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

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

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Rubi [A]  time = 0.41, 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 = {897, 1261} \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 897

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 1261

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 \operatorname {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 \operatorname {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 = 0.73, size = 249, normalized size = 0.87 \begin {gather*} \frac {2 \left (-63 e (f+g x)^3 \left (c \left (-3 d^2 g^2+12 d e f g-10 e^2 f^2\right )-e g (a e g+3 b d g-4 b e f)\right )+105 (f+g x)^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 )+315 (e f-d g)^3 \left (g (a g-b f)+c f^2\right )+315 (f+g x) (e f-d g)^2 (g (3 a e g+b d g-4 b e f)+c f (5 e f-2 d g))-45 e^2 (f+g x)^4 (-b e g-3 c d g+5 c e f)+35 c e^3 (f+g x)^5\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*(315*(e*f - d*g)^3*(c*f^2 + g*(-(b*f) + a*g)) + 315*(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))*(f + g*x) + 105*(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)^2 - 63*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)^3 - 45*e^2*(5*c*e*f - 3*c*d*g - b*e*g)*(f + g*x)^4 + 35*c*e^3*(f + g*x)^5))/(315*g^6*Sqrt[f + g*x])

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IntegrateAlgebraic [B]  time = 0.30, size = 634, normalized size = 2.22 \begin {gather*} \frac {2 \left (-315 a d^3 g^5+945 a d^2 e g^4 (f+g x)+945 a d^2 e f g^4-945 a d e^2 f^2 g^3-1890 a d e^2 f g^3 (f+g x)+315 a d e^2 g^3 (f+g x)^2+315 a e^3 f^3 g^2+945 a e^3 f^2 g^2 (f+g x)-315 a e^3 f g^2 (f+g x)^2+63 a e^3 g^2 (f+g x)^3+315 b d^3 g^4 (f+g x)+315 b d^3 f g^4-945 b d^2 e f^2 g^3-1890 b d^2 e f g^3 (f+g x)+315 b d^2 e g^3 (f+g x)^2+945 b d e^2 f^3 g^2+2835 b d e^2 f^2 g^2 (f+g x)-945 b d e^2 f g^2 (f+g x)^2+189 b d e^2 g^2 (f+g x)^3-315 b e^3 f^4 g-1260 b e^3 f^3 g (f+g x)+630 b e^3 f^2 g (f+g x)^2-252 b e^3 f g (f+g x)^3+45 b e^3 g (f+g x)^4-315 c d^3 f^2 g^3-630 c d^3 f g^3 (f+g x)+105 c d^3 g^3 (f+g x)^2+945 c d^2 e f^3 g^2+2835 c d^2 e f^2 g^2 (f+g x)-945 c d^2 e f g^2 (f+g x)^2+189 c d^2 e g^2 (f+g x)^3-945 c d e^2 f^4 g-3780 c d e^2 f^3 g (f+g x)+1890 c d e^2 f^2 g (f+g x)^2-756 c d e^2 f g (f+g x)^3+135 c d e^2 g (f+g x)^4+315 c e^3 f^5+1575 c e^3 f^4 (f+g x)-1050 c e^3 f^3 (f+g x)^2+630 c e^3 f^2 (f+g x)^3-225 c e^3 f (f+g x)^4+35 c e^3 (f+g x)^5\right )}{315 g^6 \sqrt {f+g x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(315*c*e^3*f^5 - 945*c*d*e^2*f^4*g - 315*b*e^3*f^4*g + 945*c*d^2*e*f^3*g^2 + 945*b*d*e^2*f^3*g^2 + 315*a*e^
3*f^3*g^2 - 315*c*d^3*f^2*g^3 - 945*b*d^2*e*f^2*g^3 - 945*a*d*e^2*f^2*g^3 + 315*b*d^3*f*g^4 + 945*a*d^2*e*f*g^
4 - 315*a*d^3*g^5 + 1575*c*e^3*f^4*(f + g*x) - 3780*c*d*e^2*f^3*g*(f + g*x) - 1260*b*e^3*f^3*g*(f + g*x) + 283
5*c*d^2*e*f^2*g^2*(f + g*x) + 2835*b*d*e^2*f^2*g^2*(f + g*x) + 945*a*e^3*f^2*g^2*(f + g*x) - 630*c*d^3*f*g^3*(
f + g*x) - 1890*b*d^2*e*f*g^3*(f + g*x) - 1890*a*d*e^2*f*g^3*(f + g*x) + 315*b*d^3*g^4*(f + g*x) + 945*a*d^2*e
*g^4*(f + g*x) - 1050*c*e^3*f^3*(f + g*x)^2 + 1890*c*d*e^2*f^2*g*(f + g*x)^2 + 630*b*e^3*f^2*g*(f + g*x)^2 - 9
45*c*d^2*e*f*g^2*(f + g*x)^2 - 945*b*d*e^2*f*g^2*(f + g*x)^2 - 315*a*e^3*f*g^2*(f + g*x)^2 + 105*c*d^3*g^3*(f
+ g*x)^2 + 315*b*d^2*e*g^3*(f + g*x)^2 + 315*a*d*e^2*g^3*(f + g*x)^2 + 630*c*e^3*f^2*(f + g*x)^3 - 756*c*d*e^2
*f*g*(f + g*x)^3 - 252*b*e^3*f*g*(f + g*x)^3 + 189*c*d^2*e*g^2*(f + g*x)^3 + 189*b*d*e^2*g^2*(f + g*x)^3 + 63*
a*e^3*g^2*(f + g*x)^3 - 225*c*e^3*f*(f + g*x)^4 + 135*c*d*e^2*g*(f + g*x)^4 + 45*b*e^3*g*(f + g*x)^4 + 35*c*e^
3*(f + g*x)^5))/(315*g^6*Sqrt[f + g*x])

