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

Optimal. Leaf size=532 \[ -\frac {\tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (4 c e (2 c d-b e) \left (-4 c d g^2 (a e g-2 b d g+6 b e f)+5 b^2 d e g^3+16 c^2 e^2 f^3\right )-2 g \left (-2 c e (b d-a e)-\frac {b^2 e^2}{2}+4 c^2 d^2\right ) \left (-4 c e g (a e g-2 b d g+6 b e f)+5 b^2 e^2 g^2+16 c^2 \left (d^2 g^2-3 d e f g+3 e^2 f^2\right )\right )\right )}{128 c^{7/2} e^5}+\frac {\sqrt {a+b x+c x^2} \left (2 c e g x \left (-4 c e g (a e g-2 b d g+6 b e f)+5 b^2 e^2 g^2+16 c^2 \left (d^2 g^2-3 d e f g+3 e^2 f^2\right )\right )-4 b c e^2 g^2 (a e g-2 b d g+6 b e f)+5 b^3 e^3 g^3+16 b c^2 e g \left (d^2 g^2-3 d e f g+3 e^2 f^2\right )+64 c^3 (e f-d g)^3\right )}{64 c^3 e^4}+\frac {g^2 \left (a+b x+c x^2\right )^{3/2} (-5 b e g-14 c d g+24 c e f)}{24 c^2 e^2}+\frac {(e f-d g)^3 \sqrt {a e^2-b d e+c d^2} \tanh ^{-1}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{e^5}+\frac {g^3 (d+e x) \left (a+b x+c x^2\right )^{3/2}}{4 c e^2} \]

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Rubi [A]  time = 1.70, antiderivative size = 532, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {1653, 814, 843, 621, 206, 724} \begin {gather*} \frac {\sqrt {a+b x+c x^2} \left (2 c e g x \left (-4 c e g (a e g-2 b d g+6 b e f)+5 b^2 e^2 g^2+16 c^2 \left (d^2 g^2-3 d e f g+3 e^2 f^2\right )\right )-4 b c e^2 g^2 (a e g-2 b d g+6 b e f)+5 b^3 e^3 g^3+16 b c^2 e g \left (d^2 g^2-3 d e f g+3 e^2 f^2\right )+64 c^3 (e f-d g)^3\right )}{64 c^3 e^4}-\frac {\tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (4 c e (2 c d-b e) \left (-4 c d g^2 (a e g-2 b d g+6 b e f)+5 b^2 d e g^3+16 c^2 e^2 f^3\right )-2 g \left (-2 c e (b d-a e)-\frac {b^2 e^2}{2}+4 c^2 d^2\right ) \left (-4 c e g (a e g-2 b d g+6 b e f)+5 b^2 e^2 g^2+16 c^2 \left (d^2 g^2-3 d e f g+3 e^2 f^2\right )\right )\right )}{128 c^{7/2} e^5}+\frac {g^2 \left (a+b x+c x^2\right )^{3/2} (-5 b e g-14 c d g+24 c e f)}{24 c^2 e^2}+\frac {(e f-d g)^3 \sqrt {a e^2-b d e+c d^2} \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^5}+\frac {g^3 (d+e x) \left (a+b x+c x^2\right )^{3/2}}{4 c e^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((5*b^3*e^3*g^3 + 64*c^3*(e*f - d*g)^3 - 4*b*c*e^2*g^2*(6*b*e*f - 2*b*d*g + a*e*g) + 16*b*c^2*e*g*(3*e^2*f^2 -
3*d*e*f*g + d^2*g^2) + 2*c*e*g*(5*b^2*e^2*g^2 - 4*c*e*g*(6*b*e*f - 2*b*d*g + a*e*g) + 16*c^2*(3*e^2*f^2 - 3*d
*e*f*g + d^2*g^2))*x)*Sqrt[a + b*x + c*x^2])/(64*c^3*e^4) + (g^2*(24*c*e*f - 14*c*d*g - 5*b*e*g)*(a + b*x + c*
x^2)^(3/2))/(24*c^2*e^2) + (g^3*(d + e*x)*(a + b*x + c*x^2)^(3/2))/(4*c*e^2) - ((4*c*e*(2*c*d - b*e)*(16*c^2*e
^2*f^3 + 5*b^2*d*e*g^3 - 4*c*d*g^2*(6*b*e*f - 2*b*d*g + a*e*g)) - 2*(4*c^2*d^2 - (b^2*e^2)/2 - 2*c*e*(b*d - a*
e))*g*(5*b^2*e^2*g^2 - 4*c*e*g*(6*b*e*f - 2*b*d*g + a*e*g) + 16*c^2*(3*e^2*f^2 - 3*d*e*f*g + d^2*g^2)))*ArcTan
h[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(128*c^(7/2)*e^5) + (Sqrt[c*d^2 - b*d*e + a*e^2]*(e*f - d*g)
^3*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^5

