\(\int \frac {(d+e x)^3 (a+c x^2)}{(f+g x)^{3/2}} \, dx\) [596]

Optimal result

Integrand size = 24, antiderivative size = 238 \[ \int \frac {(d+e x)^3 \left (a+c x^2\right )}{(f+g x)^{3/2}} \, dx=\frac {2 (e f-d g)^3 \left (c f^2+a g^2\right )}{g^6 \sqrt {f+g x}}+\frac {2 (e f-d g)^2 \left (3 a e g^2+c f (5 e f-2 d g)\right ) \sqrt {f+g x}}{g^6}-\frac {2 (e f-d g) \left (3 a e^2 g^2+c \left (10 e^2 f^2-8 d e f g+d^2 g^2\right )\right ) (f+g x)^{3/2}}{3 g^6}+\frac {2 e \left (a e^2 g^2+c \left (10 e^2 f^2-12 d e f g+3 d^2 g^2\right )\right ) (f+g x)^{5/2}}{5 g^6}-\frac {2 c e^2 (5 e f-3 d g) (f+g x)^{7/2}}{7 g^6}+\frac {2 c e^3 (f+g x)^{9/2}}{9 g^6} \]

[Out]

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

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 238, 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, 1275} \[ \int \frac {(d+e x)^3 \left (a+c x^2\right )}{(f+g x)^{3/2}} \, dx=\frac {2 e (f+g x)^{5/2} \left (a e^2 g^2+c \left (3 d^2 g^2-12 d e f g+10 e^2 f^2\right )\right )}{5 g^6}-\frac {2 (f+g x)^{3/2} (e f-d g) \left (3 a e^2 g^2+c \left (d^2 g^2-8 d e f g+10 e^2 f^2\right )\right )}{3 g^6}+\frac {2 \left (a g^2+c f^2\right ) (e f-d g)^3}{g^6 \sqrt {f+g x}}+\frac {2 \sqrt {f+g x} (e f-d g)^2 \left (3 a e g^2+c f (5 e f-2 d g)\right )}{g^6}-\frac {2 c e^2 (f+g x)^{7/2} (5 e f-3 d g)}{7 g^6}+\frac {2 c e^3 (f+g x)^{9/2}}{9 g^6} \]

[In]

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

[Out]

(2*(e*f - d*g)^3*(c*f^2 + a*g^2))/(g^6*Sqrt[f + g*x]) + (2*(e*f - d*g)^2*(3*a*e*g^2 + c*f*(5*e*f - 2*d*g))*Sqr
t[f + g*x])/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)^(3/2))/(3*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)^(5/2))/(5*g^6) - (2*c*e^2*(5*e*f - 3*d
*g)*(f + g*x)^(7/2))/(7*g^6) + (2*c*e^3*(f + g*x)^(9/2))/(9*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 1275

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

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

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.17 \[ \int \frac {(d+e x)^3 \left (a+c x^2\right )}{(f+g x)^{3/2}} \, dx=\frac {2 \left (63 a g^2 \left (-5 d^3 g^3+15 d^2 e g^2 (2 f+g x)+5 d e^2 g \left (-8 f^2-4 f g x+g^2 x^2\right )+e^3 \left (16 f^3+8 f^2 g x-2 f g^2 x^2+g^3 x^3\right )\right )+c \left (105 d^3 g^3 \left (-8 f^2-4 f g x+g^2 x^2\right )+189 d^2 e g^2 \left (16 f^3+8 f^2 g x-2 f g^2 x^2+g^3 x^3\right )+27 d e^2 g \left (-128 f^4-64 f^3 g x+16 f^2 g^2 x^2-8 f g^3 x^3+5 g^4 x^4\right )+5 e^3 \left (256 f^5+128 f^4 g x-32 f^3 g^2 x^2+16 f^2 g^3 x^3-10 f g^4 x^4+7 g^5 x^5\right )\right )\right )}{315 g^6 \sqrt {f+g x}} \]

[In]

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

[Out]