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fricas [A]  time = 0.62, size = 438, normalized size = 1.54 \begin {gather*} \frac {2 \, {\left (35 \, c e^{3} g^{5} x^{5} + 1280 \, c e^{3} f^{5} - 315 \, a d^{3} g^{5} - 1152 \, {\left (3 \, c d e^{2} + b e^{3}\right )} f^{4} g + 1008 \, {\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} f^{3} g^{2} - 840 \, {\left (c d^{3} + 3 \, b d^{2} e + 3 \, a d e^{2}\right )} f^{2} g^{3} + 630 \, {\left (b d^{3} + 3 \, a d^{2} e\right )} f g^{4} - 5 \, {\left (10 \, c e^{3} f g^{4} - 9 \, {\left (3 \, c d e^{2} + b e^{3}\right )} g^{5}\right )} x^{4} + {\left (80 \, c e^{3} f^{2} g^{3} - 72 \, {\left (3 \, c d e^{2} + b e^{3}\right )} f g^{4} + 63 \, {\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} g^{5}\right )} x^{3} - {\left (160 \, c e^{3} f^{3} g^{2} - 144 \, {\left (3 \, c d e^{2} + b e^{3}\right )} f^{2} g^{3} + 126 \, {\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} f g^{4} - 105 \, {\left (c d^{3} + 3 \, b d^{2} e + 3 \, a d e^{2}\right )} g^{5}\right )} x^{2} + {\left (640 \, c e^{3} f^{4} g - 576 \, {\left (3 \, c d e^{2} + b e^{3}\right )} f^{3} g^{2} + 504 \, {\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} f^{2} g^{3} - 420 \, {\left (c d^{3} + 3 \, b d^{2} e + 3 \, a d e^{2}\right )} f g^{4} + 315 \, {\left (b d^{3} + 3 \, a d^{2} e\right )} g^{5}\right )} x\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*(35*c*e^3*g^5*x^5 + 1280*c*e^3*f^5 - 315*a*d^3*g^5 - 1152*(3*c*d*e^2 + b*e^3)*f^4*g + 1008*(3*c*d^2*e +
3*b*d*e^2 + a*e^3)*f^3*g^2 - 840*(c*d^3 + 3*b*d^2*e + 3*a*d*e^2)*f^2*g^3 + 630*(b*d^3 + 3*a*d^2*e)*f*g^4 - 5*(
10*c*e^3*f*g^4 - 9*(3*c*d*e^2 + b*e^3)*g^5)*x^4 + (80*c*e^3*f^2*g^3 - 72*(3*c*d*e^2 + b*e^3)*f*g^4 + 63*(3*c*d
^2*e + 3*b*d*e^2 + a*e^3)*g^5)*x^3 - (160*c*e^3*f^3*g^2 - 144*(3*c*d*e^2 + b*e^3)*f^2*g^3 + 126*(3*c*d^2*e + 3
*b*d*e^2 + a*e^3)*f*g^4 - 105*(c*d^3 + 3*b*d^2*e + 3*a*d*e^2)*g^5)*x^2 + (640*c*e^3*f^4*g - 576*(3*c*d*e^2 + b
*e^3)*f^3*g^2 + 504*(3*c*d^2*e + 3*b*d*e^2 + a*e^3)*f^2*g^3 - 420*(c*d^3 + 3*b*d^2*e + 3*a*d*e^2)*f*g^4 + 315*
(b*d^3 + 3*a*d^2*e)*g^5)*x)*sqrt(g*x + f)/(g^7*x + f*g^6)

