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

Optimal. Leaf size=238 \[ \frac {2 (e f-d g)^3 \left (c f^2+a g^2\right )}{g^6 \sqrt {f+g x}}+\frac {2 (e f-d g)^2 \left (3 a e g^2+c f (5 e f-2 d g)\right ) \sqrt {f+g x}}{g^6}-\frac {2 (e f-d g) \left (3 a e^2 g^2+c \left (10 e^2 f^2-8 d e f g+d^2 g^2\right )\right ) (f+g x)^{3/2}}{3 g^6}+\frac {2 e \left (a e^2 g^2+c \left (10 e^2 f^2-12 d e f g+3 d^2 g^2\right )\right ) (f+g x)^{5/2}}{5 g^6}-\frac {2 c e^2 (5 e f-3 d g) (f+g x)^{7/2}}{7 g^6}+\frac {2 c e^3 (f+g x)^{9/2}}{9 g^6} \]

[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

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Rubi [A]
time = 0.17, antiderivative size = 238, normalized size of antiderivative = 1.00, 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 = {912, 1275} \begin {gather*} \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} \end {gather*}

Antiderivative was successfully verified.

[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*} \int \frac {(d+e x)^3 \left (a+c x^2\right )}{(f+g x)^{3/2}} \, dx &=\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*}

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Mathematica [A]
time = 0.21, size = 278, normalized size = 1.17 \begin {gather*} \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}} \end {gather*}

Antiderivative was successfully verified.

[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])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(437\) vs. \(2(218)=436\).
time = 0.10, size = 438, normalized size = 1.84 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/g^6*(1/9*c*e^3*(g*x+f)^(9/2)+3/7*c*d*e^2*g*(g*x+f)^(7/2)-5/7*c*e^3*f*(g*x+f)^(7/2)+1/5*a*e^3*g^2*(g*x+f)^(5/
2)+3/5*c*d^2*e*g^2*(g*x+f)^(5/2)-12/5*c*d*e^2*f*g*(g*x+f)^(5/2)+2*c*e^3*f^2*(g*x+f)^(5/2)+a*d*e^2*g^3*(g*x+f)^
(3/2)-a*e^3*f*g^2*(g*x+f)^(3/2)+1/3*c*d^3*g^3*(g*x+f)^(3/2)-3*c*d^2*e*f*g^2*(g*x+f)^(3/2)+6*c*d*e^2*f^2*g*(g*x
+f)^(3/2)-10/3*c*e^3*f^3*(g*x+f)^(3/2)+3*a*d^2*e*g^4*(g*x+f)^(1/2)-6*a*d*e^2*f*g^3*(g*x+f)^(1/2)+3*a*e^3*f^2*g
^2*(g*x+f)^(1/2)-2*c*d^3*f*g^3*(g*x+f)^(1/2)+9*c*d^2*e*f^2*g^2*(g*x+f)^(1/2)-12*c*d*e^2*f^3*g*(g*x+f)^(1/2)+5*
c*e^3*f^4*(g*x+f)^(1/2)-(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*x+f)^(1/2))

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Maxima [A]
time = 0.30, size = 322, normalized size = 1.35 \begin {gather*} \frac {2 \, {\left (\frac {35 \, {\left (g x + f\right )}^{\frac {9}{2}} c e^{3} + 45 \, {\left (3 \, c d g e^{2} - 5 \, c f e^{3}\right )} {\left (g x + f\right )}^{\frac {7}{2}} - 63 \, {\left (12 \, c d f g e^{2} - 10 \, c f^{2} e^{3} - {\left (3 \, c d^{2} e + a e^{3}\right )} g^{2}\right )} {\left (g x + f\right )}^{\frac {5}{2}} + 105 \, {\left (18 \, c d f^{2} g e^{2} - 10 \, c f^{3} e^{3} - 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 (3 \, a d^{2} g^{4} e - 12 \, c d f^{3} g e^{2} + 5 \, c f^{4} e^{3} + 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 (a d^{3} g^{5} - 3 \, a d^{2} f g^{4} e + 3 \, c d f^{4} g e^{2} - c f^{5} e^{3} - {\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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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*(3*c*d*g*e^2 - 5*c*f*e^3)*(g*x + f)^(7/2) - 63*(12*c*d*f*g*e^2 - 10*c*f^
2*e^3 - (3*c*d^2*e + a*e^3)*g^2)*(g*x + f)^(5/2) + 105*(18*c*d*f^2*g*e^2 - 10*c*f^3*e^3 - 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*(3*a*d^2*g^4*e - 12*c*d*f^3*g*e^2 + 5*c*f^4*e^3 + 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*(a*d^3*g^5 - 3*a*d^2*f*g^4*
e + 3*c*d*f^4*g*e^2 - c*f^5*e^3 - (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

