3.9.78 \(\int \frac {(f+g x)^4}{(d+e x) (a+b x+c x^2)^{3/2}} \, dx\) [878]

Optimal. Leaf size=496 \[ -\frac {2 \left (a b^3 d g^4-b^2 \left (c^2 e f^4+4 a c d f g^3+a^2 e g^4\right )+2 a c \left (a^2 e g^4+c^2 f^3 (e f-4 d g)-2 a c f g^2 (3 e f-2 d g)\right )+b c \left (c^2 d f^4+a^2 g^3 (4 e f-3 d g)+2 a c f^2 g (2 e f+3 d g)\right )+\left (2 c^4 d f^4+b^3 (b d-a e) g^4-b c g^3 \left (4 b^2 d f-3 a^2 e g-4 a b (e f-d g)\right )+2 c^2 g^2 \left (3 b^2 d f^2-3 a b f (e f-2 d g)-a^2 g (4 e f-d g)\right )+c^3 f^2 (4 a g (2 e f-3 d g)-b f (e f+4 d g))\right ) x\right )}{c^2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {a+b x+c x^2}}+\frac {g^4 \sqrt {a+b x+c x^2}}{c^2 e}+\frac {g^3 (8 c e f-2 c d g-3 b e g) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 c^{5/2} e^2}+\frac {(e f-d g)^4 \tanh ^{-1}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{e^2 \left (c d^2-b d e+a e^2\right )^{3/2}} \]

[Out]

1/2*g^3*(-3*b*e*g-2*c*d*g+8*c*e*f)*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(5/2)/e^2+(-d*g+e*f)^4
*arctanh(1/2*(b*d-2*a*e+(-b*e+2*c*d)*x)/(a*e^2-b*d*e+c*d^2)^(1/2)/(c*x^2+b*x+a)^(1/2))/e^2/(a*e^2-b*d*e+c*d^2)
^(3/2)-2*(a*b^3*d*g^4-b^2*(a^2*e*g^4+4*a*c*d*f*g^3+c^2*e*f^4)+2*a*c*(a^2*e*g^4+c^2*f^3*(-4*d*g+e*f)-2*a*c*f*g^
2*(-2*d*g+3*e*f))+b*c*(c^2*d*f^4+a^2*g^3*(-3*d*g+4*e*f)+2*a*c*f^2*g*(3*d*g+2*e*f))+(2*c^4*d*f^4+b^3*(-a*e+b*d)
*g^4-b*c*g^3*(4*b^2*d*f-3*a^2*e*g-4*a*b*(-d*g+e*f))+2*c^2*g^2*(3*b^2*d*f^2-3*a*b*f*(-2*d*g+e*f)-a^2*g*(-d*g+4*
e*f))+c^3*f^2*(4*a*g*(-3*d*g+2*e*f)-b*f*(4*d*g+e*f)))*x)/c^2/(-4*a*c+b^2)/(a*e^2-b*d*e+c*d^2)/(c*x^2+b*x+a)^(1
/2)+g^4*(c*x^2+b*x+a)^(1/2)/c^2/e

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Rubi [A]
time = 0.72, antiderivative size = 496, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {1660, 1667, 857, 635, 212, 738} \begin {gather*} -\frac {2 \left (-b^2 \left (a^2 e g^4+4 a c d f g^3+c^2 e f^4\right )+x \left (2 c^2 g^2 \left (a^2 (-g) (4 e f-d g)-3 a b f (e f-2 d g)+3 b^2 d f^2\right )-b c g^3 \left (-3 a^2 e g-4 a b (e f-d g)+4 b^2 d f\right )+b^3 g^4 (b d-a e)+c^3 f^2 (4 a g (2 e f-3 d g)-b f (4 d g+e f))+2 c^4 d f^4\right )+b c \left (a^2 g^3 (4 e f-3 d g)+2 a c f^2 g (3 d g+2 e f)+c^2 d f^4\right )+2 a c \left (a^2 e g^4-2 a c f g^2 (3 e f-2 d g)+c^2 f^3 (e f-4 d g)\right )+a b^3 d g^4\right )}{c^2 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}+\frac {g^3 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) (-3 b e g-2 c d g+8 c e f)}{2 c^{5/2} e^2}+\frac {g^4 \sqrt {a+b x+c x^2}}{c^2 e}+\frac {(e f-d g)^4 \tanh ^{-1}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{e^2 \left (a e^2-b d e+c d^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(f + g*x)^4/((d + e*x)*(a + b*x + c*x^2)^(3/2)),x]

