + c*f*(5*e*f - 2*d*g))*(f +
g*x)^(3/2))/(3*g^6) - (2*(e*f - d*g)*(3*a*e^2*g^2 + c*(10*e^2*f^2 - 8*d*e*f*g + d^2*g^2))*(f + g*x)^(5/2))/(5
*g^6) + (2*e*(a*e^2*g^2 + c*(10*e^2*f^2 - 12*d*e*f*g + 3*d^2*g^2))*(f + g*x)^(7/2))/(7*g^6) - (2*c*e^2*(5*e*f
- 3*d*g)*(f + g*x)^(9/2))/(9*g^6) + (2*c*e^3*(f + g*x)^(11/2))/(11*g^6)

Rule 912

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> With[{q = De
nominator[m]}, Dist[q/e, Subst[Int[x^(q*(m + 1) - 1)*((e*f - d*g)/e + g*(x^q/e))^n*((c*d^2 + a*e^2)/e^2 - 2*c*
d*(x^q/e^2) + c*(x^(2*q)/e^2))^p, x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*
g, 0] && NeQ[c*d^2 + a*e^2, 0] && IntegersQ[n, p] && FractionQ[m]

Rule 1167

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

Rubi steps \begin{align*} \text {integral}& = \frac {2 \text {Subst}\left (\int \left (\frac {-e f+d g}{g}+\frac {e x^2}{g}\right )^3 \left (\frac {c f^2+a g^2}{g^2}-\frac {2 c f x^2}{g^2}+\frac {c x^4}{g^2}\right ) \, dx,x,\sqrt {f+g x}\right )}{g} \\ & = \frac {2 \text {Subst}\left (\int \left (\frac {(-e f+d g)^3 \left (c f^2+a g^2\right )}{g^5}+\frac {(e f-d g)^2 \left (3 a e g^2+c f (5 e f-2 d g)\right ) x^2}{g^5}+\frac {(e f-d g) \left (-3 a e^2 g^2-c \left (10 e^2 f^2-8 d e f g+d^2 g^2\right )\right ) x^4}{g^5}+\frac {e \left (a e^2 g^2+c \left (10 e^2 f^2-12 d e f g+3 d^2 g^2\right )\right ) x^6}{g^5}-\frac {c e^2 (5 e f-3 d g) x^8}{g^5}+\frac {c e^3 x^{10}}{g^5}\right ) \, dx,x,\sqrt {f+g x}\right )}{g} \\ & = -\frac {2 (e f-d g)^3 \left (c f^2+a g^2\right ) \sqrt {f+g x}}{g^6}+\frac {2 (e f-d g)^2 \left (3 a e g^2+c f (5 e f-2 d g)\right ) (f+g x)^{3/2}}{3 g^6}-\frac {2 (e f-d g) \left (3 a e^2 g^2+c \left (10 e^2 f^2-8 d e f g+d^2 g^2\right )\right ) (f+g x)^{5/2}}{5 g^6}+\frac {2 e \left (a e^2 g^2+c \left (10 e^2 f^2-12 d e f g+3 d^2 g^2\right )\right ) (f+g x)^{7/2}}{7 g^6}-\frac {2 c e^2 (5 e f-3 d g) (f+g x)^{9/2}}{9 g^6}+\frac {2 c e^3 (f+g x)^{11/2}}{11 g^6} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.18 \[ \int \frac {(d+e x)^3 \left (a+c x^2\right )}{\sqrt {f+g x}} \, dx=\frac {2 \sqrt {f+g x} \left (99 a g^2 \left (35 d^3 g^3+35 d^2 e g^2 (-2 f+g x)+7 d e^2 g \left (8 f^2-4 f g x+3 g^2 x^2\right )+e^3 \left (-16 f^3+8 f^2 g x-6 f g^2 x^2+5 g^3 x^3\right )\right )+c \left (231 d^3 g^3 \left (8 f^2-4 f g x+3 g^2 x^2\right )+297 d^2 e g^2 \left (-16 f^3+8 f^2 g x-6 f g^2 x^2+5 g^3 x^3\right )+33 d e^2 g \left (128 f^4-64 f^3 g x+48 f^2 g^2 x^2-40 f g^3 x^3+35 g^4 x^4\right )-5 e^3 \left (256 f^5-128 f^4 g x+96 f^3 g^2 x^2-80 f^2 g^3 x^3+70 f g^4 x^4-63 g^5 x^5\right )\right )\right )}{3465 g^6} \]

