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

Optimal. Leaf size=491 \[ \frac {\tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-4 c g \left (3 b e f^2-a g (3 e f-d g)\right )+b g^2 (-4 a e g+b d g+3 b e f)+8 c^2 e f^3\right )}{8 \sqrt {c} e g^3 (e f-d g)}+\frac {\sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}{e^2 (e f-d g)}+\frac {\left (a e^2-b d e+c d^2\right )^{3/2} \tanh ^{-1}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{e^3 (e f-d g)}-\frac {(2 c d-b e) \left (a e^2-b d e+c d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {c} e^3 (e f-d g)}-\frac {\sqrt {a+b x+c x^2} \left (-g (-4 a e g-b d g+5 b e f)-2 c g x (e f-d g)+4 c e f^2\right )}{4 e g^2 (e f-d g)}-\frac {\left (a g^2-b f g+c f^2\right )^{3/2} \tanh ^{-1}\left (\frac {-2 a g+x (2 c f-b g)+b f}{2 \sqrt {a+b x+c x^2} \sqrt {a g^2-b f g+c f^2}}\right )}{g^3 (e f-d g)} \]

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Rubi [A]  time = 0.84, antiderivative size = 491, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {895, 734, 843, 621, 206, 724, 814} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-4 c g \left (3 b e f^2-a g (3 e f-d g)\right )+b g^2 (-4 a e g+b d g+3 b e f)+8 c^2 e f^3\right )}{8 \sqrt {c} e g^3 (e f-d g)}+\frac {\sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}{e^2 (e f-d g)}+\frac {\left (a e^2-b d e+c d^2\right )^{3/2} \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^3 (e f-d g)}-\frac {(2 c d-b e) \left (a e^2-b d e+c d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {c} e^3 (e f-d g)}-\frac {\sqrt {a+b x+c x^2} \left (-g (-4 a e g-b d g+5 b e f)-2 c g x (e f-d g)+4 c e f^2\right )}{4 e g^2 (e f-d g)}-\frac {\left (a g^2-b f g+c f^2\right )^{3/2} \tanh ^{-1}\left (\frac {-2 a g+x (2 c f-b g)+b f}{2 \sqrt {a+b x+c x^2} \sqrt {a g^2-b f g+c f^2}}\right )}{g^3 (e f-d g)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 734

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

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 895

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)/(((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))), x_Symbol] :> Dist[(c
*d^2 - b*d*e + a*e^2)/(e*(e*f - d*g)), Int[(a + b*x + c*x^2)^(p - 1)/(d + e*x), x], x] - Dist[1/(e*(e*f - d*g)
), Int[(Simp[c*d*f - b*e*f + a*e*g - c*(e*f - d*g)*x, x]*(a + b*x + c*x^2)^(p - 1))/(f + g*x), x], 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] && Fra
ctionQ[p] && GtQ[p, 0]

