3.9.26 \(\int \frac {(d+e x)^3 (a+b x+c x^2)}{(f+g x)^{3/2}} \, dx\) [826]

Optimal. Leaf size=285 \[ \frac {2 (e f-d g)^3 \left (c f^2-b f g+a g^2\right )}{g^6 \sqrt {f+g x}}+\frac {2 (e f-d g)^2 (c f (5 e f-2 d g)-g (4 b e f-b d g-3 a e g)) \sqrt {f+g x}}{g^6}+\frac {2 (e f-d g) \left (3 e g (2 b e f-b d g-a e g)-c \left (10 e^2 f^2-8 d e f g+d^2 g^2\right )\right ) (f+g x)^{3/2}}{3 g^6}-\frac {2 e \left (e g (4 b e f-3 b d g-a e g)-c \left (10 e^2 f^2-12 d e f g+3 d^2 g^2\right )\right ) (f+g x)^{5/2}}{5 g^6}-\frac {2 e^2 (5 c e f-3 c d g-b e g) (f+g x)^{7/2}}{7 g^6}+\frac {2 c e^3 (f+g x)^{9/2}}{9 g^6} \]

[Out]

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

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Rubi [A]
time = 0.26, antiderivative size = 285, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {911, 1275} \begin {gather*} -\frac {2 e (f+g x)^{5/2} \left (e g (-a e g-3 b d g+4 b e f)-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 e g (-a e g-b d g+2 b e f)-c \left (d^2 g^2-8 d e f g+10 e^2 f^2\right )\right )}{3 g^6}+\frac {2 (e f-d g)^3 \left (a g^2-b f g+c f^2\right )}{g^6 \sqrt {f+g x}}+\frac {2 \sqrt {f+g x} (e f-d g)^2 (c f (5 e f-2 d g)-g (-3 a e g-b d g+4 b e f))}{g^6}-\frac {2 e^2 (f+g x)^{7/2} (-b e g-3 c d g+5 c e f)}{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 + b*x + c*x^2))/(f + g*x)^(3/2),x]

[Out]

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

Rule 911

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> With[{q = Denominator[m]}, Dist[q/e, Subst[Int[x^(q*(m + 1) - 1)*((e*f - d*g)/e + g*(x^q/e))^n*((c*d^2 - b*d
*e + a*e^2)/e^2 - (2*c*d - b*e)*(x^q/e^2) + c*(x^(2*q)/e^2))^p, x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, b, c
, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + 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+b x+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-b f g+a g^2}{g^2}-\frac {(2 c f-b g) 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 (c f (5 e f-2 d g)-g (4 b e f-b d g-3 a e g))}{g^5}+\frac {(-e f+d g)^3 \left (c f^2-b f g+a g^2\right )}{g^5 x^2}+\frac {(e f-d g) \left (3 e g (2 b e f-b d g-a e g)-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 (-e g (4 b e f-3 b d g-a e g)+c \left (10 e^2 f^2-12 d e f g+3 d^2 g^2\right )\right ) x^4}{g^5}+\frac {e^2 (-5 c e f+3 c d g+b e 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-b f g+a g^2\right )}{g^6 \sqrt {f+g x}}+\frac {2 (e f-d g)^2 (c f (5 e f-2 d g)-g (4 b e f-b d g-3 a e g)) \sqrt {f+g x}}{g^6}+\frac {2 (e f-d g) \left (3 e g (2 b e f-b d g-a e g)-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 (e g (4 b e f-3 b d g-a e g)-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 e^2 (5 c e f-3 c d g-b e 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 = 0.36, size = 406, normalized size = 1.42 \begin {gather*} \frac {2 \left (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 )+9 g \left (7 a g \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 )+b \left (35 d^3 g^3 (2 f+g x)+35 d^2 e g^2 \left (-8 f^2-4 f g x+g^2 x^2\right )+21 d e^2 g \left (16 f^3+8 f^2 g x-2 f g^2 x^2+g^3 x^3\right )+e^3 \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 )\right )\right )\right )}{315 g^6 \sqrt {f+g x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(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)) + 9*g*(7*a*g*(-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)) + b*(35*d^3*
g^3*(2*f + g*x) + 35*d^2*e*g^2*(-8*f^2 - 4*f*g*x + g^2*x^2) + 21*d*e^2*g*(16*f^3 + 8*f^2*g*x - 2*f*g^2*x^2 + g
^3*x^3) + e^3*(-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)))))/(315*g^6*Sqrt[f + g*x])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(649\) vs. \(2(265)=530\).
time = 0.11, size = 650, normalized size = 2.28 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/g^6*(1/9*c*e^3*(g*x+f)^(9/2)-3*b*d*e^2*f*g^2*(g*x+f)^(3/2)+b*d^3*g^4*(g*x+f)^(1/2)+1/7*b*e^3*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)+2*c*e^3*f^2*(g*x+f)^(5/2)+1/3*c*d^3*g^3*(g*x+f)^(3/2)-1
0/3*c*e^3*f^3*(g*x+f)^(3/2)+5*c*e^3*f^4*(g*x+f)^(1/2)-6*b*d^2*e*f*g^3*(g*x+f)^(1/2)-12*c*d*e^2*f^3*g*(g*x+f)^(
1/2)+6*c*d*e^2*f^2*g*(g*x+f)^(3/2)-6*a*d*e^2*f*g^3*(g*x+f)^(1/2)+9*c*d^2*e*f^2*g^2*(g*x+f)^(1/2)+9*b*d*e^2*f^2
*g^2*(g*x+f)^(1/2)-12/5*c*d*e^2*f*g*(g*x+f)^(5/2)-3*c*d^2*e*f*g^2*(g*x+f)^(3/2)-(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-b*d^3*f*g^4+3*b*d^2*e*f^2*g^3-3*b*d*e^2*f^3*g^2+b*e^3*f^4*g+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)+3/5*c*d^2*e*g^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)+3*a*d^2*e*g^4*(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)+
3/7*c*d*e^2*g*(g*x+f)^(7/2)+b*d^2*e*g^3*(g*x+f)^(3/2)-4*b*e^3*f^3*g*(g*x+f)^(1/2)+3/5*b*d*e^2*g^2*(g*x+f)^(5/2
)-4/5*b*e^3*f*g*(g*x+f)^(5/2)+2*b*e^3*f^2*g*(g*x+f)^(3/2))