[In]

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

[Out]

(2*Sqrt[f + g*x]*(99*a*g^2*(35*d^3*g^3 + 35*d^2*e*g^2*(-2*f + g*x) + 7*d*e^2*g*(8*f^2 - 4*f*g*x + 3*g^2*x^2) +
e^3*(-16*f^3 + 8*f^2*g*x - 6*f*g^2*x^2 + 5*g^3*x^3)) + c*(231*d^3*g^3*(8*f^2 - 4*f*g*x + 3*g^2*x^2) + 297*d^2
*e*g^2*(-16*f^3 + 8*f^2*g*x - 6*f*g^2*x^2 + 5*g^3*x^3) + 33*d*e^2*g*(128*f^4 - 64*f^3*g*x + 48*f^2*g^2*x^2 - 4
0*f*g^3*x^3 + 35*g^4*x^4) - 5*e^3*(256*f^5 - 128*f^4*g*x + 96*f^3*g^2*x^2 - 80*f^2*g^3*x^3 + 70*f*g^4*x^4 - 63
*g^5*x^5))))/(3465*g^6)

Maple [A] (verified)

Time = 0.46 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.01

method	result	size
derivativedivides	\(\frac {\frac {2 c \,e^{3} \left (g x +f \right )^{\frac {11}{2}}}{11}+\frac {2 \left (3 \left (d g -e f \right ) e^{2} c -2 f c \,e^{3}\right ) \left (g x +f \right )^{\frac {9}{2}}}{9}+\frac {2 \left (3 \left (d g -e f \right )^{2} e c -6 \left (d g -e f \right ) e^{2} c f +e^{3} \left (a \,g^{2}+c \,f^{2}\right )\right ) \left (g x +f \right )^{\frac {7}{2}}}{7}+\frac {2 \left (\left (d g -e f \right )^{3} c -6 \left (d g -e f \right )^{2} e c f +3 \left (d g -e f \right ) e^{2} \left (a \,g^{2}+c \,f^{2}\right )\right ) \left (g x +f \right )^{\frac {5}{2}}}{5}+\frac {2 \left (-2 \left (d g -e f \right )^{3} c f +3 \left (d g -e f \right )^{2} e \left (a \,g^{2}+c \,f^{2}\right )\right ) \left (g x +f \right )^{\frac {3}{2}}}{3}+2 \left (d g -e f \right )^{3} \left (a \,g^{2}+c \,f^{2}\right ) \sqrt {g x +f}}{g^{6}}\)	\(243\)
default	\(\frac {\frac {2 c \,e^{3} \left (g x +f \right )^{\frac {11}{2}}}{11}+\frac {2 \left (3 \left (d g -e f \right ) e^{2} c -2 f c \,e^{3}\right ) \left (g x +f \right )^{\frac {9}{2}}}{9}+\frac {2 \left (3 \left (d g -e f \right )^{2} e c -6 \left (d g -e f \right ) e^{2} c f +e^{3} \left (a \,g^{2}+c \,f^{2}\right )\right ) \left (g x +f \right )^{\frac {7}{2}}}{7}+\frac {2 \left (\left (d g -e f \right )^{3} c -6 \left (d g -e f \right )^{2} e c f +3 \left (d g -e f \right ) e^{2} \left (a \,g^{2}+c \,f^{2}\right )\right ) \left (g x +f \right )^{\frac {5}{2}}}{5}+\frac {2 \left (-2 \left (d g -e f \right )^{3} c f +3 \left (d g -e f \right )^{2} e \left (a \,g^{2}+c \,f^{2}\right )\right ) \left (g x +f \right )^{\frac {3}{2}}}{3}+2 \left (d g -e f \right )^{3} \left (a \,g^{2}+c \,f^{2}\right ) \sqrt {g x +f}}{g^{6}}\)	\(243\)
pseudoelliptic	\(\frac {2 \left (\left (\frac {x^{3} \left (\frac {7 c \,x^{2}}{11}+a \right ) e^{3}}{7}+\frac {3 x^{2} \left (\frac {5 c \,x^{2}}{9}+a \right ) d \,e^{2}}{5}+d^{2} x \left (\frac {3 c \,x^{2}}{7}+a \right ) e +d^{3} \left (\frac {c \,x^{2}}{5}+a \right )\right ) g^{5}-2 \left (\left (\frac {5}{99} c \,x^{4}+\frac {3}{35} a \,x^{2}\right ) e^{3}+\frac {2 \left (\frac {10 c \,x^{2}}{21}+a \right ) x d \,e^{2}}{5}+d^{2} \left (\frac {9 c \,x^{2}}{35}+a \right ) e +\frac {2 c \,d^{3} x}{15}\right ) f \,g^{4}+\frac {8 f^{2} \left (\frac {\left (\frac {50 c \,x^{2}}{99}+a \right ) x \,e^{3}}{7}+d \left (\frac {2 c \,x^{2}}{7}+a \right ) e^{2}+\frac {3 c \,d^{2} e x}{7}+\frac {c \,d^{3}}{3}\right ) g^{3}}{5}-\frac {16 e \left (\left (\frac {10 c \,x^{2}}{33}+a \right ) e^{2}+\frac {4 c d e x}{3}+3 c \,d^{2}\right ) f^{3} g^{2}}{35}+\frac {128 e^{2} c \,f^{4} \left (\frac {5 e x}{33}+d \right ) g}{105}-\frac {256 c \,e^{3} f^{5}}{693}\right ) \sqrt {g x +f}}{g^{6}}\)	\(246\)
