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

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

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Rubi [A]  time = 0.27, 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 = {898, 1261} \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 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 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+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+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 \operatorname {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 = 0.24, size = 207, normalized size = 0.87 \begin {gather*} \frac {2 \left (63 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 )-105 (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 )+315 \left (a g^2+c f^2\right ) (e f-d g)^3+315 (f+g x) (e f-d g)^2 \left (3 a e g^2+c f (5 e f-2 d g)\right )-45 c e^2 (f+g x)^4 (5 e f-3 d g)+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 + c*x^2))/(f + g*x)^(3/2),x]

[Out]

(2*(315*(e*f - d*g)^3*(c*f^2 + a*g^2) + 315*(e*f - d*g)^2*(3*a*e*g^2 + c*f*(5*e*f - 2*d*g))*(f + g*x) - 105*(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 + 63*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 - 45*c*e^2*(5*e*f - 3*d*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 [A]  time = 0.18, size = 427, normalized size = 1.79 \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 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 + 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 + 945*c*d^2*e*f^3*g^2 + 315*a*e^3*f^3*g^2 - 315*c*d^3*f^2*g^3 - 945*a*d*
e^2*f^2*g^3 + 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) + 28
35*c*d^2*e*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*a*d*e^2*f*g^3*(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 - 945*c*d^2*
e*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*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 + 189*c*d^2*e*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 + 35*c*e^3*(f + g*x)^5))/(315*g^6*Sqrt[f + g*x])

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fricas [A]  time = 0.39, size = 333, normalized size = 1.40 \begin {gather*} \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 )}} \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*(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)

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giac [B]  time = 0.21, 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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maple [A]  time = 0.01, size = 365, normalized size = 1.53 \begin {gather*} -\frac {2 \left (-35 e^{3} c \,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 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}+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+a)/(g*x+f)^(3/2),x)

[Out]

-2/315/(g*x+f)^(1/2)*(-35*c*e^3*g^5*x^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)/g^6

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maxima [A]  time = 0.46, size = 334, normalized size = 1.40 \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 - 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} \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*(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

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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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sympy [A]  time = 110.87, 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}} \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}} \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}} \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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