Rule 206

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

Rule 621

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 724

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 814

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*(a + b*x + c*x^
2)^p)/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), x] - Dist[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), Int[(d + e*x)^m*(a
+ b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2*a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p -
c*d - 2*c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c^2*d^2*(1 + 2*p) - c*e*(b*
d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0
] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] ||  !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])
) &&  !ILtQ[m + 2*p, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 843

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 1653

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)^3 \sqrt {a+b x+c x^2}}{d+e x} \, dx &=\frac {g^3 (d+e x) \left (a+b x+c x^2\right )^{3/2}}{4 c e^2}+\frac {\int \frac {\sqrt {a+b x+c x^2} \left (\frac {1}{2} e \left (8 c e^2 f^3-d (3 b d+2 a e) g^3\right )-e g \left (e (4 b d+a e) g^2-3 c \left (4 e^2 f^2-d^2 g^2\right )\right ) x+\frac {1}{2} e^2 g^2 (24 c e f-14 c d g-5 b e g) x^2\right )}{d+e x} \, dx}{4 c e^3}\\ &=\frac {g^2 (24 c e f-14 c d g-5 b e g) \left (a+b x+c x^2\right )^{3/2}}{24 c^2 e^2}+\frac {g^3 (d+e x) \left (a+b x+c x^2\right )^{3/2}}{4 c e^2}+\frac {\int \frac {\left (\frac {3}{4} e^3 \left (16 c^2 e^2 f^3+5 b^2 d e g^3-4 c d g^2 (6 b e f-2 b d g+a e g)\right )+\frac {3}{4} e^3 g \left (5 b^2 e^2 g^2-4 c e g (6 b e f-2 b d g+a e g)+16 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{d+e x} \, dx}{12 c^2 e^5}\\ &=\frac {\left (5 b^3 e^3 g^3+64 c^3 (e f-d g)^3-4 b c e^2 g^2 (6 b e f-2 b d g+a e g)+16 b c^2 e g \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )+2 c e g \left (5 b^2 e^2 g^2-4 c e g (6 b e f-2 b d g+a e g)+16 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{64 c^3 e^4}+\frac {g^2 (24 c e f-14 c d g-5 b e g) \left (a+b x+c x^2\right )^{3/2}}{24 c^2 e^2}+\frac {g^3 (d+e x) \left (a+b x+c x^2\right )^{3/2}}{4 c e^2}-\frac {\int \frac {\frac {3}{8} e^3 \left (4 c e (b d-2 a e) \left (16 c^2 e^2 f^3+5 b^2 d e g^3-4 c d g^2 (6 b e f-2 b d g+a e g)\right )-d \left (4 b c d-b^2 e-4 a c e\right ) g \left (5 b^2 e^2 g^2-4 c e g (6 b e f-2 b d g+a e g)+16 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )\right )+\frac {3}{8} e^3 \left (4 c e (2 c d-b e) \left (16 c^2 e^2 f^3+5 b^2 d e g^3-4 c d g^2 (6 b e f-2 b d g+a e g)\right )-2 \left (4 c^2 d^2-\frac {b^2 e^2}{2}-2 c e (b d-a e)\right ) g \left (5 b^2 e^2 g^2-4 c e g (6 b e f-2 b d