gosper	\(\frac {2 \sqrt {g x +f}\, \left (315 c \,e^{3} x^{5} g^{5}+1155 c d \,e^{2} g^{5} x^{4}-350 c \,e^{3} f \,g^{4} x^{4}+495 a \,e^{3} g^{5} x^{3}+1485 c \,d^{2} e \,g^{5} x^{3}-1320 c d \,e^{2} f \,g^{4} x^{3}+400 c \,e^{3} f^{2} g^{3} x^{3}+2079 a d \,e^{2} g^{5} x^{2}-594 a \,e^{3} f \,g^{4} x^{2}+693 c \,d^{3} g^{5} x^{2}-1782 c \,d^{2} e f \,g^{4} x^{2}+1584 c d \,e^{2} f^{2} g^{3} x^{2}-480 c \,e^{3} f^{3} g^{2} x^{2}+3465 a \,d^{2} e \,g^{5} x -2772 a d \,e^{2} f \,g^{4} x +792 a \,e^{3} f^{2} g^{3} x -924 c \,d^{3} f \,g^{4} x +2376 c \,d^{2} e \,f^{2} g^{3} x -2112 c d \,e^{2} f^{3} g^{2} x +640 c \,e^{3} f^{4} g x +3465 a \,d^{3} g^{5}-6930 a \,d^{2} e f \,g^{4}+5544 a d \,e^{2} f^{2} g^{3}-1584 a \,e^{3} f^{3} g^{2}+1848 c \,d^{3} f^{2} g^{3}-4752 c \,d^{2} e \,f^{3} g^{2}+4224 c d \,e^{2} f^{4} g -1280 c \,e^{3} f^{5}\right )}{3465 g^{6}}\)	\(365\)
trager	\(\frac {2 \sqrt {g x +f}\, \left (315 c \,e^{3} x^{5} g^{5}+1155 c d \,e^{2} g^{5} x^{4}-350 c \,e^{3} f \,g^{4} x^{4}+495 a \,e^{3} g^{5} x^{3}+1485 c \,d^{2} e \,g^{5} x^{3}-1320 c d \,e^{2} f \,g^{4} x^{3}+400 c \,e^{3} f^{2} g^{3} x^{3}+2079 a d \,e^{2} g^{5} x^{2}-594 a \,e^{3} f \,g^{4} x^{2}+693 c \,d^{3} g^{5} x^{2}-1782 c \,d^{2} e f \,g^{4} x^{2}+1584 c d \,e^{2} f^{2} g^{3} x^{2}-480 c \,e^{3} f^{3} g^{2} x^{2}+3465 a \,d^{2} e \,g^{5} x -2772 a d \,e^{2} f \,g^{4} x +792 a \,e^{3} f^{2} g^{3} x -924 c \,d^{3} f \,g^{4} x +2376 c \,d^{2} e \,f^{2} g^{3} x -2112 c d \,e^{2} f^{3} g^{2} x +640 c \,e^{3} f^{4} g x +3465 a \,d^{3} g^{5}-6930 a \,d^{2} e f \,g^{4}+5544 a d \,e^{2} f^{2} g^{3}-1584 a \,e^{3} f^{3} g^{2}+1848 c \,d^{3} f^{2} g^{3}-4752 c \,d^{2} e \,f^{3} g^{2}+4224 c d \,e^{2} f^{4} g -1280 c \,e^{3} f^{5}\right )}{3465 g^{6}}\)	\(365\)
risch	\(\frac {2 \sqrt {g x +f}\, \left (315 c \,e^{3} x^{5} g^{5}+1155 c d \,e^{2} g^{5} x^{4}-350 c \,e^{3} f \,g^{4} x^{4}+495 a \,e^{3} g^{5} x^{3}+1485 c \,d^{2} e \,g^{5} x^{3}-1320 c d \,e^{2} f \,g^{4} x^{3}+400 c \,e^{3} f^{2} g^{3} x^{3}+2079 a d \,e^{2} g^{5} x^{2}-594 a \,e^{3} f \,g^{4} x^{2}+693 c \,d^{3} g^{5} x^{2}-1782 c \,d^{2} e f \,g^{4} x^{2}+1584 c d \,e^{2} f^{2} g^{3} x^{2}-480 c \,e^{3} f^{3} g^{2} x^{2}+3465 a \,d^{2} e \,g^{5} x -2772 a d \,e^{2} f \,g^{4} x +792 a \,e^{3} f^{2} g^{3} x -924 c \,d^{3} f \,g^{4} x +2376 c \,d^{2} e \,f^{2} g^{3} x -2112 c d \,e^{2} f^{3} g^{2} x +640 c \,e^{3} f^{4} g x +3465 a \,d^{3} g^{5}-6930 a \,d^{2} e f \,g^{4}+5544 a d \,e^{2} f^{2} g^{3}-1584 a \,e^{3} f^{3} g^{2}+1848 c \,d^{3} f^{2} g^{3}-4752 c \,d^{2} e \,f^{3} g^{2}+4224 c d \,e^{2} f^{4} g -1280 c \,e^{3} f^{5}\right )}{3465 g^{6}}\)	\(365\)