(2*(63*a*g^2*(-5*d^3*g^3 + 15*d^2*e*g^2*(2*f + g*x) + 5*d*e^2*g*(-8*f^2 - 4*f*g*x + g^2*x^2) + e^3*(16*f^3 + 8
*f^2*g*x - 2*f*g^2*x^2 + g^3*x^3)) + c*(105*d^3*g^3*(-8*f^2 - 4*f*g*x + g^2*x^2) + 189*d^2*e*g^2*(16*f^3 + 8*f
^2*g*x - 2*f*g^2*x^2 + g^3*x^3) + 27*d*e^2*g*(-128*f^4 - 64*f^3*g*x + 16*f^2*g^2*x^2 - 8*f*g^3*x^3 + 5*g^4*x^4
) + 5*e^3*(256*f^5 + 128*f^4*g*x - 32*f^3*g^2*x^2 + 16*f^2*g^3*x^3 - 10*f*g^4*x^4 + 7*g^5*x^5))))/(315*g^6*Sqr
t[f + g*x])

Maple [A] (verified)

Time = 0.48 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.04

method	result	size
pseudoelliptic	\(-\frac {2 \left (\left (-\frac {x^{3} \left (\frac {5 c \,x^{2}}{9}+a \right ) e^{3}}{5}-x^{2} d \left (\frac {3 c \,x^{2}}{7}+a \right ) e^{2}-3 \left (\frac {c \,x^{2}}{5}+a \right ) x \,d^{2} e +d^{3} \left (-\frac {c \,x^{2}}{3}+a \right )\right ) g^{5}-6 f \left (-\frac {\left (\frac {25 c \,x^{2}}{63}+a \right ) x^{2} e^{3}}{15}-\frac {2 x \left (\frac {6 c \,x^{2}}{35}+a \right ) d \,e^{2}}{3}+d^{2} \left (-\frac {c \,x^{2}}{5}+a \right ) e -\frac {2 c \,d^{3} x}{9}\right ) g^{4}+8 f^{2} \left (\left (-\frac {2}{63} c \,x^{3}-\frac {1}{5} a x \right ) e^{3}+d \left (-\frac {6 c \,x^{2}}{35}+a \right ) e^{2}-\frac {3 c \,d^{2} e x}{5}+\frac {c \,d^{3}}{3}\right ) g^{3}-\frac {16 e \left (\left (-\frac {10 c \,x^{2}}{63}+a \right ) e^{2}-\frac {12 c d e x}{7}+3 c \,d^{2}\right ) f^{3} g^{2}}{5}+\frac {384 e^{2} c \left (-\frac {5 e x}{27}+d \right ) f^{4} g}{35}-\frac {256 c \,e^{3} f^{5}}{63}\right )}{\sqrt {g x +f}\, g^{6}}\)	\(247\)
risch	\(\frac {2 \left (35 c \,e^{3} x^{4} g^{4}+135 c d \,e^{2} g^{4} x^{3}-85 c \,e^{3} f \,g^{3} x^{3}+63 a \,e^{3} g^{4} x^{2}+189 c \,d^{2} e \,g^{4} x^{2}-351 c d \,e^{2} f \,g^{3} x^{2}+165 c \,e^{3} f^{2} g^{2} x^{2}+315 a d \,e^{2} g^{4} x -189 a \,e^{3} f \,g^{3} x +105 c \,d^{3} g^{4} x -567 c \,d^{2} e f \,g^{3} x +783 c d \,e^{2} f^{2} g^{2} x -325 c \,e^{3} f^{3} g x +945 a \,d^{2} e \,g^{4}-1575 a d \,e^{2} f \,g^{3}+693 a \,e^{3} f^{2} g^{2}-525 c \,d^{3} f \,g^{3}+2079 c \,d^{2} e \,f^{2} g^{2}-2511 c d \,e^{2} f^{3} g +965 c \,e^{3} f^{4}\right ) \sqrt {g x +f}}{315 g^{6}}-\frac {2 \left (a \,d^{3} g^{5}-3 a \,d^{2} e f \,g^{4}+3 a d \,e^{2} f^{2} g^{3}-a \,e^{3} f^{3} g^{2}+c \,d^{3} f^{2} g^{3}-3 c \,d^{2} e \,f^{3} g^{2}+3 c d \,e^{2} f^{4} g -c \,e^{3} f^{5}\right )}{g^{6} \sqrt {g x +f}}\)	\(353\)