Rubi steps

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

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Mathematica [A]  time = 1.07, size = 323, normalized size = 0.66 \begin {gather*} \frac {\frac {\tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right ) \left (-12 c e g (-a e g+b d g+b e f)+3 b^2 e^2 g^2+8 c^2 \left (d^2 g^2+d e f g+e^2 f^2\right )\right )}{\sqrt {c}}+\frac {2 \left (-4 g^3 \left (e (a e-b d)+c d^2\right )^{3/2} \tanh ^{-1}\left (\frac {2 a e-b d+b e x-2 c d x}{2 \sqrt {a+x (b+c x)} \sqrt {e (a e-b d)+c d^2}}\right )+e g \sqrt {a+x (b+c x)} (e f-d g) (5 b e g+c (-4 d g-4 e f+2 e g x))+4 e^3 \left (g (a g-b f)+c f^2\right )^{3/2} \tanh ^{-1}\left (\frac {2 a g-b f+b g x-2 c f x}{2 \sqrt {a+x (b+c x)} \sqrt {g (a g-b f)+c f^2}}\right )\right )}{e f-d g}}{8 e^3 g^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [B]  time = 78.21, size = 2557, normalized size = 5.21 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a + b*x + c*x^2]*(-4*a*b^2*e*f - 4*a*b^2*d*g + 5*b^3*e*g*x^2) + c*Sqrt[a + b*x + c*x^2]*(-16*a^2*e*f - 1
6*a^2*d*g - 16*a*b*e*f*x - 16*a*b*d*g*x + 16*a*b*e*g*x^2 + 20*b^2*e*g*x^3) + Sqrt[c]*(16*a^2*b*e*f + 16*a^2*b*
d*g + 12*a*b^2*e*f*x + 12*a*b^2*d*g*x - 2*b^3*e*f*x^2 - 2*b^3*d*g*x^2 - 19*a*b^2*e*g*x^2 - 14*b^3*e*g*x^3) + c
^2*Sqrt[a + b*x + c*x^2]*(-32*a*e*f*x^2 - 32*a*d*g*x^2 - 48*b*e*f*x^3 - 48*b*d*g*x^3 + 8*a*e*g*x^3 + 20*b*e*g*
x^4) + c^(3/2)*(16*a^2*e*f*x + 16*a^2*d*g*x + 40*a*b*e*f*x^2 + 40*a*b*d*g*x^2 + 4*a^2*e*g*x^2 + 16*b^2*e*f*x^3
+ 16*b^2*d*g*x^3 - 24*a*b*e*g*x^3 - 27*b^2*e*g*x^4) + c^3*Sqrt[a + b*x + c*x^2]*(-64*e*f*x^4 - 64*d*g*x^4 + 2
4*e*g*x^5) + c^(5/2)*(64*a*e*f*x^3 + 64*a*d*g*x^3 + 80*b*e*f*x^4 + 80*b*d*g*x^4 - 20*a*e*g*x^4 - 32*b*e*g*x^5)
+ c^(7/2)*(64*e*f*x^5 + 64*d*g*x^5 - 24*e*g*x^6))/(8*b^2*e^2*g^2*x^2 + 64*c^2*e^2*g^2*x^4 + 8*c*e^2*g^2*x^2*(
4*a + 8*b*x) - 32*b*Sqrt[c]*e^2*g^2*x^2*Sqrt[a + b*x + c*x^2] - 64*c^(3/2)*e^2*g^2*x^3*Sqrt[a + b*x + c*x^2])
- (2*c^2*d^4*ArcTan[(Sqrt[c]*d)/Sqrt[-(c*d^2) + b*d*e - a*e^2] + (Sqrt[c]*e*x)/Sqrt[-(c*d^2) + b*d*e - a*e^2]
- (e*Sqrt[a + b*x + c*x^2])/Sqrt[-(c*d^2) + b*d*e - a*e^2]])/(e^3*Sqrt[-(c*d^2) + b*d*e - a*e^2]*(e*f - d*g))
+ (4*b*c*d^3*ArcTan[(Sqrt[c]*d)/Sqrt[-(c*d^2) + b*d*e - a*e^2] + (Sqrt[c]*e*x)/Sqrt[-(c*d^2) + b*d*e - a*e^2]
- (e*Sqrt[a + b*x + c*x^2])/Sqrt[-(c*d^2) + b*d*e - a*e^2]])/(e^2*Sqrt[-(c*d^2) + b*d*e - a*e^2]*(e*f - d*g))
- (4*a*c*d^2*ArcTan[(Sqrt[c]*d)/Sqrt[-(c*d^2) + b*d*e - a*e^2] + (Sqrt[c]*e*x)/Sqrt[-(c*d^2) + b*d*e - a*e^2]