[In]

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

[Out]

2/g^6*(1/11*c*e^3*(g*x+f)^(11/2)+1/9*(3*(d*g-e*f)*e^2*c-2*f*c*e^3)*(g*x+f)^(9/2)+1/7*(3*(d*g-e*f)^2*e*c-6*(d*g
-e*f)*e^2*c*f+e^3*(a*g^2+c*f^2))*(g*x+f)^(7/2)+1/5*((d*g-e*f)^3*c-6*(d*g-e*f)^2*e*c*f+3*(d*g-e*f)*e^2*(a*g^2+c
*f^2))*(g*x+f)^(5/2)+1/3*(-2*(d*g-e*f)^3*c*f+3*(d*g-e*f)^2*e*(a*g^2+c*f^2))*(g*x+f)^(3/2)+(d*g-e*f)^3*(a*g^2+c
*f^2)*(g*x+f)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.35 \[ \int \frac {(d+e x)^3 \left (a+c x^2\right )}{\sqrt {f+g x}} \, dx=\frac {2 \, {\left (315 \, c e^{3} g^{5} x^{5} - 1280 \, c e^{3} f^{5} + 4224 \, c d e^{2} f^{4} g - 6930 \, a d^{2} e f g^{4} + 3465 \, a d^{3} g^{5} - 1584 \, {\left (3 \, c d^{2} e + a e^{3}\right )} f^{3} g^{2} + 1848 \, {\left (c d^{3} + 3 \, a d e^{2}\right )} f^{2} g^{3} - 35 \, {\left (10 \, c e^{3} f g^{4} - 33 \, c d e^{2} g^{5}\right )} x^{4} + 5 \, {\left (80 \, c e^{3} f^{2} g^{3} - 264 \, c d e^{2} f g^{4} + 99 \, {\left (3 \, c d^{2} e + a e^{3}\right )} g^{5}\right )} x^{3} - 3 \, {\left (160 \, c e^{3} f^{3} g^{2} - 528 \, c d e^{2} f^{2} g^{3} + 198 \, {\left (3 \, c d^{2} e + a e^{3}\right )} f g^{4} - 231 \, {\left (c d^{3} + 3 \, a d e^{2}\right )} g^{5}\right )} x^{2} + {\left (640 \, c e^{3} f^{4} g - 2112 \, c d e^{2} f^{3} g^{2} + 3465 \, a d^{2} e g^{5} + 792 \, {\left (3 \, c d^{2} e + a e^{3}\right )} f^{2} g^{3} - 924 \, {\left (c d^{3} + 3 \, a d e^{2}\right )} f g^{4}\right )} x\right )} \sqrt {g x + f}}{3465 \, g^{6}} \]

