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

Optimal. Leaf size=240 \[ \frac{2 e (f+g x)^{7/2} \left (a e^2 g^2+c \left (3 d^2 g^2-12 d e f g+10 e^2 f^2\right )\right )}{7 g^6}-\frac{2 (f+g x)^{5/2} (e f-d g) \left (3 a e^2 g^2+c \left (d^2 g^2-8 d e f g+10 e^2 f^2\right )\right )}{5 g^6}-\frac{2 \sqrt{f+g x} \left (a g^2+c f^2\right ) (e f-d g)^3}{g^6}+\frac{2 (f+g x)^{3/2} (e f-d g)^2 \left (3 a e g^2+c f (5 e f-2 d g)\right )}{3 g^6}-\frac{2 c e^2 (f+g x)^{9/2} (5 e f-3 d g)}{9 g^6}+\frac{2 c e^3 (f+g x)^{11/2}}{11 g^6} \]

[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)

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Rubi [A]  time = 0.344977, antiderivative size = 240, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {898, 1153} \[ \frac{2 e (f+g x)^{7/2} \left (a e^2 g^2+c \left (3 d^2 g^2-12 d e f g+10 e^2 f^2\right )\right )}{7 g^6}-\frac{2 (f+g x)^{5/2} (e f-d g) \left (3 a e^2 g^2+c \left (d^2 g^2-8 d e f g+10 e^2 f^2\right )\right )}{5 g^6}-\frac{2 \sqrt{f+g x} \left (a g^2+c f^2\right ) (e f-d g)^3}{g^6}+\frac{2 (f+g x)^{3/2} (e f-d g)^2 \left (3 a e g^2+c f (5 e f-2 d g)\right )}{3 g^6}-\frac{2 c e^2 (f+g x)^{9/2} (5 e f-3 d g)}{9 g^6}+\frac{2 c e^3 (f+g x)^{11/2}}{11 g^6} \]

Antiderivative was successfully verified.

[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 898

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 1153

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*} \int \frac{(d+e x)^3 \left (a+c x^2\right )}{\sqrt{f+g x}} \, dx &=\frac{2 \operatorname{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 \operatorname{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]  time = 0.254791, size = 207, normalized size = 0.86 \[ \frac{2 \sqrt{f+g x} \left (495 e (f+g x)^3 \left (a e^2 g^2+c \left (3 d^2 g^2-12 d e f g+10 e^2 f^2\right )\right )-693 (f+g x)^2 (e f-d g) \left (3 a e^2 g^2+c \left (d^2 g^2-8 d e f g+10 e^2 f^2\right )\right )-3465 \left (a g^2+c f^2\right ) (e f-d g)^3+1155 (f+g x) (e f-d g)^2 \left (3 a e g^2+c f (5 e f-2 d g)\right )-385 c e^2 (f+g x)^4 (5 e f-3 d g)+315 c e^3 (f+g x)^5\right )}{3465 g^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[f + g*x]*(-3465*(e*f - d*g)^3*(c*f^2 + a*g^2) + 1155*(e*f - d*g)^2*(3*a*e*g^2 + c*f*(5*e*f - 2*d*g))*(
f + g*x) - 693*(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)^2 + 495*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)^3 - 385*c*e^2*(5*e*f - 3*d*g)*(f + g*x)^4 + 315*c*e^3*(f
+ g*x)^5))/(3465*g^6)

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Maple [A]  time = 0.047, size = 365, normalized size = 1.5 \begin{align*}{\frac{630\,{e}^{3}c{x}^{5}{g}^{5}+2310\,cd{e}^{2}{g}^{5}{x}^{4}-700\,c{e}^{3}f{g}^{4}{x}^{4}+990\,a{e}^{3}{g}^{5}{x}^{3}+2970\,c{d}^{2}e{g}^{5}{x}^{3}-2640\,cd{e}^{2}f{g}^{4}{x}^{3}+800\,c{e}^{3}{f}^{2}{g}^{3}{x}^{3}+4158\,ad{e}^{2}{g}^{5}{x}^{2}-1188\,a{e}^{3}f{g}^{4}{x}^{2}+1386\,c{d}^{3}{g}^{5}{x}^{2}-3564\,c{d}^{2}ef{g}^{4}{x}^{2}+3168\,cd{e}^{2}{f}^{2}{g}^{3}{x}^{2}-960\,c{e}^{3}{f}^{3}{g}^{2}{x}^{2}+6930\,a{d}^{2}e{g}^{5}x-5544\,ad{e}^{2}f{g}^{4}x+1584\,a{e}^{3}{f}^{2}{g}^{3}x-1848\,c{d}^{3}f{g}^{4}x+4752\,c{d}^{2}e{f}^{2}{g}^{3}x-4224\,cd{e}^{2}{f}^{3}{g}^{2}x+1280\,c{e}^{3}{f}^{4}gx+6930\,{d}^{3}a{g}^{5}-13860\,a{d}^{2}ef{g}^{4}+11088\,ad{e}^{2}{f}^{2}{g}^{3}-3168\,a{e}^{3}{f}^{3}{g}^{2}+3696\,c{d}^{3}{f}^{2}{g}^{3}-9504\,c{d}^{2}e{f}^{3}{g}^{2}+8448\,cd{e}^{2}{f}^{4}g-2560\,c{e}^{3}{f}^{5}}{3465\,{g}^{6}}\sqrt{gx+f}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3465*(g*x+f)^(1/2)*(315*c*e^3*g^5*x^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)/g^6