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giac [B]  time = 0.24, 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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maple [B]  time = 0.01, size = 540, normalized size = 1.89 \begin {gather*} -\frac {2 \left (-35 e^{3} c \,x^{5} g^{5}-45 b \,e^{3} g^{5} x^{4}-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 b d \,e^{2} g^{5} x^{3}+72 b \,e^{3} f \,g^{4} 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}-315 b \,d^{2} e \,g^{5} x^{2}+378 b d \,e^{2} f \,g^{4} x^{2}-144 b \,e^{3} f^{2} g^{3} 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 -315 b \,d^{3} g^{5} x +1260 b \,d^{2} e f \,g^{4} x -1512 b d \,e^{2} f^{2} g^{3} x +576 b \,e^{3} f^{3} g^{2} 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 d^{3} a \,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}-630 b \,d^{3} f \,g^{4}+2520 b \,d^{2} e \,f^{2} g^{3}-3024 b d \,e^{2} f^{3} g^{2}+1152 b \,e^{3} f^{4} g +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}} \end {gather*}

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)

[Out]

-2/315/(g*x+f)^(1/2)*(-35*c*e^3*g^5*x^5-45*b*e^3*g^5*x^4-135*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*b*d*e^2*g^5*x^3+72*b*e^3*f*g^4*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^2-315*b*d^2*e*g^5*x^2+378*b*d*e^2*f*g^4*x^2-144*b*e^3*f^2*g^3*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-315*b*d^3*g^5*x+1260*b*d^2*e*f*g^4*x-1512*b*d*e^2*f^2*g^3*x+576*b*e^3*f^3*g^2*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+2
520*a*d*e^2*f^2*g^3-1008*a*e^3*f^3*g^2-630*b*d^3*f*g^4+2520*b*d^2*e*f^2*g^3-3024*b*d*e^2*f^3*g^2+1152*b*e^3*f^
4*g+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)/g^6

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maxima [A]  time = 0.46, size = 437, normalized size = 1.53 \begin {gather*} \frac {2 \, {\left (\frac {35 \, {\left (g x + f\right )}^{\frac {9}{2}} c e^{3} - 45 \, {\left (5 \, c e^{3} f - {\left (3 \, c d e^{2} + b e^{3}\right )} g\right )} {\left (g x + f\right )}^{\frac {7}{2}} + 63 \, {\left (10 \, c e^{3} f^{2} - 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 e^{3} f^{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 e^{3} f^{4} - 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 (c e^{3} f^{5} - a d^{3} g^{5} - {\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*e^3*f - (3*c*d*e^2 + b*e^3)*g)*(g*x + f)^(7/2) + 63*(10*c*e^3*f^2 -
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*e^3*f^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*e^3*f^4 - 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*(c*e^3*f^5 - a*d^3*g^5 -
(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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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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sympy [A]  time = 158.56, 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}} \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}} \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}} \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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