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Maxima [A]
time = 0.28, size = 419, normalized size = 1.47 \begin {gather*} \frac {2 \, {\left (\frac {35 \, {\left (g x + f\right )}^{\frac {9}{2}} c e^{3} - 45 \, {\left (5 \, c f e^{3} - {\left (3 \, c d e^{2} + b e^{3}\right )} g\right )} {\left (g x + f\right )}^{\frac {7}{2}} + 63 \, {\left (10 \, c f^{2} e^{3} - 4 \, {\left (3 \, c d e^{2} + b e^{3}\right )} f g + {\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} g^{2}\right )} {\left (g x + f\right )}^{\frac {5}{2}} - 105 \, {\left (10 \, c f^{3} e^{3} - 6 \, {\left (3 \, c d e^{2} + b e^{3}\right )} f^{2} g + 3 \, {\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} f g^{2} - {\left (c d^{3} + 3 \, b d^{2} e + 3 \, a d e^{2}\right )} g^{3}\right )} {\left (g x + f\right )}^{\frac {3}{2}} + 315 \, {\left (5 \, c f^{4} e^{3} - 4 \, {\left (3 \, c d e^{2} + b e^{3}\right )} f^{3} g + 3 \, {\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} f^{2} g^{2} - 2 \, {\left (c d^{3} + 3 \, b d^{2} e + 3 \, a d e^{2}\right )} f g^{3} + {\left (b d^{3} + 3 \, a d^{2} e\right )} g^{4}\right )} \sqrt {g x + f}}{g^{5}} - \frac {315 \, {\left (a d^{3} g^{5} - c f^{5} e^{3} + {\left (3 \, c d e^{2} + b e^{3}\right )} f^{4} g - {\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} f^{3} g^{2} + {\left (c d^{3} + 3 \, b d^{2} e + 3 \, a d e^{2}\right )} f^{2} g^{3} - {\left (b d^{3} + 3 \, a d^{2} e\right )} f g^{4}\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+b*x+a)/(g*x+f)^(3/2),x, algorithm="maxima")

[Out]