- (e*Sqrt[a + b*x + c*x^2])/Sqrt[-(c*d^2) + b*d*e - a*e^2]])/(e*Sqrt[-(c*d^2) + b*d*e - a*e^2]*(e*f - d*g)) +
(-1/2*(a*d^2*Sqrt[a + b*x + c*x^2])/(e^2*(e*f - d*g)) + (a*f^2*Sqrt[a + b*x + c*x^2])/(2*g^2*(e*f - d*g)) - (5
*b*d*x^2*Sqrt[a + b*x + c*x^2])/(8*e*(e*f - d*g)) + (5*b*f*x^2*Sqrt[a + b*x + c*x^2])/(8*g*(e*f - d*g)) - (4*a
*b*d*x^2*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]])/(Sqrt[
-(c*d^2) + b*d*e - a*e^2]*(e*f - d*g)) + (2*b^2*d^2*x^2*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*Sqrt[-(c*d^2) + b*d*e - a*e^2]*(e*f - d*g)) + (2*a^2*e*x^2*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]])/(Sqrt[-(c*d^2) + b*d*e
- a*e^2]*(e*f - d*g)) + (4*a*b*f*x^2*ArcTan[(-(Sqrt[c]*f) - Sqrt[c]*g*x + g*Sqrt[a + b*x + c*x^2])/Sqrt[-(c*f
^2) + b*f*g - a*g^2]])/((e*f - d*g)*Sqrt[-(c*f^2) + b*f*g - a*g^2]) - (2*b^2*f^2*x^2*ArcTan[(-(Sqrt[c]*f) - Sq
rt[c]*g*x + g*Sqrt[a + b*x + c*x^2])/Sqrt[-(c*f^2) + b*f*g - a*g^2]])/(g*(e*f - d*g)*Sqrt[-(c*f^2) + b*f*g - a
*g^2]) - (2*a^2*g*x^2*ArcTan[(-(Sqrt[c]*f) - Sqrt[c]*g*x + g*Sqrt[a + b*x + c*x^2])/Sqrt[-(c*f^2) + b*f*g - a*
g^2]])/((e*f - d*g)*Sqrt[-(c*f^2) + b*f*g - a*g^2]))/x^2 + (2*c^2*f^4*ArcTan[(Sqrt[c]*f)/Sqrt[-(c*f^2) + b*f*g
- a*g^2] + (Sqrt[c]*g*x)/Sqrt[-(c*f^2) + b*f*g - a*g^2] - (g*Sqrt[a + b*x + c*x^2])/Sqrt[-(c*f^2) + b*f*g - a
*g^2]])/(g^3*(e*f - d*g)*Sqrt[-(c*f^2) + b*f*g - a*g^2]) - (4*b*c*f^3*ArcTan[(Sqrt[c]*f)/Sqrt[-(c*f^2) + b*f*g
- a*g^2] + (Sqrt[c]*g*x)/Sqrt[-(c*f^2) + b*f*g - a*g^2] - (g*Sqrt[a + b*x + c*x^2])/Sqrt[-(c*f^2) + b*f*g - a
*g^2]])/(g^2*(e*f - d*g)*Sqrt[-(c*f^2) + b*f*g - a*g^2]) + (4*a*c*f^2*ArcTan[(Sqrt[c]*f)/Sqrt[-(c*f^2) + b*f*g
- a*g^2] + (Sqrt[c]*g*x)/Sqrt[-(c*f^2) + b*f*g - a*g^2] - (g*Sqrt[a + b*x + c*x^2])/Sqrt[-(c*f^2) + b*f*g - a
*g^2]])/(g*(e*f - d*g)*Sqrt[-(c*f^2) + b*f*g - a*g^2]) - (c^(3/2)*f^2*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[a + b*x +
c*x^2]])/(e*g^3) - (c^(3/2)*d*f*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[a + b*x + c*x^2]])/(e^2*g^2) - (c^(3/2)*d^2*Lo
g[b + 2*c*x - 2*Sqrt[c]*Sqrt[a + b*x + c*x^2]])/(e^3*g) - (3*b^2*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[a + b*x + c*x^
2]])/(8*Sqrt[c]*e*g) + ((a*Sqrt[c]*f)/(2*e*g^2) + (a*Sqrt[c]*d)/(2*e^2*g) - (a*Sqrt[c]*x)/(8*e*g) - (3*b*Sqrt[
c]*x^2)/(4*e*g) - (c^(3/2)*x^3)/(8*e*g) + (3*b*Sqrt[c]*f*x*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[a + b*x + c*x^2]])/(
2*e*g^2) + (3*b*Sqrt[c]*d*x*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[a + b*x + c*x^2]])/(2*e^2*g) - (3*a*Sqrt[c]*x*Log[b
+ 2*c*x - 2*Sqrt[c]*Sqrt[a + b*x + c*x^2]])/(2*e*g))/x