gosper	\(-\frac {2 \left (-35 c \,e^{3} x^{5} g^{5}-135 c d \,e^{2} g^{5} x^{4}+50 c \,e^{3} f \,g^{4} x^{4}-63 a \,e^{3} g^{5} x^{3}-189 c \,d^{2} e \,g^{5} x^{3}+216 c d \,e^{2} f \,g^{4} x^{3}-80 c \,e^{3} f^{2} g^{3} x^{3}-315 a d \,e^{2} g^{5} x^{2}+126 a \,e^{3} f \,g^{4} x^{2}-105 c \,d^{3} g^{5} x^{2}+378 c \,d^{2} e f \,g^{4} x^{2}-432 c d \,e^{2} f^{2} g^{3} x^{2}+160 c \,e^{3} f^{3} g^{2} x^{2}-945 a \,d^{2} e \,g^{5} x +1260 a d \,e^{2} f \,g^{4} x -504 a \,e^{3} f^{2} g^{3} x +420 c \,d^{3} f \,g^{4} x -1512 c \,d^{2} e \,f^{2} g^{3} x +1728 c d \,e^{2} f^{3} g^{2} x -640 c \,e^{3} f^{4} g x +315 a \,d^{3} g^{5}-1890 a \,d^{2} e f \,g^{4}+2520 a d \,e^{2} f^{2} g^{3}-1008 a \,e^{3} f^{3} g^{2}+840 c \,d^{3} f^{2} g^{3}-3024 c \,d^{2} e \,f^{3} g^{2}+3456 c d \,e^{2} f^{4} g -1280 c \,e^{3} f^{5}\right )}{315 \sqrt {g x +f}\, g^{6}}\)	\(365\)
trager	\(-\frac {2 \left (-35 c \,e^{3} x^{5} g^{5}-135 c d \,e^{2} g^{5} x^{4}+50 c \,e^{3} f \,g^{4} x^{4}-63 a \,e^{3} g^{5} x^{3}-189 c \,d^{2} e \,g^{5} x^{3}+216 c d \,e^{2} f \,g^{4} x^{3}-80 c \,e^{3} f^{2} g^{3} x^{3}-315 a d \,e^{2} g^{5} x^{2}+126 a \,e^{3} f \,g^{4} x^{2}-105 c \,d^{3} g^{5} x^{2}+378 c \,d^{2} e f \,g^{4} x^{2}-432 c d \,e^{2} f^{2} g^{3} x^{2}+160 c \,e^{3} f^{3} g^{2} x^{2}-945 a \,d^{2} e \,g^{5} x +1260 a d \,e^{2} f \,g^{4} x -504 a \,e^{3} f^{2} g^{3} x +420 c \,d^{3} f \,g^{4} x -1512 c \,d^{2} e \,f^{2} g^{3} x +1728 c d \,e^{2} f^{3} g^{2} x -640 c \,e^{3} f^{4} g x +315 a \,d^{3} g^{5}-1890 a \,d^{2} e f \,g^{4}+2520 a d \,e^{2} f^{2} g^{3}-1008 a \,e^{3} f^{3} g^{2}+840 c \,d^{3} f^{2} g^{3}-3024 c \,d^{2} e \,f^{3} g^{2}+3456 c d \,e^{2} f^{4} g -1280 c \,e^{3} f^{5}\right )}{315 \sqrt {g x +f}\, g^{6}}\)	\(365\)
derivativedivides	\(\frac {\frac {2 c \,e^{3} \left (g x +f \right )^{\frac {9}{2}}}{9}+\frac {6 c d \,e^{2} g \left (g x +f \right )^{\frac {7}{2}}}{7}-\frac {10 c \,e^{3} f \left (g x +f \right )^{\frac {7}{2}}}{7}+\frac {2 a \,e^{3} g^{2} \left (g x +f \right )^{\frac {5}{2}}}{5}+\frac {6 c \,d^{2} e \,g^{2} \left (g x +f \right )^{\frac {5}{2}}}{5}-\frac {24 c d \,e^{2} f g \left (g x +f \right )^{\frac {5}{2}}}{5}+4 c \,e^{3} f^{2} \left (g x +f \right )^{\frac {5}{2}}+2 a d \,e^{2} g^{3} \left (g x +f \right )^{\frac {3}{2}}-2 a \,e^{3} f \,g^{2} \left (g x +f \right )^{\frac {3}{2}}+\frac {2 c \,d^{3} g^{3} \left (g x +f \right )^{\frac {3}{2}}}{3}-6 c \,d^{2} e f \,g^{2} \left (g x +f \right )^{\frac {3}{2}}+12 c d \,e^{2} f^{2} g \left (g x +f \right )^{\frac {3}{2}}-\frac {20 c \,e^{3} f^{3} \left (g x +f \right )^{\frac {3}{2}}}{3}+6 a \,d^{2} e \,g^{4} \sqrt {g x +f}-12 a d \,e^{2} f \,g^{3} \sqrt {g x +f}+6 a \,e^{3} f^{2} g^{2} \sqrt {g x +f}-4 c \,d^{3} f \,g^{3} \sqrt {g x +f}+18 c \,d^{2} e \,f^{2} g^{2} \sqrt {g x +f}-24 c d \,e^{2} f^{3} g \sqrt {g x +f}+10 c \,e^{3} f^{4} \sqrt {g x +f}-\frac {2 \left (a \,d^{3} g^{5}-3 a \,d^{2} e f \,g^{4}+3 a d \,e^{2} f^{2} g^{3}-a \,e^{3} f^{3} g^{2}+c \,d^{3} f^{2} g^{3}-3 c \,d^{2} e \,f^{3} g^{2}+3 c d \,e^{2} f^{4} g -c \,e^{3} f^{5}\right )}{\sqrt {g x +f}}}{g^{6}}\)	\(438\)
default	\(\frac {\frac {2 c \,e^{3} \left (g x +f \right )^{\frac {9}{2}}}{9}+\frac {6 c d \,e^{2} g \left (g x +f \right )^{\frac {7}{2}}}{7}-\frac {10 c \,e^{3} f \left (g x +f \right )^{\frac {7}{2}}}{7}+\frac {2 a \,e^{3} g^{2} \left (g x +f \right )^{\frac {5}{2}}}{5}+\frac {6 c \,d^{2} e \,g^{2} \left (g x +f \right )^{\frac {5}{2}}}{5}-\frac {24 c d \,e^{2} f g \left (g x +f \right )^{\frac {5}{2}}}{5}+4 c \,e^{3} f^{2} \left (g x +f \right )^{\frac {5}{2}}+2 a d \,e^{2} g^{3} \left (g x +f \right )^{\frac {3}{2}}-2 a \,e^{3} f \,g^{2} \left (g x +f \right )^{\frac {3}{2}}+\frac {2 c \,d^{3} g^{3} \left (g x +f \right )^{\frac {3}{2}}}{3}-6 c \,d^{2} e f \,g^{2} \left (g x +f \right )^{\frac {3}{2}}+12 c d \,e^{2} f^{2} g \left (g x +f \right )^{\frac {3}{2}}-\frac {20 c \,e^{3} f^{3} \left (g x +f \right )^{\frac {3}{2}}}{3}+6 a \,d^{2} e \,g^{4} \sqrt {g x +f}-12 a d \,e^{2} f \,g^{3} \sqrt {g x +f}+6 a \,e^{3} f^{2} g^{2} \sqrt {g x +f}-4 c \,d^{3} f \,g^{3} \sqrt {g x +f}+18 c \,d^{2} e \,f^{2} g^{2} \sqrt {g x +f}-24 c d \,e^{2} f^{3} g \sqrt {g x +f}+10 c \,e^{3} f^{4} \sqrt {g x +f}-\frac {2 \left (a \,d^{3} g^{5}-3 a \,d^{2} e f \,g^{4}+3 a d \,e^{2} f^{2} g^{3}-a \,e^{3} f^{3} g^{2}+c \,d^{3} f^{2} g^{3}-3 c \,d^{2} e \,f^{3} g^{2}+3 c d \,e^{2} f^{4} g -c \,e^{3} f^{5}\right )}{\sqrt {g x +f}}}{g^{6}}\)	\(438\)