2/315*((35*(g*x + f)^(9/2)*c*e^3 - 45*(5*c*f*e^3 - (3*c*d*e^2 + b*e^3)*g)*(g*x + f)^(7/2) + 63*(10*c*f^2*e^3 -
4*(3*c*d*e^2 + b*e^3)*f*g + (3*c*d^2*e + 3*b*d*e^2 + a*e^3)*g^2)*(g*x + f)^(5/2) - 105*(10*c*f^3*e^3 - 6*(3*c
*d*e^2 + b*e^3)*f^2*g + 3*(3*c*d^2*e + 3*b*d*e^2 + a*e^3)*f*g^2 - (c*d^3 + 3*b*d^2*e + 3*a*d*e^2)*g^3)*(g*x +
f)^(3/2) + 315*(5*c*f^4*e^3 - 4*(3*c*d*e^2 + b*e^3)*f^3*g + 3*(3*c*d^2*e + 3*b*d*e^2 + a*e^3)*f^2*g^2 - 2*(c*d
^3 + 3*b*d^2*e + 3*a*d*e^2)*f*g^3 + (b*d^3 + 3*a*d^2*e)*g^4)*sqrt(g*x + f))/g^5 - 315*(a*d^3*g^5 - c*f^5*e^3 +
(3*c*d*e^2 + b*e^3)*f^4*g - (3*c*d^2*e + 3*b*d*e^2 + a*e^3)*f^3*g^2 + (c*d^3 + 3*b*d^2*e + 3*a*d*e^2)*f^2*g^3
- (b*d^3 + 3*a*d^2*e)*f*g^4)/(sqrt(g*x + f)*g^5))/g

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Fricas [A]
time = 1.16, size = 466, normalized size = 1.64 \begin {gather*} \frac {2 \, {\left (105 \, c d^{3} g^{5} x^{2} - 840 \, c d^{3} f^{2} g^{3} + 630 \, b d^{3} f g^{4} - 315 \, a d^{3} g^{5} - 105 \, {\left (4 \, c d^{3} f g^{4} - 3 \, b d^{3} g^{5}\right )} x + {\left (35 \, c g^{5} x^{5} + 1280 \, c f^{5} - 1152 \, b f^{4} g + 1008 \, a f^{3} g^{2} - 5 \, {\left (10 \, c f g^{4} - 9 \, b g^{5}\right )} x^{4} + {\left (80 \, c f^{2} g^{3} - 72 \, b f g^{4} + 63 \, a g^{5}\right )} x^{3} - 2 \, {\left (80 \, c f^{3} g^{2} - 72 \, b f^{2} g^{3} + 63 \, a f g^{4}\right )} x^{2} + 8 \, {\left (80 \, c f^{4} g - 72 \, b f^{3} g^{2} + 63 \, a f^{2} g^{3}\right )} x\right )} e^{3} + 9 \, {\left (15 \, c d g^{5} x^{4} - 384 \, c d f^{4} g + 336 \, b d f^{3} g^{2} - 280 \, a d f^{2} g^{3} - 3 \, {\left (8 \, c d f g^{4} - 7 \, b d g^{5}\right )} x^{3} + {\left (48 \, c d f^{2} g^{3} - 42 \, b d f g^{4} + 35 \, a d g^{5}\right )} x^{2} - 4 \, {\left (48 \, c d f^{3} g^{2} - 42 \, b d f^{2} g^{3} + 35 \, a d f g^{4}\right )} x\right )} e^{2} + 63 \, {\left (3 \, c d^{2} g^{5} x^{3} + 48 \, c d^{2} f^{3} g^{2} - 40 \, b d^{2} f^{2} g^{3} + 30 \, a d^{2} f g^{4} - {\left (6 \, c d^{2} f g^{4} - 5 \, b d^{2} g^{5}\right )} x^{2} + {\left (24 \, c d^{2} f^{2} g^{3} - 20 \, b d^{2} f g^{4} + 15 \, 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+b*x+a)/(g*x+f)^(3/2),x, algorithm="fricas")

[Out]