[Out]

(-2*(a*b^3*d*g^4 - b^2*(c^2*e*f^4 + 4*a*c*d*f*g^3 + a^2*e*g^4) + 2*a*c*(a^2*e*g^4 + c^2*f^3*(e*f - 4*d*g) - 2*
a*c*f*g^2*(3*e*f - 2*d*g)) + b*c*(c^2*d*f^4 + a^2*g^3*(4*e*f - 3*d*g) + 2*a*c*f^2*g*(2*e*f + 3*d*g)) + (2*c^4*
d*f^4 + b^3*(b*d - a*e)*g^4 - b*c*g^3*(4*b^2*d*f - 3*a^2*e*g - 4*a*b*(e*f - d*g)) + 2*c^2*g^2*(3*b^2*d*f^2 - 3
*a*b*f*(e*f - 2*d*g) - a^2*g*(4*e*f - d*g)) + c^3*f^2*(4*a*g*(2*e*f - 3*d*g) - b*f*(e*f + 4*d*g)))*x))/(c^2*(b
^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*Sqrt[a + b*x + c*x^2]) + (g^4*Sqrt[a + b*x + c*x^2])/(c^2*e) + (g^3*(8*c*e
*f - 2*c*d*g - 3*b*e*g)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(2*c^(5/2)*e^2) + ((e*f - d*g)
^4*ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2])])/(e^2*(c*d^2
- b*d*e + a*e^2)^(3/2))

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 635

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 738

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 857

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + b*x + c*x^
2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&&  !IGtQ[m, 0]

Rule 1660

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = Polynomi
alQuotient[(d + e*x)^m*Pq, a + b*x + c*x^2, x], f = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2,
x], x, 0], g = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2, x], x, 1]}, Simp[(b*f - 2*a*g + (2*
c*f - b*g)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)), Int[(d
+ e*x)^m*(a + b*x + c*x^2)^(p + 1)*ExpandToSum[((p + 1)*(b^2 - 4*a*c)*Q)/(d + e*x)^m - ((2*p + 3)*(2*c*f - b*
g))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2
- b*d*e + a*e^2, 0] && LtQ[p, -1] && ILtQ[m, 0]

Rule 1667

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq
, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x)^(m + q - 1)*((a + b*x + c*x^2)^(p + 1)/(c*e^(q - 1)*(m
+ q + 2*p + 1))), x] + Dist[1/(c*e^q*(m + q + 2*p + 1)), Int[(d + e*x)^m*(a + b*x + c*x^2)^p*ExpandToSum[c*e^
q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(b*d*e*(p + 1) + a*e^2*(m + q
- 1) - c*d^2*(m + q + 2*p + 1) - e*(2*c*d - b*e)*(m + q + p)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p +
1, 0]] /; FreeQ[{a, b, c, d, e, m, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2
, 0] &&  !(IGtQ[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))