[In]

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

[Out]

-2*((-1/5*x^3*(5/9*c*x^2+a)*e^3-x^2*d*(3/7*c*x^2+a)*e^2-3*(1/5*c*x^2+a)*x*d^2*e+d^3*(-1/3*c*x^2+a))*g^5-6*f*(-
1/15*(25/63*c*x^2+a)*x^2*e^3-2/3*x*(6/35*c*x^2+a)*d*e^2+d^2*(-1/5*c*x^2+a)*e-2/9*c*d^3*x)*g^4+8*f^2*((-2/63*c*
x^3-1/5*a*x)*e^3+d*(-6/35*c*x^2+a)*e^2-3/5*c*d^2*e*x+1/3*c*d^3)*g^3-16/5*e*((-10/63*c*x^2+a)*e^2-12/7*c*d*e*x+
3*c*d^2)*f^3*g^2+384/35*e^2*c*(-5/27*e*x+d)*f^4*g-256/63*c*e^3*f^5)/(g*x+f)^(1/2)/g^6

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.40 \[ \int \frac {(d+e x)^3 \left (a+c x^2\right )}{(f+g x)^{3/2}} \, dx=\frac {2 \, {\left (35 \, c e^{3} g^{5} x^{5} + 1280 \, c e^{3} f^{5} - 3456 \, c d e^{2} f^{4} g + 1890 \, a d^{2} e f g^{4} - 315 \, a d^{3} g^{5} + 1008 \, {\left (3 \, c d^{2} e + a e^{3}\right )} f^{3} g^{2} - 840 \, {\left (c d^{3} + 3 \, a d e^{2}\right )} f^{2} g^{3} - 5 \, {\left (10 \, c e^{3} f g^{4} - 27 \, c d e^{2} g^{5}\right )} x^{4} + {\left (80 \, c e^{3} f^{2} g^{3} - 216 \, c d e^{2} f g^{4} + 63 \, {\left (3 \, c d^{2} e + a e^{3}\right )} g^{5}\right )} x^{3} - {\left (160 \, c e^{3} f^{3} g^{2} - 432 \, c d e^{2} f^{2} g^{3} + 126 \, {\left (3 \, c d^{2} e + a e^{3}\right )} f g^{4} - 105 \, {\left (c d^{3} + 3 \, a d e^{2}\right )} g^{5}\right )} x^{2} + {\left (640 \, c e^{3} f^{4} g - 1728 \, c d e^{2} f^{3} g^{2} + 945 \, a d^{2} e g^{5} + 504 \, {\left (3 \, c d^{2} e + a e^{3}\right )} f^{2} g^{3} - 420 \, {\left (c d^{3} + 3 \, a d e^{2}\right )} f g^{4}\right )} x\right )} \sqrt {g x + f}}{315 \, {\left (g^{7} x + f g^{6}\right )}} \]

[In]

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

[Out]

2/315*(35*c*e^3*g^5*x^5 + 1280*c*e^3*f^5 - 3456*c*d*e^2*f^4*g + 1890*a*d^2*e*f*g^4 - 315*a*d^3*g^5 + 1008*(3*c
*d^2*e + a*e^3)*f^3*g^2 - 840*(c*d^3 + 3*a*d*e^2)*f^2*g^3 - 5*(10*c*e^3*f*g^4 - 27*c*d*e^2*g^5)*x^4 + (80*c*e^
3*f^2*g^3 - 216*c*d*e^2*f*g^4 + 63*(3*c*d^2*e + a*e^3)*g^5)*x^3 - (160*c*e^3*f^3*g^2 - 432*c*d*e^2*f^2*g^3 + 1
26*(3*c*d^2*e + a*e^3)*f*g^4 - 105*(c*d^3 + 3*a*d*e^2)*g^5)*x^2 + (640*c*e^3*f^4*g - 1728*c*d*e^2*f^3*g^2 + 94
5*a*d^2*e*g^5 + 504*(3*c*d^2*e + a*e^3)*f^2*g^3 - 420*(c*d^3 + 3*a*d*e^2)*f*g^4)*x)*sqrt(g*x + f)/(g^7*x + f*g
^6)