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Maxima [A]  time = 0.985909, size = 440, normalized size = 1.83 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]  time = 1.80019, size = 747, normalized size = 3.11 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [A]  time = 97.7774, size = 1040, normalized size = 4.33 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-(2*a*d**3*f/sqrt(f + g*x) + 2*a*d**3*(-f/sqrt(f + g*x) - sqrt(f + g*x)) + 6*a*d**2*e*f*(-f/sqrt(f
+ g*x) - sqrt(f + g*x))/g + 6*a*d**2*e*(f**2/sqrt(f + g*x) + 2*f*sqrt(f + g*x) - (f + g*x)**(3/2)/3)/g + 6*a*d
*e**2*f*(f**2/sqrt(f + g*x) + 2*f*sqrt(f + g*x) - (f + g*x)**(3/2)/3)/g**2 + 6*a*d*e**2*(-f**3/sqrt(f + g*x) -
3*f**2*sqrt(f + g*x) + f*(f + g*x)**(3/2) - (f + g*x)**(5/2)/5)/g**2 + 2*a*e**3*f*(-f**3/sqrt(f + g*x) - 3*f*
*2*sqrt(f + g*x) + f*(f + g*x)**(3/2) - (f + g*x)**(5/2)/5)/g**3 + 2*a*e**3*(f**4/sqrt(f + g*x) + 4*f**3*sqrt(
f + g*x) - 2*f**2*(f + g*x)**(3/2) + 4*f*(f + g*x)**(5/2)/5 - (f + g*x)**(7/2)/7)/g**3 + 2*c*d**3*f*(f**2/sqrt
(f + g*x) + 2*f*sqrt(f + g*x) - (f + g*x)**(3/2)/3)/g**2 + 2*c*d**3*(-f**3/sqrt(f + g*x) - 3*f**2*sqrt(f + g*x
) + f*(f + g*x)**(3/2) - (f + g*x)**(5/2)/5)/g**2 + 6*c*d**2*e*f*(-f**3/sqrt(f + g*x) - 3*f**2*sqrt(f + g*x) +
f*(f + g*x)**(3/2) - (f + g*x)**(5/2)/5)/g**3 + 6*c*d**2*e*(f**4/sqrt(f + g*x) + 4*f**3*sqrt(f + g*x) - 2*f**
2*(f + g*x)**(3/2) + 4*f*(f + g*x)**(5/2)/5 - (f + g*x)**(7/2)/7)/g**3 + 6*c*d*e**2*f*(f**4/sqrt(f + g*x) + 4*
f**3*sqrt(f + g*x) - 2*f**2*(f + g*x)**(3/2) + 4*f*(f + g*x)**(5/2)/5 - (f + g*x)**(7/2)/7)/g**4 + 6*c*d*e**2*
(-f**5/sqrt(f + g*x) - 5*f**4*sqrt(f + g*x) + 10*f**3*(f + g*x)**(3/2)/3 - 2*f**2*(f + g*x)**(5/2) + 5*f*(f +
g*x)**(7/2)/7 - (f + g*x)**(9/2)/9)/g**4 + 2*c*e**3*f*(-f**5/sqrt(f + g*x) - 5*f**4*sqrt(f + g*x) + 10*f**3*(f
+ g*x)**(3/2)/3 - 2*f**2*(f + g*x)**(5/2) + 5*f*(f + g*x)**(7/2)/7 - (f + g*x)**(9/2)/9)/g**5 + 2*c*e**3*(f**
6/sqrt(f + g*x) + 6*f**5*sqrt(f + g*x) - 5*f**4*(f + g*x)**(3/2) + 4*f**3*(f + g*x)**(5/2) - 15*f**2*(f + g*x)
**(7/2)/7 + 2*f*(f + g*x)**(9/2)/3 - (f + g*x)**(11/2)/11)/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), Tr
ue))

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Giac [A]  time = 1.17097, size = 510, normalized size = 2.12 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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