Rubi steps

\begin {align*} \int \frac {(f+g x)^4}{(d+e x) \left (a+b x+c x^2\right )^{3/2}} \, dx &=-\frac {2 \left (a b^3 d g^4-b^2 \left (c^2 e f^4+4 a c d f g^3+a^2 e g^4\right )+2 a c \left (a^2 e g^4+c^2 f^3 (e f-4 d g)-2 a c f g^2 (3 e f-2 d g)\right )+b c \left (c^2 d f^4+a^2 g^3 (4 e f-3 d g)+2 a c f^2 g (2 e f+3 d g)\right )+\left (2 c^4 d f^4+b^3 (b d-a e) g^4-b c g^3 \left (4 b^2 d f-3 a^2 e g-4 a b (e f-d g)\right )+2 c^2 g^2 \left (3 b^2 d f^2-3 a b f (e f-2 d g)-a^2 g (4 e f-d g)\right )+c^3 f^2 (4 a g (2 e f-3 d g)-b f (e f+4 d g))\right ) x\right )}{c^2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {a+b x+c x^2}}-\frac {2 \int \frac {-\frac {\left (b^2-4 a c\right ) \left (b d (b d-a e) g^4-c d g^3 (4 b d f-4 a e f+a d g)+c^2 f^2 \left (e^2 f^2-4 d e f g+6 d^2 g^2\right )\right )}{2 c^2 \left (c d^2-b d e+a e^2\right )}-\frac {\left (b^2-4 a c\right ) g^3 (4 c f-b g) x}{2 c^2}-\frac {\left (b^2-4 a c\right ) g^4 x^2}{2 c}}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{b^2-4 a c}\\ &=-\frac {2 \left (a b^3 d g^4-b^2 \left (c^2 e f^4+4 a c d f g^3+a^2 e g^4\right )+2 a c \left (a^2 e g^4+c^2 f^3 (e f-4 d g)-2 a c f g^2 (3 e f-2 d g)\right )+b c \left (c^2 d f^4+a^2 g^3 (4 e f-3 d g)+2 a c f^2 g (2 e f+3 d g)\right )+\left (2 c^4 d f^4+b^3 (b d-a e) g^4-b c g^3 \left (4 b^2 d f-3 a^2 e g-4 a b (e f-d g)\right )+2 c^2 g^2 \left (3 b^2 d f^2-3 a b f (e f-2 d g)-a^2 g (4 e f-d g)\right )+c^3 f^2 (4 a g (2 e f-3 d g)-b f (e f+4 d g))\right ) x\right )}{c^2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {a+b x+c x^2}}+\frac {g^4 \sqrt {a+b x+c x^2}}{c^2 e}-\frac {2 \int \frac {-\frac {\left (b^2-4 a c\right ) e \left (2 c^2 e^3 f^4-8 c^2 d e^2 f^3 g+12 c^2 d^2 e f^2 g^2-8 b c d^2 e f g^3+8 a c d e^2 f g^3-b c d^3 g^4+3 b^2 d^2 e g^4-2 a c d^2 e g^4-3 a b d e^2 g^4\right )}{4 c \left (c d^2-b d e+a e^2\right )}-\frac {\left (b^2-4 a c\right ) e g^3 (8 c e f-2 c d g-3 b e g) x}{4 c}}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{c \left (b^2-4 a c\right ) e^2}\\ &=-\frac {2 \left (a b^3 d g^4-b^2 \left (c^2 e f^4+4 a c d f g^3+a^2 e g^4\right )+2 a c \left (a^2 e g^4+c^2 f^3 (e f-4 d g)-2 a c f g^2 (3 e f-2 d g)\right )+b c \left (c^2 d f^4+a^2 g^3 (4 e f-3 d g)+2 a c f^2 g (2 e f+3 d g)\right )+\left (2 c^4 d f^4+b^3 (b d-a e) g^4-b c g^3 \left (4 b^2 d f-3 a^2 e g-4 a b (e f-d g)\right )+2 c^2 g^2 \left (3 b^2 d f^2-3 a b f (e f-2 d g)-a^2 g (4 e f-d g)\right )+c^3 f^2 (4 a g (2 e f-3 d g)-b f (e f+4 d g))\right ) x\right )}{c^2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {a+b x+c x^2}}+\frac {g^4 \sqrt {a+b x+c x^2}}{c^2 e}+\frac {(e f-d g)^4 \int \frac {1}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{e^2 \left (c d^2-b d e+a e^2\right )}+\frac {\left (g^3 (8 c e f-2 c d g-3 b e g)\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{2 c^2 e^2}\\ &=-\frac {2 \left (a b^3 d g^4-b^2 \left (c^2 e f^4+4 a c d f g^3+a^2 e g^4\right )+2 a c \left (a^2 e g^4+c^2 f^3 (e f-4 d g)-2 a c f g^2 (3 e f-2 d g)\right )+b c \left (c^2 d f^4+a^2 g^3 (4 e f-3 d g)+2 a c f^2 g (2 e f+3 d g)\right )+\left (2 c^4 d f^4+b^3 (b d-a e) g^4-b c g^3 \left (4 b^2 d f-3 a^2 e g-4 a b (e f-d g)\right )+2 c^2 g^2 \left (3 b^2 d f^2-3 a b f (e f-2 d g)-a^2 g (4 e f-d g)\right )+c^3 f^2 (4 a g (2 e f-3 d g)-b f (e f+4 d g))\right ) x\right )}{c^2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {a+b x+c x^2}}+\frac {g^4 \sqrt {a+b x+c x^2}}{c^2 e}-\frac {\left (2 (e f-d g)^4\right ) \text {Subst}\left (\int \frac {1}{4 c d^2-4 b d e+4 a e^2-x^2} \, dx,x,\frac {-b d+2 a e-(2 c d-b e) x}{\sqrt {a+b x+c x^2}}\right )}{e^2 \left (c d^2-b d e+a e^2\right )}+\frac {\left (g^3 (8 c e f-2 c d g-3 b e g)\right ) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{c^2 e^2}\\ &=-\frac {2 \left (a b^3 d g^4-b^2 \left (c^2 e f^4+4 a c d f g^3+a^2 e g^4\right )+2 a c \left (a^2 e g^4+c^2 f^3 (e f-4 d g)-2 a c f g^2 (3 e f-2 d g)\right )+b c \left (c^2 d f^4+a^2 g^3 (4 e f-3 d g)+2 a c f^2 g (2 e f+3 d g)\right )+\left (2 c^4 d f^4+b^3 (b d-a e) g^4-b c g^3 \left (4 b^2 d f-3 a^2 e g-4 a b (e f-d g)\right )+2 c^2 g^2 \left (3 b^2 d f^2-3 a b f (e f-2 d g)-a^2 g (4 e f-d g)\right )+c^3 f^2 (4 a g (2 e f-3 d g)-b f (e f+4 d g))\right ) x\right )}{c^2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {a+b x+c x^2}}+\frac {g^4 \sqrt {a+b x+c x^2}}{c^2 e}+\frac {g^3 (8 c e f-2 c d g-3 b e g) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 c^{5/2} e^2}+\frac {(e f-d g)^4 \tanh ^{-1}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{e^2 \left (c d^2-b d e+a e^2\right )^{3/2}}\\ \end {align*}