Sympy [A] (verification not implemented)

Time = 11.71 (sec) , antiderivative size = 420, normalized size of antiderivative = 1.76 \[ \int \frac {(d+e x)^3 \left (a+c x^2\right )}{(f+g x)^{3/2}} \, dx=\begin {cases} \frac {2 \left (\frac {c e^{3} \left (f + g x\right )^{\frac {9}{2}}}{9 g^{5}} + \frac {\left (f + g x\right )^{\frac {7}{2}} \cdot \left (3 c d e^{2} g - 5 c e^{3} f\right )}{7 g^{5}} + \frac {\left (f + g x\right )^{\frac {5}{2}} \left (a e^{3} g^{2} + 3 c d^{2} e g^{2} - 12 c d e^{2} f g + 10 c e^{3} f^{2}\right )}{5 g^{5}} + \frac {\left (f + g x\right )^{\frac {3}{2}} \cdot \left (3 a d e^{2} g^{3} - 3 a e^{3} f g^{2} + c d^{3} g^{3} - 9 c d^{2} e f g^{2} + 18 c d e^{2} f^{2} g - 10 c e^{3} f^{3}\right )}{3 g^{5}} + \frac {\sqrt {f + g x} \left (3 a d^{2} e g^{4} - 6 a d e^{2} f g^{3} + 3 a e^{3} f^{2} g^{2} - 2 c d^{3} f g^{3} + 9 c d^{2} e f^{2} g^{2} - 12 c d e^{2} f^{3} g + 5 c e^{3} f^{4}\right )}{g^{5}} - \frac {\left (a g^{2} + c f^{2}\right ) \left (d g - e f\right )^{3}}{g^{5} \sqrt {f + g x}}\right )}{g} & \text {for}\: g \neq 0 \\\frac {a d^{3} x + \frac {3 a d^{2} e x^{2}}{2} + \frac {3 c d e^{2} x^{5}}{5} + \frac {c e^{3} x^{6}}{6} + \frac {x^{4} \left (a e^{3} + 3 c d^{2} e\right )}{4} + \frac {x^{3} \cdot \left (3 a d e^{2} + c d^{3}\right )}{3}}{f^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((2*(c*e**3*(f + g*x)**(9/2)/(9*g**5) + (f + g*x)**(7/2)*(3*c*d*e**2*g - 5*c*e**3*f)/(7*g**5) + (f +
g*x)**(5/2)*(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)/(5*g**5) + (f + g*x)**(3/2)*(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)/(3*g*
*5) + sqrt(f + g*x)*(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)/g**5 - (a*g**2 + c*f**2)*(d*g - e*f)**3/(g**5*sqrt(f + g*x)))/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)/f**(3/2), True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.40 \[ \int \frac {(d+e x)^3 \left (a+c x^2\right )}{(f+g x)^{3/2}} \, dx=\frac {2 \, {\left (\frac {35 \, {\left (g x + f\right )}^{\frac {9}{2}} c e^{3} - 45 \, {\left (5 \, c e^{3} f - 3 \, c d e^{2} g\right )} {\left (g x + f\right )}^{\frac {7}{2}} + 63 \, {\left (10 \, c e^{3} f^{2} - 12 \, c d e^{2} f g + {\left (3 \, c d^{2} e + a e^{3}\right )} g^{2}\right )} {\left (g x + f\right )}^{\frac {5}{2}} - 105 \, {\left (10 \, c e^{3} f^{3} - 18 \, c d e^{2} f^{2} g + 3 \, {\left (3 \, c d^{2} e + a e^{3}\right )} f g^{2} - {\left (c d^{3} + 3 \, a d e^{2}\right )} g^{3}\right )} {\left (g x + f\right )}^{\frac {3}{2}} + 315 \, {\left (5 \, c e^{3} f^{4} - 12 \, c d e^{2} f^{3} g + 3 \, a d^{2} e g^{4} + 3 \, {\left (3 \, c d^{2} e + a e^{3}\right )} f^{2} g^{2} - 2 \, {\left (c d^{3} + 3 \, a d e^{2}\right )} f g^{3}\right )} \sqrt {g x + f}}{g^{5}} + \frac {315 \, {\left (c e^{3} f^{5} - 3 \, c d e^{2} f^{4} g + 3 \, a d^{2} e f g^{4} - a d^{3} g^{5} + {\left (3 \, c d^{2} e + a e^{3}\right )} f^{3} g^{2} - {\left (c d^{3} + 3 \, a d e^{2}\right )} f^{2} g^{3}\right )}}{\sqrt {g x + f} g^{5}}\right )}}{315 \, g} \]