g+a e g)+16 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )\right ) x}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{48 c^3 e^7}\\ &=\frac {\left (5 b^3 e^3 g^3+64 c^3 (e f-d g)^3-4 b c e^2 g^2 (6 b e f-2 b d g+a e g)+16 b c^2 e g \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )+2 c e g \left (5 b^2 e^2 g^2-4 c e g (6 b e f-2 b d g+a e g)+16 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{64 c^3 e^4}+\frac {g^2 (24 c e f-14 c d g-5 b e g) \left (a+b x+c x^2\right )^{3/2}}{24 c^2 e^2}+\frac {g^3 (d+e x) \left (a+b x+c x^2\right )^{3/2}}{4 c e^2}+\frac {\left (\left (c d^2-b d e+a e^2\right ) (e f-d g)^3\right ) \int \frac {1}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{e^5}-\frac {\left (4 c e (2 c d-b e) \left (16 c^2 e^2 f^3+5 b^2 d e g^3-4 c d g^2 (6 b e f-2 b d g+a e g)\right )-2 \left (4 c^2 d^2-\frac {b^2 e^2}{2}-2 c e (b d-a e)\right ) g \left (5 b^2 e^2 g^2-4 c e g (6 b e f-2 b d g+a e g)+16 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{128 c^3 e^5}\\ &=\frac {\left (5 b^3 e^3 g^3+64 c^3 (e f-d g)^3-4 b c e^2 g^2 (6 b e f-2 b d g+a e g)+16 b c^2 e g \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )+2 c e g \left (5 b^2 e^2 g^2-4 c e g (6 b e f-2 b d g+a e g)+16 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{64 c^3 e^4}+\frac {g^2 (24 c e f-14 c d g-5 b e g) \left (a+b x+c x^2\right )^{3/2}}{24 c^2 e^2}+\frac {g^3 (d+e x) \left (a+b x+c x^2\right )^{3/2}}{4 c e^2}-\frac {\left (2 \left (c d^2-b d e+a e^2\right ) (e f-d g)^3\right ) \operatorname {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^5}-\frac {\left (4 c e (2 c d-b e) \left (16 c^2 e^2 f^3+5 b^2 d e g^3-4 c d g^2 (6 b e f-2 b d g+a e g)\right )-2 \left (4 c^2 d^2-\frac {b^2 e^2}{2}-2 c e (b d-a e)\right ) g \left (5 b^2 e^2 g^2-4 c e g (6 b e f-2 b d g+a e g)+16 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{64 c^3 e^5}\\ &=\frac {\left (5 b^3 e^3 g^3+64 c^3 (e f-d g)^3-4 b c e^2 g^2 (6 b e f-2 b d g+a e g)+16 b c^2 e g \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )+2 c e g \left (5 b^2 e^2 g^2-4 c e g (6 b e f-2 b d g+a e g)+16 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{64 c^3 e^4}+\frac {g^2 (24 c e f-14 c d g-5 b e g) \left (a+b x+c x^2\right )^{3/2}}{24 c^2 e^2}+\frac {g^3 (d+e x) \left (a+b x+c x^2\right )^{3/2}}{4 c e^2}-\frac {\left (4 c e (2 c d-b e) \left (16 c^2 e^2 f^3+5 b^2 d e g^3-4 c d g^2 (6 b e f-2 b d g+a e g)\right )-2 \left (4 c^2 d^2-\frac {b^2 e^2}{2}-2 c e (b d-a e)\right ) g \left (5 b^2 e^2 g^2-4 c e g (6 b e f-2 b d g+a e g)+16 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{128 c^{7/2} e^5}+\frac {\sqrt {c d^2-b d e+a e^2} (e f-d g)^3 \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^5}\\ \end {align*}