[In]

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

[Out]

2/3465*(315*c*e^3*g^5*x^5 - 1280*c*e^3*f^5 + 4224*c*d*e^2*f^4*g - 6930*a*d^2*e*f*g^4 + 3465*a*d^3*g^5 - 1584*(
3*c*d^2*e + a*e^3)*f^3*g^2 + 1848*(c*d^3 + 3*a*d*e^2)*f^2*g^3 - 35*(10*c*e^3*f*g^4 - 33*c*d*e^2*g^5)*x^4 + 5*(
80*c*e^3*f^2*g^3 - 264*c*d*e^2*f*g^4 + 99*(3*c*d^2*e + a*e^3)*g^5)*x^3 - 3*(160*c*e^3*f^3*g^2 - 528*c*d*e^2*f^
2*g^3 + 198*(3*c*d^2*e + a*e^3)*f*g^4 - 231*(c*d^3 + 3*a*d*e^2)*g^5)*x^2 + (640*c*e^3*f^4*g - 2112*c*d*e^2*f^3
*g^2 + 3465*a*d^2*e*g^5 + 792*(3*c*d^2*e + a*e^3)*f^2*g^3 - 924*(c*d^3 + 3*a*d*e^2)*f*g^4)*x)*sqrt(g*x + f)/g^
6

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 502 vs. \(2 (241) = 482\).