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Fricas [A]
time = 3.15, size = 330, normalized size = 1.39 \begin {gather*} \frac {2 \, {\left (105 \, c d^{3} g^{5} x^{2} - 420 \, c d^{3} f g^{4} x - 840 \, c d^{3} f^{2} g^{3} - 315 \, a d^{3} g^{5} + {\left (35 \, c g^{5} x^{5} - 50 \, c f g^{4} x^{4} + 1280 \, c f^{5} + 1008 \, a f^{3} g^{2} + {\left (80 \, c f^{2} g^{3} + 63 \, a g^{5}\right )} x^{3} - 2 \, {\left (80 \, c f^{3} g^{2} + 63 \, a f g^{4}\right )} x^{2} + 8 \, {\left (80 \, c f^{4} g + 63 \, a f^{2} g^{3}\right )} x\right )} e^{3} + 9 \, {\left (15 \, c d g^{5} x^{4} - 24 \, c d f g^{4} x^{3} - 384 \, c d f^{4} g - 280 \, a d f^{2} g^{3} + {\left (48 \, c d f^{2} g^{3} + 35 \, a d g^{5}\right )} x^{2} - 4 \, {\left (48 \, c d f^{3} g^{2} + 35 \, a d f g^{4}\right )} x\right )} e^{2} + 189 \, {\left (c d^{2} g^{5} x^{3} - 2 \, c d^{2} f g^{4} x^{2} + 16 \, c d^{2} f^{3} g^{2} + 10 \, a d^{2} f g^{4} + {\left (8 \, c d^{2} f^{2} g^{3} + 5 \, a d^{2} g^{5}\right )} x\right )} e\right )} \sqrt {g x + f}}{315 \, {\left (g^{7} x + f g^{6}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(105*c*d^3*g^5*x^2 - 420*c*d^3*f*g^4*x - 840*c*d^3*f^2*g^3 - 315*a*d^3*g^5 + (35*c*g^5*x^5 - 50*c*f*g^4*
x^4 + 1280*c*f^5 + 1008*a*f^3*g^2 + (80*c*f^2*g^3 + 63*a*g^5)*x^3 - 2*(80*c*f^3*g^2 + 63*a*f*g^4)*x^2 + 8*(80*
c*f^4*g + 63*a*f^2*g^3)*x)*e^3 + 9*(15*c*d*g^5*x^4 - 24*c*d*f*g^4*x^3 - 384*c*d*f^4*g - 280*a*d*f^2*g^3 + (48*
c*d*f^2*g^3 + 35*a*d*g^5)*x^2 - 4*(48*c*d*f^3*g^2 + 35*a*d*f*g^4)*x)*e^2 + 189*(c*d^2*g^5*x^3 - 2*c*d^2*f*g^4*
x^2 + 16*c*d^2*f^3*g^2 + 10*a*d^2*f*g^4 + (8*c*d^2*f^2*g^3 + 5*a*d^2*g^5)*x)*e)*sqrt(g*x + f)/(g^7*x + f*g^6)