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Mathematica [A]  time = 1.01, size = 559, normalized size = 1.05 \begin {gather*} \frac {-\frac {24 e^2 g (b g-2 c f) (e f-d g) \left (2 \sqrt {c} (b+2 c x) \sqrt {a+x (b+c x)}-\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )\right )}{c^{5/2}}-\frac {48 e g \left (b^2-4 a c\right ) (e f-d g)^2 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )}{c^{3/2}}+\frac {e^3 g \left (3 \left (-4 c g (a g+4 b f)+5 b^2 g^2+16 c^2 f^2\right ) \left (2 \sqrt {c} (b+2 c x) \sqrt {a+x (b+c x)}-\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )\right )+80 c^{3/2} g (a+x (b+c x))^{3/2} (2 c f-b g)\right )}{c^{7/2}}+\frac {192 (e f-d g)^3 \left (2 \sqrt {c} \sqrt {e (a e-b d)+c d^2} \tanh ^{-1}\left (\frac {-2 a e+b (d-e x)+2 c d x}{2 \sqrt {a+x (b+c x)} \sqrt {e (a e-b d)+c d^2}}\right )+(b e-2 c d) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )\right )}{\sqrt {c} e}+\frac {128 e^2 g^2 (a+x (b+c x))^{3/2} (e f-d g)}{c}+\frac {96 e g (b+2 c x) \sqrt {a+x (b+c x)} (e f-d g)^2}{c}+384 \sqrt {a+x (b+c x)} (e f-d g)^3+\frac {96 e^3 g^2 (f+g x) (a+x (b+c x))^{3/2}}{c}}{384 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(384*(e*f - d*g)^3*Sqrt[a + x*(b + c*x)] + (96*e*g*(e*f - d*g)^2*(b + 2*c*x)*Sqrt[a + x*(b + c*x)])/c + (128*e
^2*g^2*(e*f - d*g)*(a + x*(b + c*x))^(3/2))/c + (96*e^3*g^2*(f + g*x)*(a + x*(b + c*x))^(3/2))/c - (48*(b^2 -
4*a*c)*e*g*(e*f - d*g)^2*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/c^(3/2) - (24*e^2*g*(-2*c*f +
b*g)*(e*f - d*g)*(2*Sqrt[c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)] - (b^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*
Sqrt[a + x*(b + c*x)])]))/c^(5/2) + (e^3*g*(80*c^(3/2)*g*(2*c*f - b*g)*(a + x*(b + c*x))^(3/2) + 3*(16*c^2*f^2
+ 5*b^2*g^2 - 4*c*g*(4*b*f + a*g))*(2*Sqrt[c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)] - (b^2 - 4*a*c)*ArcTanh[(b +
2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])))/c^(7/2) + (192*(e*f - d*g)^3*((-2*c*d + b*e)*ArcTanh[(b + 2*c*x)/
(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])] + 2*Sqrt[c]*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*ArcTanh[(-2*a*e + 2*c*d*x + b*(d
- e*x))/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])]))/(Sqrt[c]*e))/(384*e^4)