Time = 1.00 (sec) , antiderivative size = 502, normalized size of antiderivative = 2.09 \[ \int \frac {(d+e x)^3 \left (a+c x^2\right )}{\sqrt {f+g x}} \, dx=\begin {cases} \frac {2 \left (\frac {c e^{3} \left (f + g x\right )^{\frac {11}{2}}}{11 g^{5}} + \frac {\left (f + g x\right )^{\frac {9}{2}} \cdot \left (3 c d e^{2} g - 5 c e^{3} f\right )}{9 g^{5}} + \frac {\left (f + g x\right )^{\frac {7}{2}} \left (a e^{3} g^{2} + 3 c d^{2} e g^{2} - 12 c d e^{2} f g + 10 c e^{3} f^{2}\right )}{7 g^{5}} + \frac {\left (f + g x\right )^{\frac {5}{2}} \cdot \left (3 a d e^{2} g^{3} - 3 a e^{3} f g^{2} + c d^{3} g^{3} - 9 c d^{2} e f g^{2} + 18 c d e^{2} f^{2} g - 10 c e^{3} f^{3}\right )}{5 g^{5}} + \frac {\left (f + g x\right )^{\frac {3}{2}} \cdot \left (3 a d^{2} e g^{4} - 6 a d e^{2} f g^{3} + 3 a e^{3} f^{2} g^{2} - 2 c d^{3} f g^{3} + 9 c d^{2} e f^{2} g^{2} - 12 c d e^{2} f^{3} g + 5 c e^{3} f^{4}\right )}{3 g^{5}} + \frac {\sqrt {f + g x} \left (a d^{3} g^{5} - 3 a d^{2} e f g^{4} + 3 a d e^{2} f^{2} g^{3} - a e^{3} f^{3} g^{2} + c d^{3} f^{2} g^{3} - 3 c d^{2} e f^{3} g^{2} + 3 c d e^{2} f^{4} g - c e^{3} f^{5}\right )}{g^{5}}\right )}{g} & \text {for}\: g \neq 0 \\\frac {a d^{3} x + \frac {3 a d^{2} e x^{2}}{2} + \frac {3 c d e^{2} x^{5}}{5} + \frac {c e^{3} x^{6}}{6} + \frac {x^{4} \left (a e^{3} + 3 c d^{2} e\right )}{4} + \frac {x^{3} \cdot \left (3 a d e^{2} + c d^{3}\right )}{3}}{\sqrt {f}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((2*(c*e**3*(f + g*x)**(11/2)/(11*g**5) + (f + g*x)**(9/2)*(3*c*d*e**2*g - 5*c*e**3*f)/(9*g**5) + (f
+ g*x)**(7/2)*(a*e**3*g**2 + 3*c*d**2*e*g**2 - 12*c*d*e**2*f*g + 10*c*e**3*f**2)/(7*g**5) + (f + g*x)**(5/2)*(
3*a*d*e**2*g**3 - 3*a*e**3*f*g**2 + c*d**3*g**3 - 9*c*d**2*e*f*g**2 + 18*c*d*e**2*f**2*g - 10*c*e**3*f**3)/(5*
g**5) + (f + g*x)**(3/2)*(3*a*d**2*e*g**4 - 6*a*d*e**2*f*g**3 + 3*a*e**3*f**2*g**2 - 2*c*d**3*f*g**3 + 9*c*d**
2*e*f**2*g**2 - 12*c*d*e**2*f**3*g + 5*c*e**3*f**4)/(3*g**5) + sqrt(f + g*x)*(a*d**3*g**5 - 3*a*d**2*e*f*g**4
+ 3*a*d*e**2*f**2*g**3 - a*e**3*f**3*g**2 + c*d**3*f**2*g**3 - 3*c*d**2*e*f**3*g**2 + 3*c*d*e**2*f**4*g - c*e*
*3*f**5)/g**5)/g, Ne(g, 0)), ((a*d**3*x + 3*a*d**2*e*x**2/2 + 3*c*d*e**2*x**5/5 + c*e**3*x**6/6 + x**4*(a*e**3
+ 3*c*d**2*e)/4 + x**3*(3*a*d*e**2 + c*d**3)/3)/sqrt(f), True))

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.36 \[ \int \frac {(d+e x)^3 \left (a+c x^2\right )}{\sqrt {f+g x}} \, dx=\frac {2 \, {\left (315 \, {\left (g x + f\right )}^{\frac {11}{2}} c e^{3} - 385 \, {\left (5 \, c e^{3} f - 3 \, c d e^{2} g\right )} {\left (g x + f\right )}^{\frac {9}{2}} + 495 \, {\left (10 \, c e^{3} f^{2} - 12 \, c d e^{2} f g + {\left (3 \, c d^{2} e + a e^{3}\right )} g^{2}\right )} {\left (g x + f\right )}^{\frac {7}{2}} - 693 \, {\left (10 \, c e^{3} f^{3} - 18 \, c d e^{2} f^{2} g + 3 \, {\left (3 \, c d^{2} e + a e^{3}\right )} f g^{2} - {\left (c d^{3} + 3 \, a d e^{2}\right )} g^{3}\right )} {\left (g x + f\right )}^{\frac {5}{2}} + 1155 \, {\left (5 \, c e^{3} f^{4} - 12 \, c d e^{2} f^{3} g + 3 \, a d^{2} e g^{4} + 3 \, {\left (3 \, c d^{2} e + a e^{3}\right )} f^{2} g^{2} - 2 \, {\left (c d^{3} + 3 \, a d e^{2}\right )} f g^{3}\right )} {\left (g x + f\right )}^{\frac {3}{2}} - 3465 \, {\left (c e^{3} f^{5} - 3 \, c d e^{2} f^{4} g + 3 \, a d^{2} e f g^{4} - a d^{3} g^{5} + {\left (3 \, c d^{2} e + a e^{3}\right )} f^{3} g^{2} - {\left (c d^{3} + 3 \, a d e^{2}\right )} f^{2} g^{3}\right )} \sqrt {g x + f}\right )}}{3465 \, g^{6}} \]