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Mathematica [A]
time = 11.49, size = 587, normalized size = 1.18 \begin {gather*} \frac {-\frac {2 e \left (-3 b^4 d e g^4 x+b^3 g^3 (3 a e g (-d+e x)+c d x (8 e f+d g-e g x))+b^2 \left (3 a^2 e^2 g^4+c^2 \left (2 e^2 f^4-12 d e f^2 g^2 x+d^2 g^4 x^2\right )+a c g^3 \left (d^2 g+e^2 x (-8 f+g x)+4 d e (2 f+3 g x)\right )\right )-2 b c \left (a^2 e g^3 (4 e f-5 d g+5 e g x)+c^2 e f^3 (-e f x+d (f-4 g x))+2 a c g \left (d^2 g^3 x+e^2 f^2 (2 f-3 g x)+d e g \left (3 f^2+6 f g x-g^2 x^2\right )\right )\right )-4 c \left (2 a^3 e^2 g^4+c^3 d e f^4 x+a c^2 \left (d^2 g^4 x^2-2 d e f^2 g (2 f+3 g x)+e^2 f^3 (f+4 g x)\right )+a^2 c g^2 \left (d^2 g^2+d e g (4 f+g x)+e^2 \left (-6 f^2-4 f g x+g^2 x^2\right )\right )\right )\right )}{c^2 \left (b^2-4 a c\right ) \left (-c d^2+e (b d-a e)\right ) \sqrt {a+x (b+c x)}}+\frac {2 (e f-d g)^4 \log (d+e x)}{\left (c d^2+e (-b d+a e)\right )^{3/2}}+\frac {g^3 (8 c e f-2 c d g-3 b e g) \log \left (b+2 c x+2 \sqrt {c} \sqrt {a+x (b+c x)}\right )}{c^{5/2}}-\frac {2 (e f-d g)^4 \log \left (-b d+2 a e-2 c d x+b e x+2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}\right )}{\left (c d^2+e (-b d+a e)\right )^{3/2}}}{2 e^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(f + g*x)^4/((d + e*x)*(a + b*x + c*x^2)^(3/2)),x]