[In]

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

[Out]

2/315*((35*(g*x + f)^(9/2)*c*e^3 - 45*(5*c*e^3*f - 3*c*d*e^2*g)*(g*x + f)^(7/2) + 63*(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)^(5/2) - 105*(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)^(3/2) + 315*(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)*sqrt(g*x + f))/g^5 + 315*(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^
5))/g

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 465 vs. \(2 (218) = 436\).

Time = 0.28 (sec) , antiderivative size = 465, normalized size of antiderivative = 1.95 \[ \int \frac {(d+e x)^3 \left (a+c x^2\right )}{(f+g x)^{3/2}} \, dx=\frac {2 \, {\left (c e^{3} f^{5} - 3 \, c d e^{2} f^{4} g + 3 \, c d^{2} e f^{3} g^{2} + a e^{3} f^{3} g^{2} - c d^{3} f^{2} g^{3} - 3 \, a d e^{2} f^{2} g^{3} + 3 \, a d^{2} e f g^{4} - a d^{3} g^{5}\right )}}{\sqrt {g x + f} g^{6}} + \frac {2 \, {\left (35 \, {\left (g x + f\right )}^{\frac {9}{2}} c e^{3} g^{48} - 225 \, {\left (g x + f\right )}^{\frac {7}{2}} c e^{3} f g^{48} + 630 \, {\left (g x + f\right )}^{\frac {5}{2}} c e^{3} f^{2} g^{48} - 1050 \, {\left (g x + f\right )}^{\frac {3}{2}} c e^{3} f^{3} g^{48} + 1575 \, \sqrt {g x + f} c e^{3} f^{4} g^{48} + 135 \, {\left (g x + f\right )}^{\frac {7}{2}} c d e^{2} g^{49} - 756 \, {\left (g x + f\right )}^{\frac {5}{2}} c d e^{2} f g^{49} + 1890 \, {\left (g x + f\right )}^{\frac {3}{2}} c d e^{2} f^{2} g^{49} - 3780 \, \sqrt {g x + f} c d e^{2} f^{3} g^{49} + 189 \, {\left (g x + f\right )}^{\frac {5}{2}} c d^{2} e g^{50} + 63 \, {\left (g x + f\right )}^{\frac {5}{2}} a e^{3} g^{50} - 945 \, {\left (g x + f\right )}^{\frac {3}{2}} c d^{2} e f g^{50} - 315 \, {\left (g x + f\right )}^{\frac {3}{2}} a e^{3} f g^{50} + 2835 \, \sqrt {g x + f} c d^{2} e f^{2} g^{50} + 945 \, \sqrt {g x + f} a e^{3} f^{2} g^{50} + 105 \, {\left (g x + f\right )}^{\frac {3}{2}} c d^{3} g^{51} + 315 \, {\left (g x + f\right )}^{\frac {3}{2}} a d e^{2} g^{51} - 630 \, \sqrt {g x + f} c d^{3} f g^{51} - 1890 \, \sqrt {g x + f} a d e^{2} f g^{51} + 945 \, \sqrt {g x + f} a d^{2} e g^{52}\right )}}{315 \, g^{54}} \]

[In]

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

[Out]