[In]

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

[Out]

2/3465*(315*(g*x + f)^(11/2)*c*e^3 - 385*(5*c*e^3*f - 3*c*d*e^2*g)*(g*x + f)^(9/2) + 495*(10*c*e^3*f^2 - 12*c*
d*e^2*f*g + (3*c*d^2*e + a*e^3)*g^2)*(g*x + f)^(7/2) - 693*(10*c*e^3*f^3 - 18*c*d*e^2*f^2*g + 3*(3*c*d^2*e + a
*e^3)*f*g^2 - (c*d^3 + 3*a*d*e^2)*g^3)*(g*x + f)^(5/2) + 1155*(5*c*e^3*f^4 - 12*c*d*e^2*f^3*g + 3*a*d^2*e*g^4
+ 3*(3*c*d^2*e + a*e^3)*f^2*g^2 - 2*(c*d^3 + 3*a*d*e^2)*f*g^3)*(g*x + f)^(3/2) - 3465*(c*e^3*f^5 - 3*c*d*e^2*f
^4*g + 3*a*d^2*e*f*g^4 - a*d^3*g^5 + (3*c*d^2*e + a*e^3)*f^3*g^2 - (c*d^3 + 3*a*d*e^2)*f^2*g^3)*sqrt(g*x + f))
/g^6

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 380, normalized size of antiderivative = 1.58 \[ \int \frac {(d+e x)^3 \left (a+c x^2\right )}{\sqrt {f+g x}} \, dx=\frac {2 \, {\left (3465 \, \sqrt {g x + f} a d^{3} + \frac {3465 \, {\left ({\left (g x + f\right )}^{\frac {3}{2}} - 3 \, \sqrt {g x + f} f\right )} a d^{2} e}{g} + \frac {231 \, {\left (3 \, {\left (g x + f\right )}^{\frac {5}{2}} - 10 \, {\left (g x + f\right )}^{\frac {3}{2}} f + 15 \, \sqrt {g x + f} f^{2}\right )} c d^{3}}{g^{2}} + \frac {693 \, {\left (3 \, {\left (g x + f\right )}^{\frac {5}{2}} - 10 \, {\left (g x + f\right )}^{\frac {3}{2}} f + 15 \, \sqrt {g x + f} f^{2}\right )} a d e^{2}}{g^{2}} + \frac {297 \, {\left (5 \, {\left (g x + f\right )}^{\frac {7}{2}} - 21 \, {\left (g x + f\right )}^{\frac {5}{2}} f + 35 \, {\left (g x + f\right )}^{\frac {3}{2}} f^{2} - 35 \, \sqrt {g x + f} f^{3}\right )} c d^{2} e}{g^{3}} + \frac {99 \, {\left (5 \, {\left (g x + f\right )}^{\frac {7}{2}} - 21 \, {\left (g x + f\right )}^{\frac {5}{2}} f + 35 \, {\left (g x + f\right )}^{\frac {3}{2}} f^{2} - 35 \, \sqrt {g x + f} f^{3}\right )} a e^{3}}{g^{3}} + \frac {33 \, {\left (35 \, {\left (g x + f\right )}^{\frac {9}{2}} - 180 \, {\left (g x + f\right )}^{\frac {7}{2}} f + 378 \, {\left (g x + f\right )}^{\frac {5}{2}} f^{2} - 420 \, {\left (g x + f\right )}^{\frac {3}{2}} f^{3} + 315 \, \sqrt {g x + f} f^{4}\right )} c d e^{2}}{g^{4}} + \frac {5 \, {\left (63 \, {\left (g x + f\right )}^{\frac {11}{2}} - 385 \, {\left (g x + f\right )}^{\frac {9}{2}} f + 990 \, {\left (g x + f\right )}^{\frac {7}{2}} f^{2} - 1386 \, {\left (g x + f\right )}^{\frac {5}{2}} f^{3} + 1155 \, {\left (g x + f\right )}^{\frac {3}{2}} f^{4} - 693 \, \sqrt {g x + f} f^{5}\right )} c e^{3}}{g^{5}}\right )}}{3465 \, g} \]