2/315*(105*c*d^3*g^5*x^2 - 840*c*d^3*f^2*g^3 + 630*b*d^3*f*g^4 - 315*a*d^3*g^5 - 105*(4*c*d^3*f*g^4 - 3*b*d^3*
g^5)*x + (35*c*g^5*x^5 + 1280*c*f^5 - 1152*b*f^4*g + 1008*a*f^3*g^2 - 5*(10*c*f*g^4 - 9*b*g^5)*x^4 + (80*c*f^2
*g^3 - 72*b*f*g^4 + 63*a*g^5)*x^3 - 2*(80*c*f^3*g^2 - 72*b*f^2*g^3 + 63*a*f*g^4)*x^2 + 8*(80*c*f^4*g - 72*b*f^
3*g^2 + 63*a*f^2*g^3)*x)*e^3 + 9*(15*c*d*g^5*x^4 - 384*c*d*f^4*g + 336*b*d*f^3*g^2 - 280*a*d*f^2*g^3 - 3*(8*c*
d*f*g^4 - 7*b*d*g^5)*x^3 + (48*c*d*f^2*g^3 - 42*b*d*f*g^4 + 35*a*d*g^5)*x^2 - 4*(48*c*d*f^3*g^2 - 42*b*d*f^2*g
^3 + 35*a*d*f*g^4)*x)*e^2 + 63*(3*c*d^2*g^5*x^3 + 48*c*d^2*f^3*g^2 - 40*b*d^2*f^2*g^3 + 30*a*d^2*f*g^4 - (6*c*
d^2*f*g^4 - 5*b*d^2*g^5)*x^2 + (24*c*d^2*f^2*g^3 - 20*b*d^2*f*g^4 + 15*a*d^2*g^5)*x)*e)*sqrt(g*x + f)/(g^7*x +
f*g^6)

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Sympy [A]
time = 47.05, size = 452, normalized size = 1.59 \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 (2 b e^{3} g + 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 b d e^{2} g^{2} - 8 b e^{3} f g + 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} + 6 b d^{2} e g^{3} - 18 b d e^{2} f g^{2} + 12 b e^{3} f^{2} g + 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} + 2 b d^{3} g^{4} - 12 b d^{2} e f g^{3} + 18 b d e^{2} f^{2} g^{2} - 8 b e^{3} f^{3} g - 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 (d g - e f\right )^{3} \left (a g^{2} - b f g + c f^{2}\right )}{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+b*x+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)*(2*b*e**3*g + 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*b*d*e**2*g**2 - 8*b*e**3*f*g + 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 + 6*b*d**2*e*g**3 - 18*b*d*e**2*f*g**2 +
12*b*e**3*f**2*g + 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 + 2*b*d**3*g**4 - 12*b*d**2*e*f*g**3 + 18*b
*d*e**2*f**2*g**2 - 8*b*e**3*f**3*g - 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*(d*g - e*f)**3*(a*g**2 - b*f*g + c*f**2)/(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. 669 vs. \(2 (271) = 542\).
time = 2.97, size = 669, normalized size = 2.35 \begin {gather*} -\frac {2 \, {\left (c d^{3} f^{2} g^{3} - b d^{3} f g^{4} + a d^{3} g^{5} - 3 \, c d^{2} f^{3} g^{2} e + 3 \, b d^{2} f^{2} g^{3} e - 3 \, a d^{2} f g^{4} e + 3 \, c d f^{4} g e^{2} - 3 \, b d f^{3} g^{2} e^{2} + 3 \, a d f^{2} g^{3} e^{2} - c f^{5} e^{3} + b f^{4} g 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} + 315 \, \sqrt {g x + f} b d^{3} g^{52} + 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 + 315 \, {\left (g x + f\right )}^{\frac {3}{2}} b d^{2} g^{51} e - 1890 \, \sqrt {g x + f} b d^{2} f g^{51} 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} + 189 \, {\left (g x + f\right )}^{\frac {5}{2}} b d g^{50} e^{2} - 945 \, {\left (g x + f\right )}^{\frac {3}{2}} b d f g^{50} e^{2} + 2835 \, \sqrt {g x + f} b d f^{2} g^{50} 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} + 45 \, {\left (g x + f\right )}^{\frac {7}{2}} b g^{49} e^{3} - 252 \, {\left (g x + f\right )}^{\frac {5}{2}} b f g^{49} e^{3} + 630 \, {\left (g x + f\right )}^{\frac {3}{2}} b f^{2} g^{49} e^{3} - 1260 \, \sqrt {g x + f} b f^{3} g^{49} 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+b*x+a)/(g*x+f)^(3/2),x, algorithm="giac")

[Out]