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IntegrateAlgebraic [A]  time = 4.59, size = 788, normalized size = 1.48 \begin {gather*} \frac {\log \left (-2 \sqrt {c} \sqrt {a+b x+c x^2}+b+2 c x\right ) \left (16 a^2 c^2 e^4 g^3-24 a b^2 c e^4 g^3-32 a b c^2 d e^3 g^3+96 a b c^2 e^4 f g^2-64 a c^3 d^2 e^2 g^3+192 a c^3 d e^3 f g^2-192 a c^3 e^4 f^2 g+5 b^4 e^4 g^3+8 b^3 c d e^3 g^3-24 b^3 c e^4 f g^2+16 b^2 c^2 d^2 e^2 g^3-48 b^2 c^2 d e^3 f g^2+48 b^2 c^2 e^4 f^2 g+64 b c^3 d^3 e g^3-192 b c^3 d^2 e^2 f g^2+192 b c^3 d e^3 f^2 g-64 b c^3 e^4 f^3-128 c^4 d^4 g^3+384 c^4 d^3 e f g^2-384 c^4 d^2 e^2 f^2 g+128 c^4 d e^3 f^3\right )}{128 c^{7/2} e^5}+\frac {\sqrt {a+b x+c x^2} \left (-52 a b c e^3 g^3-64 a c^2 d e^2 g^3+192 a c^2 e^3 f g^2+24 a c^2 e^3 g^3 x+15 b^3 e^3 g^3+24 b^2 c d e^2 g^3-72 b^2 c e^3 f g^2-10 b^2 c e^3 g^3 x+48 b c^2 d^2 e g^3-144 b c^2 d e^2 f g^2-16 b c^2 d e^2 g^3 x+144 b c^2 e^3 f^2 g+48 b c^2 e^3 f g^2 x+8 b c^2 e^3 g^3 x^2-192 c^3 d^3 g^3+576 c^3 d^2 e f g^2+96 c^3 d^2 e g^3 x-576 c^3 d e^2 f^2 g-288 c^3 d e^2 f g^2 x-64 c^3 d e^2 g^3 x^2+192 c^3 e^3 f^3+288 c^3 e^3 f^2 g x+192 c^3 e^3 f g^2 x^2+48 c^3 e^3 g^3 x^3\right )}{192 c^3 e^4}-\frac {2 \left (d^3 g^3-3 d^2 e f g^2+3 d e^2 f^2 g-e^3 f^3\right ) \sqrt {-a e^2+b d e-c d^2} \tan ^{-1}\left (\frac {-e \sqrt {a+b x+c x^2}+\sqrt {c} d+\sqrt {c} e x}{\sqrt {-a e^2+b d e-c d^2}}\right )}{e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a + b*x + c*x^2]*(192*c^3*e^3*f^3 - 576*c^3*d*e^2*f^2*g + 144*b*c^2*e^3*f^2*g + 576*c^3*d^2*e*f*g^2 - 14
4*b*c^2*d*e^2*f*g^2 - 72*b^2*c*e^3*f*g^2 + 192*a*c^2*e^3*f*g^2 - 192*c^3*d^3*g^3 + 48*b*c^2*d^2*e*g^3 + 24*b^2
*c*d*e^2*g^3 - 64*a*c^2*d*e^2*g^3 + 15*b^3*e^3*g^3 - 52*a*b*c*e^3*g^3 + 288*c^3*e^3*f^2*g*x - 288*c^3*d*e^2*f*
g^2*x + 48*b*c^2*e^3*f*g^2*x + 96*c^3*d^2*e*g^3*x - 16*b*c^2*d*e^2*g^3*x - 10*b^2*c*e^3*g^3*x + 24*a*c^2*e^3*g
^3*x + 192*c^3*e^3*f*g^2*x^2 - 64*c^3*d*e^2*g^3*x^2 + 8*b*c^2*e^3*g^3*x^2 + 48*c^3*e^3*g^3*x^3))/(192*c^3*e^4)
- (2*Sqrt[-(c*d^2) + b*d*e - a*e^2]*(-(e^3*f^3) + 3*d*e^2*f^2*g - 3*d^2*e*f*g^2 + d^3*g^3)*ArcTan[(Sqrt[c]*d
+ Sqrt[c]*e*x - e*Sqrt[a + b*x + c*x^2])/Sqrt[-(c*d^2) + b*d*e - a*e^2]])/e^5 + ((128*c^4*d*e^3*f^3 - 64*b*c^3
*e^4*f^3 - 384*c^4*d^2*e^2*f^2*g + 192*b*c^3*d*e^3*f^2*g + 48*b^2*c^2*e^4*f^2*g - 192*a*c^3*e^4*f^2*g + 384*c^
4*d^3*e*f*g^2 - 192*b*c^3*d^2*e^2*f*g^2 - 48*b^2*c^2*d*e^3*f*g^2 + 192*a*c^3*d*e^3*f*g^2 - 24*b^3*c*e^4*f*g^2
+ 96*a*b*c^2*e^4*f*g^2 - 128*c^4*d^4*g^3 + 64*b*c^3*d^3*e*g^3 + 16*b^2*c^2*d^2*e^2*g^3 - 64*a*c^3*d^2*e^2*g^3
+ 8*b^3*c*d*e^3*g^3 - 32*a*b*c^2*d*e^3*g^3 + 5*b^4*e^4*g^3 - 24*a*b^2*c*e^4*g^3 + 16*a^2*c^2*e^4*g^3)*Log[b +
2*c*x - 2*Sqrt[c]*Sqrt[a + b*x + c*x^2]])/(128*c^(7/2)*e^5)