2*(c*e^3*f^5 - 3*c*d*e^2*f^4*g + 3*c*d^2*e*f^3*g^2 + a*e^3*f^3*g^2 - c*d^3*f^2*g^3 - 3*a*d*e^2*f^2*g^3 + 3*a*d
^2*e*f*g^4 - a*d^3*g^5)/(sqrt(g*x + f)*g^6) + 2/315*(35*(g*x + f)^(9/2)*c*e^3*g^48 - 225*(g*x + f)^(7/2)*c*e^3
*f*g^48 + 630*(g*x + f)^(5/2)*c*e^3*f^2*g^48 - 1050*(g*x + f)^(3/2)*c*e^3*f^3*g^48 + 1575*sqrt(g*x + f)*c*e^3*
f^4*g^48 + 135*(g*x + f)^(7/2)*c*d*e^2*g^49 - 756*(g*x + f)^(5/2)*c*d*e^2*f*g^49 + 1890*(g*x + f)^(3/2)*c*d*e^
2*f^2*g^49 - 3780*sqrt(g*x + f)*c*d*e^2*f^3*g^49 + 189*(g*x + f)^(5/2)*c*d^2*e*g^50 + 63*(g*x + f)^(5/2)*a*e^3
*g^50 - 945*(g*x + f)^(3/2)*c*d^2*e*f*g^50 - 315*(g*x + f)^(3/2)*a*e^3*f*g^50 + 2835*sqrt(g*x + f)*c*d^2*e*f^2
*g^50 + 945*sqrt(g*x + f)*a*e^3*f^2*g^50 + 105*(g*x + f)^(3/2)*c*d^3*g^51 + 315*(g*x + f)^(3/2)*a*d*e^2*g^51 -
630*sqrt(g*x + f)*c*d^3*f*g^51 - 1890*sqrt(g*x + f)*a*d*e^2*f*g^51 + 945*sqrt(g*x + f)*a*d^2*e*g^52)/g^54

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.23 \[ \int \frac {(d+e x)^3 \left (a+c x^2\right )}{(f+g x)^{3/2}} \, dx=\frac {{\left (f+g\,x\right )}^{5/2}\,\left (6\,c\,d^2\,e\,g^2-24\,c\,d\,e^2\,f\,g+20\,c\,e^3\,f^2+2\,a\,e^3\,g^2\right )}{5\,g^6}-\frac {2\,c\,d^3\,f^2\,g^3+2\,a\,d^3\,g^5-6\,c\,d^2\,e\,f^3\,g^2-6\,a\,d^2\,e\,f\,g^4+6\,c\,d\,e^2\,f^4\,g+6\,a\,d\,e^2\,f^2\,g^3-2\,c\,e^3\,f^5-2\,a\,e^3\,f^3\,g^2}{g^6\,\sqrt {f+g\,x}}+\frac {2\,c\,e^3\,{\left (f+g\,x\right )}^{9/2}}{9\,g^6}+\frac {2\,\sqrt {f+g\,x}\,{\left (d\,g-e\,f\right )}^2\,\left (5\,c\,e\,f^2-2\,c\,d\,f\,g+3\,a\,e\,g^2\right )}{g^6}+\frac {2\,{\left (f+g\,x\right )}^{3/2}\,\left (d\,g-e\,f\right )\,\left (c\,d^2\,g^2-8\,c\,d\,e\,f\,g+10\,c\,e^2\,f^2+3\,a\,e^2\,g^2\right )}{3\,g^6}+\frac {2\,c\,e^2\,{\left (f+g\,x\right )}^{7/2}\,\left (3\,d\,g-5\,e\,f\right )}{7\,g^6} \]

[In]

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

[Out]

((f + g*x)^(5/2)*(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))/(5*g^6) - (2*a*d^3*g^5 - 2*c*e
^3*f^5 - 2*a*e^3*f^3*g^2 + 2*c*d^3*f^2*g^3 - 6*a*d^2*e*f*g^4 + 6*c*d*e^2*f^4*g + 6*a*d*e^2*f^2*g^3 - 6*c*d^2*e
*f^3*g^2)/(g^6*(f + g*x)^(1/2)) + (2*c*e^3*(f + g*x)^(9/2))/(9*g^6) + (2*(f + g*x)^(1/2)*(d*g - e*f)^2*(3*a*e*
g^2 + 5*c*e*f^2 - 2*c*d*f*g))/g^6 + (2*(f + g*x)^(3/2)*(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))/(3*g^6) + (2*c*e^2*(f + g*x)^(7/2)*(3*d*g - 5*e*f))/(7*g^6)