[Out]

((-2*e*(-3*b^4*d*e*g^4*x + b^3*g^3*(3*a*e*g*(-d + e*x) + c*d*x*(8*e*f + d*g - e*g*x)) + b^2*(3*a^2*e^2*g^4 + c
^2*(2*e^2*f^4 - 12*d*e*f^2*g^2*x + d^2*g^4*x^2) + a*c*g^3*(d^2*g + e^2*x*(-8*f + g*x) + 4*d*e*(2*f + 3*g*x)))
- 2*b*c*(a^2*e*g^3*(4*e*f - 5*d*g + 5*e*g*x) + c^2*e*f^3*(-(e*f*x) + d*(f - 4*g*x)) + 2*a*c*g*(d^2*g^3*x + e^2
*f^2*(2*f - 3*g*x) + d*e*g*(3*f^2 + 6*f*g*x - g^2*x^2))) - 4*c*(2*a^3*e^2*g^4 + c^3*d*e*f^4*x + a*c^2*(d^2*g^4
*x^2 - 2*d*e*f^2*g*(2*f + 3*g*x) + e^2*f^3*(f + 4*g*x)) + a^2*c*g^2*(d^2*g^2 + d*e*g*(4*f + g*x) + e^2*(-6*f^2
- 4*f*g*x + g^2*x^2)))))/(c^2*(b^2 - 4*a*c)*(-(c*d^2) + e*(b*d - a*e))*Sqrt[a + x*(b + c*x)]) + (2*(e*f - d*g
)^4*Log[d + e*x])/(c*d^2 + e*(-(b*d) + a*e))^(3/2) + (g^3*(8*c*e*f - 2*c*d*g - 3*b*e*g)*Log[b + 2*c*x + 2*Sqrt
[c]*Sqrt[a + x*(b + c*x)]])/c^(5/2) - (2*(e*f - d*g)^4*Log[-(b*d) + 2*a*e - 2*c*d*x + b*e*x + 2*Sqrt[c*d^2 + e
*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)]])/(c*d^2 + e*(-(b*d) + a*e))^(3/2))/(2*e^2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1025\) vs. \(2(474)=948\).
time = 0.16, size = 1026, normalized size = 2.07