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fricas [F(-1)]  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)^3*(c*x^2+b*x+a)^(1/2)/(e*x+d),x, algorithm="fricas")

[Out]

Timed out

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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)^3*(c*x^2+b*x+a)^(1/2)/(e*x+d),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Erro
r: Bad Argument Type

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maple [B]  time = 0.04, size = 3941, normalized size = 7.41 \begin {gather*} \text {output too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x+f)^3*(c*x^2+b*x+a)^(1/2)/(e*x+d),x)

[Out]

-1/e^4*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*g^3*d^3+1/e*((x+d/e)^2*c+(b*e-2*c*d)/
e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*f^3+3/e^3*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)
^(1/2)*d^2*f*g^2-1/16*g^3/e/c^2*a*(c*x^2+b*x+a)^(1/2)*b+1/8*g^3/e^2*b^2/c^2*(c*x^2+b*x+a)^(1/2)*d-3/8*g^2/e*b^
2/c^2*(c*x^2+b*x+a)^(1/2)*f-3/8*g/e*f^2/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*b^2-3/2/e^2*ln((1/
2*(b*e-2*c*d)/e+(x+d/e)*c)/c^(1/2)+((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/c^(1/2)*
b*d*f^2*g-3/e^3/((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)*((x+d/e)^2*c+(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))*a*d^2*f*g^
2+3/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)*((x+d/e)^2*c+(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))*a*d*f^2*g+3/e^4/((
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)*((x+d/e)^2*c+(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))*b*d^3*f*g^2-3/e^3/((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)*((
x+d/e)^2*c+(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))*b*d^2*f^2*g-3/e^5/((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)*((x+d/e)^2
*c+(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))*c*d^4*f*g^2+3/e^4/((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)*((x+d/e)^2*c+(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))*c*d^3*f^2*g-3/4*g^2/e^2*d*f/c*(c*x^2+b*x+a)^(1/2)*b-
3/2*g^2/e^2*d*f/c^(1/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a+3/8*g^2/e^2*d*f/c^(3/2)*ln((1/2*b+c*x)/c
^(1/2)+(c*x^2+b*x+a)^(1/2))*b^2-3/4*g^2/e*b/c*x*(c*x^2+b*x+a)^(1/2)*f+1/4*g^3/e^2*b/c^(3/2)*ln((1/2*b+c*x)/c^(
1/2)+(c*x^2+b*x+a)^(1/2))*a*d-1/2/e^4*ln((1/2*(b*e-2*c*d)/e+(x+d/e)*c)/c^(1/2)+((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d
/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/c^(1/2)*b*g^3*d^3-3/e^4*ln((1/2*(b*e-2*c*d)/e+(x+d/e)*c)/c^(1/2)+((x+d/e)^
2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))*c^(1/2)*d^3*f*g^2+3/e^3*ln((1/2*(b*e-2*c*d)/e+(x+d/e
)*c)/c^(1/2)+((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))*c^(1/2)*d^2*f^2*g+1/e^4/((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
)*((x+d/e)^2*c+(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))*a*g^3*d^3-1/e^5/((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)*((x+d/e)