-2*(c*d^3*f^2*g^3 - b*d^3*f*g^4 + a*d^3*g^5 - 3*c*d^2*f^3*g^2*e + 3*b*d^2*f^2*g^3*e - 3*a*d^2*f*g^4*e + 3*c*d*
f^4*g*e^2 - 3*b*d*f^3*g^2*e^2 + 3*a*d*f^2*g^3*e^2 - c*f^5*e^3 + b*f^4*g*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 + 315*sqrt(g*x + f)*b*d^3*g^52 + 1
89*(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 + 3
15*(g*x + f)^(3/2)*b*d^2*g^51*e - 1890*sqrt(g*x + f)*b*d^2*f*g^51*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 + 189*(g*x + f)^(5/2)*b*d*g^50*e^2 - 945*(g*x + f)^(3/2)*b*d*f*g^50*e^2 + 2835*
sqrt(g*x + f)*b*d*f^2*g^50*e^2 + 315*(g*x + f)^(3/2)*a*d*g^51*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 + 45*(g*x + f)^(7/2)*b*g^49*e^3 - 252*(g*x + f)^(5
/2)*b*f*g^49*e^3 + 630*(g*x + f)^(3/2)*b*f^2*g^49*e^3 - 1260*sqrt(g*x + f)*b*f^3*g^49*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 = 0.12, size = 394, normalized size = 1.38 \begin {gather*} \frac {{\left (f+g\,x\right )}^{7/2}\,\left (2\,b\,e^3\,g-10\,c\,e^3\,f+6\,c\,d\,e^2\,g\right )}{7\,g^6}-\frac {2\,c\,d^3\,f^2\,g^3-2\,b\,d^3\,f\,g^4+2\,a\,d^3\,g^5-6\,c\,d^2\,e\,f^3\,g^2+6\,b\,d^2\,e\,f^2\,g^3-6\,a\,d^2\,e\,f\,g^4+6\,c\,d\,e^2\,f^4\,g-6\,b\,d\,e^2\,f^3\,g^2+6\,a\,d\,e^2\,f^2\,g^3-2\,c\,e^3\,f^5+2\,b\,e^3\,f^4\,g-2\,a\,e^3\,f^3\,g^2}{g^6\,\sqrt {f+g\,x}}+\frac {{\left (f+g\,x\right )}^{5/2}\,\left (6\,c\,d^2\,e\,g^2-24\,c\,d\,e^2\,f\,g+6\,b\,d\,e^2\,g^2+20\,c\,e^3\,f^2-8\,b\,e^3\,f\,g+2\,a\,e^3\,g^2\right )}{5\,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+3\,b\,d\,e\,g^2+10\,c\,e^2\,f^2-6\,b\,e^2\,f\,g+3\,a\,e^2\,g^2\right )}{3\,g^6}+\frac {2\,\sqrt {f+g\,x}\,{\left (d\,g-e\,f\right )}^2\,\left (3\,a\,e\,g^2+b\,d\,g^2+5\,c\,e\,f^2-4\,b\,e\,f\,g-2\,c\,d\,f\,g\right )}{g^6}+\frac {2\,c\,e^3\,{\left (f+g\,x\right )}^{9/2}}{9\,g^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((f + g*x)^(7/2)*(2*b*e^3*g - 10*c*e^3*f + 6*c*d*e^2*g))/(7*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 - 2*b*d^3*f*g^4 + 2*b*e^3*f^4*g - 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*b*d*e^2*f^3*g^2 + 6*b*d^2*e*f^2*g^3 - 6*c*d^2*e*f^3*g^2)/(g^6*(f + g*x)^(1/2)) + ((f + g*x)^(5/2)*(2*a*e^3*g
^2 + 20*c*e^3*f^2 - 8*b*e^3*f*g + 6*b*d*e^2*g^2 + 6*c*d^2*e*g^2 - 24*c*d*e^2*f*g))/(5*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 + 3*b*d*e*g^2 - 6*b*e^2*f*g - 8*c*d*e*f*g))/(3*g^6) + (2
*(f + g*x)^(1/2)*(d*g - e*f)^2*(3*a*e*g^2 + b*d*g^2 + 5*c*e*f^2 - 4*b*e*f*g - 2*c*d*f*g))/g^6 + (2*c*e^3*(f +
g*x)^(9/2))/(9*g^6)

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