method	result	size
default	\(-\frac {g \left (-e^{3} g^{3} \left (\frac {x^{2}}{c \sqrt {c \,x^{2}+b x +a}}-\frac {3 b \left (-\frac {x}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}\right )}{2 c}-\frac {2 a \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{c}\right )+\left (d \,e^{2} g^{3}-4 e^{3} f \,g^{2}\right ) \left (-\frac {x}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}\right )+\left (-d^{2} e \,g^{3}+4 d \,e^{2} f \,g^{2}-6 e^{3} f^{2} g \right ) \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )+\frac {2 d^{3} g^{3} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {8 d^{2} e f \,g^{2} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\frac {12 d \,e^{2} f^{2} g \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {8 e^{3} f^{3} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{e^{4}}+\frac {\left (d^{4} g^{4}-4 d^{3} e f \,g^{3}+6 d^{2} e^{2} f^{2} g^{2}-4 d \,e^{3} f^{3} g +e^{4} f^{4}\right ) \left (\frac {e^{2}}{\left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (e b -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}-\frac {\left (e b -2 c d \right ) e \left (2 c \left (x +\frac {d}{e}\right )+\frac {e b -2 c d}{e}\right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (\frac {4 c \left (a \,e^{2}-b d e +c \,d^{2}\right )}{e^{2}}-\frac {\left (e b -2 c d \right )^{2}}{e^{2}}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (e b -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}-\frac {e^{2} \ln \left (\frac {\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (e b -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (e b -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}\right )}{e^{5}}\)	\(1026\)
risch	\(\text {Expression too large to display}\)	\(4958\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x+f)^4/(e*x+d)/(c*x^2+b*x+a)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-g/e^4*(-e^3*g^3*(x^2/c/(c*x^2+b*x+a)^(1/2)-3/2*b/c*(-x/c/(c*x^2+b*x+a)^(1/2)-1/2*b/c*(-1/c/(c*x^2+b*x+a)^(1/2
)-b/c*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2))+1/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))-2*a/c*
(-1/c/(c*x^2+b*x+a)^(1/2)-b/c*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)))+(d*e^2*g^3-4*e^3*f*g^2)*(-x/c/(c*x^2
+b*x+a)^(1/2)-1/2*b/c*(-1/c/(c*x^2+b*x+a)^(1/2)-b/c*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2))+1/c^(3/2)*ln((1
/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))+(-d^2*e*g^3+4*d*e^2*f*g^2-6*e^3*f^2*g)*(-1/c/(c*x^2+b*x+a)^(1/2)-b/c*(
2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2))+2*d^3*g^3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)-8*d^2*e*f*g^2*(2
*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+12*d*e^2*f^2*g*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)-8*e^3*f^3*(2*
c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2))+(d^4*g^4-4*d^3*e*f*g^3+6*d^2*e^2*f^2*g^2-4*d*e^3*f^3*g+e^4*f^4)/e^5*(1
/(a*e^2-b*d*e+c*d^2)*e^2/(c*(x+d/e)^2+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)-(b*e-2*c*d)*e/(a*e^
2-b*d*e+c*d^2)*(2*c*(x+d/e)+(b*e-2*c*d)/e)/(4*c*(a*e^2-b*d*e+c*d^2)/e^2-(b*e-2*c*d)^2/e^2)/(c*(x+d/e)^2+(b*e-2
*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)-1/(a*e^2-b*d*e+c*d^2)*e^2/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2
*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(x+d/e)^2+(b*e-2*c*d)/e*(x
+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e)))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^4/(e*x+d)/(c*x^2+b*x+a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume((%e^-1*b-2*%e^-2*c*d)^2>0)', s
ee `assume?`

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^4/(e*x+d)/(c*x^2+b*x+a)^(3/2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (f + g x\right )^{4}}{\left (d + e x\right ) \left (a + b x + c x^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)**4/(e*x+d)/(c*x**2+b*x+a)**(3/2),x)

[Out]

Integral((f + g*x)**4/((d + e*x)*(a + b*x + c*x**2)**(3/2)), x)

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^4/(e*x+d)/(c*x^2+b*x+a)^(3/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Error: Bad Argument Type

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (f+g\,x\right )}^4}{\left (d+e\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f + g*x)^4/((d + e*x)*(a + b*x + c*x^2)^(3/2)),x)

[Out]

int((f + g*x)^4/((d + e*x)*(a + b*x + c*x^2)^(3/2)), x)

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