^2*c+(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))*b*d^4*g^3+1/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)*((x+d/e)^2*c+(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))*b*d*f^3+1/e^6/((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)*((x+d/e)^2*c+(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))*c*d^5*g^3-1/e^3/((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)*((x+d/e)^2*c+(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))*c*d^2*f^3+3/16*g^2/e*b^3/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2
))*f-3/2*g^2/e^2*d*f*x*(c*x^2+b*x+a)^(1/2)-1/8*g^3/e/c*a*x*(c*x^2+b*x+a)^(1/2)-3/e^2*((x+d/e)^2*c+(b*e-2*c*d)/
e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*d*f^2*g+1/2/e*ln((1/2*(b*e-2*c*d)/e+(x+d/e)*c)/c^(1/2)+((x+d/e)^2*c+(
b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/c^(1/2)*b*f^3+1/e^5*ln((1/2*(b*e-2*c*d)/e+(x+d/e)*c)/c^(1
/2)+((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))*c^(1/2)*d^4*g^3-1/e^2*ln((1/2*(b*e-2*c*
d)/e+(x+d/e)*c)/c^(1/2)+((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))*c^(1/2)*d*f^3-1/e/(
(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)*((x+d/e)^2*c+(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))*a*f^3+5/32*g^3/e*b^2/c^2*x*
(c*x^2+b*x+a)^(1/2)+3/16*g^3/e*b^2/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a+1/4*g^3/e^3*d^2/c*(c*
x^2+b*x+a)^(1/2)*b+1/2*g^3/e^3*d^2/c^(1/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a-1/8*g^3/e^3*d^2/c^(3/
2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*b^2+3/4*g/e*f^2/c*(c*x^2+b*x+a)^(1/2)*b+3/2*g/e*f^2/c^(1/2)*ln(
(1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a-1/16*g^3/e^2*b^3/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)
)*d-3/4*g^2/e*b/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a*f+1/4*g^3/e^2*b/c*x*(c*x^2+b*x+a)^(1/2)*
d+3/2/e^3*ln((1/2*(b*e-2*c*d)/e+(x+d/e)*c)/c^(1/2)+((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)
^(1/2))/c^(1/2)*b*d^2*f*g^2+3/2*g/e*f^2*x*(c*x^2+b*x+a)^(1/2)+1/2*g^3/e^3*d^2*x*(c*x^2+b*x+a)^(1/2)+1/4*g^3/e*
x*(c*x^2+b*x+a)^(3/2)/c-5/24*g^3/e*b/c^2*(c*x^2+b*x+a)^(3/2)+5/64*g^3/e*b^3/c^3*(c*x^2+b*x+a)^(1/2)-5/128*g^3/
e*b^4/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-1/8*g^3/e/c^(3/2)*a^2*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+
b*x+a)^(1/2))-1/3*g^3/e^2*(c*x^2+b*x+a)^(3/2)/c*d+g^2/e*(c*x^2+b*x+a)^(3/2)/c*f

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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)^3*(c*x^2+b*x+a)^(1/2)/(e*x+d),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(b*e-2*c*d>0)', see `assume?` f
or more details)Is b*e-2*c*d zero or nonzero?

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((f + g*x)^3*(a + b*x + c*x^2)^(1/2))/(d + e*x),x)

[Out]

int(((f + g*x)^3*(a + b*x + c*x^2)^(1/2))/(d + e*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)**3*(c*x**2+b*x+a)**(1/2)/(e*x+d),x)

[Out]

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

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