[In]

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

[Out]

2/3465*(3465*sqrt(g*x + f)*a*d^3 + 3465*((g*x + f)^(3/2) - 3*sqrt(g*x + f)*f)*a*d^2*e/g + 231*(3*(g*x + f)^(5/
2) - 10*(g*x + f)^(3/2)*f + 15*sqrt(g*x + f)*f^2)*c*d^3/g^2 + 693*(3*(g*x + f)^(5/2) - 10*(g*x + f)^(3/2)*f +
15*sqrt(g*x + f)*f^2)*a*d*e^2/g^2 + 297*(5*(g*x + f)^(7/2) - 21*(g*x + f)^(5/2)*f + 35*(g*x + f)^(3/2)*f^2 - 3
5*sqrt(g*x + f)*f^3)*c*d^2*e/g^3 + 99*(5*(g*x + f)^(7/2) - 21*(g*x + f)^(5/2)*f + 35*(g*x + f)^(3/2)*f^2 - 35*
sqrt(g*x + f)*f^3)*a*e^3/g^3 + 33*(35*(g*x + f)^(9/2) - 180*(g*x + f)^(7/2)*f + 378*(g*x + f)^(5/2)*f^2 - 420*
(g*x + f)^(3/2)*f^3 + 315*sqrt(g*x + f)*f^4)*c*d*e^2/g^4 + 5*(63*(g*x + f)^(11/2) - 385*(g*x + f)^(9/2)*f + 99
0*(g*x + f)^(7/2)*f^2 - 1386*(g*x + f)^(5/2)*f^3 + 1155*(g*x + f)^(3/2)*f^4 - 693*sqrt(g*x + f)*f^5)*c*e^3/g^5
)/g

Mupad [B] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.92 \[ \int \frac {(d+e x)^3 \left (a+c x^2\right )}{\sqrt {f+g x}} \, dx=\frac {{\left (f+g\,x\right )}^{7/2}\,\left (6\,c\,d^2\,e\,g^2-24\,c\,d\,e^2\,f\,g+20\,c\,e^3\,f^2+2\,a\,e^3\,g^2\right )}{7\,g^6}+\frac {2\,\sqrt {f+g\,x}\,\left (c\,f^2+a\,g^2\right )\,{\left (d\,g-e\,f\right )}^3}{g^6}+\frac {2\,c\,e^3\,{\left (f+g\,x\right )}^{11/2}}{11\,g^6}+\frac {2\,{\left (f+g\,x\right )}^{3/2}\,{\left (d\,g-e\,f\right )}^2\,\left (5\,c\,e\,f^2-2\,c\,d\,f\,g+3\,a\,e\,g^2\right )}{3\,g^6}+\frac {2\,{\left (f+g\,x\right )}^{5/2}\,\left (d\,g-e\,f\right )\,\left (c\,d^2\,g^2-8\,c\,d\,e\,f\,g+10\,c\,e^2\,f^2+3\,a\,e^2\,g^2\right )}{5\,g^6}+\frac {2\,c\,e^2\,{\left (f+g\,x\right )}^{9/2}\,\left (3\,d\,g-5\,e\,f\right )}{9\,g^6} \]

[In]

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

[Out]

((f + g*x)^(7/2)*(2*a*e^3*g^2 + 20*c*e^3*f^2 + 6*c*d^2*e*g^2 - 24*c*d*e^2*f*g))/(7*g^6) + (2*(f + g*x)^(1/2)*(
a*g^2 + c*f^2)*(d*g - e*f)^3)/g^6 + (2*c*e^3*(f + g*x)^(11/2))/(11*g^6) + (2*(f + g*x)^(3/2)*(d*g - e*f)^2*(3*
a*e*g^2 + 5*c*e*f^2 - 2*c*d*f*g))/(3*g^6) + (2*(f + g*x)^(5/2)*(d*g - e*f)*(3*a*e^2*g^2 + c*d^2*g^2 + 10*c*e^2
*f^2 - 8*c*d*e*f*g))/(5*g^6) + (2*c*e^2*(f + g*x)^(9/2)*(3*d*g - 5*e*f))/(9*g^6)