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Sympy [A]
time = 25.73, size = 328, normalized size = 1.38 \begin {gather*} \frac {2 c e^{3} \left (f + g x\right )^{\frac {9}{2}}}{9 g^{6}} + \frac {\left (f + g x\right )^{\frac {7}{2}} \cdot \left (6 c d e^{2} g - 10 c e^{3} f\right )}{7 g^{6}} + \frac {\left (f + g x\right )^{\frac {5}{2}} \cdot \left (2 a e^{3} g^{2} + 6 c d^{2} e g^{2} - 24 c d e^{2} f g + 20 c e^{3} f^{2}\right )}{5 g^{6}} + \frac {\left (f + g x\right )^{\frac {3}{2}} \cdot \left (6 a d e^{2} g^{3} - 6 a e^{3} f g^{2} + 2 c d^{3} g^{3} - 18 c d^{2} e f g^{2} + 36 c d e^{2} f^{2} g - 20 c e^{3} f^{3}\right )}{3 g^{6}} + \frac {\sqrt {f + g x} \left (6 a d^{2} e g^{4} - 12 a d e^{2} f g^{3} + 6 a e^{3} f^{2} g^{2} - 4 c d^{3} f g^{3} + 18 c d^{2} e f^{2} g^{2} - 24 c d e^{2} f^{3} g + 10 c e^{3} f^{4}\right )}{g^{6}} - \frac {2 \left (a g^{2} + c f^{2}\right ) \left (d g - e f\right )^{3}}{g^{6} \sqrt {f + g x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*c*e**3*(f + g*x)**(9/2)/(9*g**6) + (f + g*x)**(7/2)*(6*c*d*e**2*g - 10*c*e**3*f)/(7*g**6) + (f + g*x)**(5/2)
*(2*a*e**3*g**2 + 6*c*d**2*e*g**2 - 24*c*d*e**2*f*g + 20*c*e**3*f**2)/(5*g**6) + (f + g*x)**(3/2)*(6*a*d*e**2*
g**3 - 6*a*e**3*f*g**2 + 2*c*d**3*g**3 - 18*c*d**2*e*f*g**2 + 36*c*d*e**2*f**2*g - 20*c*e**3*f**3)/(3*g**6) +
sqrt(f + g*x)*(6*a*d**2*e*g**4 - 12*a*d*e**2*f*g**3 + 6*a*e**3*f**2*g**2 - 4*c*d**3*f*g**3 + 18*c*d**2*e*f**2*
g**2 - 24*c*d*e**2*f**3*g + 10*c*e**3*f**4)/g**6 - 2*(a*g**2 + c*f**2)*(d*g - e*f)**3/(g**6*sqrt(f + g*x))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 453 vs. \(2 (222) = 444\).
time = 1.02, size = 453, normalized size = 1.90 \begin {gather*} -\frac {2 \, {\left (c d^{3} f^{2} g^{3} + a d^{3} g^{5} - 3 \, c d^{2} f^{3} g^{2} e - 3 \, a d^{2} f g^{4} e + 3 \, c d f^{4} g e^{2} + 3 \, a d f^{2} g^{3} e^{2} - c f^{5} e^{3} - a f^{3} g^{2} e^{3}\right )}}{\sqrt {g x + f} g^{6}} + \frac {2 \, {\left (105 \, {\left (g x + f\right )}^{\frac {3}{2}} c d^{3} g^{51} - 630 \, \sqrt {g x + f} c d^{3} f g^{51} + 189 \, {\left (g x + f\right )}^{\frac {5}{2}} c d^{2} g^{50} e - 945 \, {\left (g x + f\right )}^{\frac {3}{2}} c d^{2} f g^{50} e + 2835 \, \sqrt {g x + f} c d^{2} f^{2} g^{50} e + 945 \, \sqrt {g x + f} a d^{2} g^{52} e + 135 \, {\left (g x + f\right )}^{\frac {7}{2}} c d g^{49} e^{2} - 756 \, {\left (g x + f\right )}^{\frac {5}{2}} c d f g^{49} e^{2} + 1890 \, {\left (g x + f\right )}^{\frac {3}{2}} c d f^{2} g^{49} e^{2} - 3780 \, \sqrt {g x + f} c d f^{3} g^{49} e^{2} + 315 \, {\left (g x + f\right )}^{\frac {3}{2}} a d g^{51} e^{2} - 1890 \, \sqrt {g x + f} a d f g^{51} e^{2} + 35 \, {\left (g x + f\right )}^{\frac {9}{2}} c g^{48} e^{3} - 225 \, {\left (g x + f\right )}^{\frac {7}{2}} c f g^{48} e^{3} + 630 \, {\left (g x + f\right )}^{\frac {5}{2}} c f^{2} g^{48} e^{3} - 1050 \, {\left (g x + f\right )}^{\frac {3}{2}} c f^{3} g^{48} e^{3} + 1575 \, \sqrt {g x + f} c f^{4} g^{48} e^{3} + 63 \, {\left (g x + f\right )}^{\frac {5}{2}} a g^{50} e^{3} - 315 \, {\left (g x + f\right )}^{\frac {3}{2}} a f g^{50} e^{3} + 945 \, \sqrt {g x + f} a f^{2} g^{50} e^{3}\right )}}{315 \, g^{54}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.09, size = 292, normalized size = 1.23 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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