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [B]  time = 0.02, size = 4226, normalized size = 8.61 \begin {gather*} \text {output too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4/(d*g-e*f)*((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*b-1/8/(d*g-e*f)/c*((x+d/e)^
2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b^2+1/16/(d*g-e*f)/c^(3/2)*ln(((x+d/e)*c+1/2*(b*e-2*c
*d)/e)/c^(1/2)+((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))*b^3+1/(d*g-e*f)/((a*e^2-b*d*
e+c*d^2)/e^2)^(1/2)*ln(((b*e-2*c*d)*(x+d/e)/e+2*(a*e^2-b*d*e+c*d^2)/e^2+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((x+
d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e))*a^2+1/4/(d*g-e*f)*((x+f/g)^2*c+(b*g-2*
c*f)*(x+f/g)/g+(a*g^2-b*f*g+c*f^2)/g^2)^(1/2)*x*b+1/8/(d*g-e*f)/c*((x+f/g)^2*c+(b*g-2*c*f)*(x+f/g)/g+(a*g^2-b*
f*g+c*f^2)/g^2)^(1/2)*b^2-1/16/(d*g-e*f)/c^(3/2)*ln(((x+f/g)*c+1/2*(b*g-2*c*f)/g)/c^(1/2)+((x+f/g)^2*c+(b*g-2*
c*f)*(x+f/g)/g+(a*g^2-b*f*g+c*f^2)/g^2)^(1/2))*b^3-1/(d*g-e*f)/((a*g^2-b*f*g+c*f^2)/g^2)^(1/2)*ln(((b*g-2*c*f)
*(x+f/g)/g+2*(a*g^2-b*f*g+c*f^2)/g^2+2*((a*g^2-b*f*g+c*f^2)/g^2)^(1/2)*((x+f/g)^2*c+(b*g-2*c*f)*(x+f/g)/g+(a*g
^2-b*f*g+c*f^2)/g^2)^(1/2))/(x+f/g))*a^2-1/3/(d*g-e*f)*((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/
e^2)^(3/2)+1/3/(d*g-e*f)*((x+f/g)^2*c+(b*g-2*c*f)*(x+f/g)/g+(a*g^2-b*f*g+c*f^2)/g^2)^(3/2)+1/(d*g-e*f)/e^2/((a
*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln(((b*e-2*c*d)*(x+d/e)/e+2*(a*e^2-b*d*e+c*d^2)/e^2+2*((a*e^2-b*d*e+c*d^2)/e^2)^(
1/2)*((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e))*b^2*d^2+1/(d*g-e*f)/e^4/((a*e
^2-b*d*e+c*d^2)/e^2)^(1/2)*ln(((b*e-2*c*d)*(x+d/e)/e+2*(a*e^2-b*d*e+c*d^2)/e^2+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/
2)*((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e))*c^2*d^4+1/2/(d*g-e*f)/e*((x+d/e
)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*c*d+3/2/(d*g-e*f)/e*ln(((x+d/e)*c+1/2*(b*e-2*c*d)
/e)/c^(1/2)+((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))*c^(1/2)*d*a+3/8/(d*g-e*f)/e*ln(
((x+d/e)*c+1/2*(b*e-2*c*d)/e)/c^(1/2)+((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/c^(1/
2)*b^2*d-3/2/(d*g-e*f)/e^2*ln(((x+d/e)*c+1/2*(b*e-2*c*d)/e)/c^(1/2)+((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-
b*d*e+c*d^2)/e^2)^(1/2))*c^(1/2)*d^2*b-1/2/(d*g-e*f)/g*((x+f/g)^2*c+(b*g-2*c*f)*(x+f/g)/g+(a*g^2-b*f*g+c*f^2)/
g^2)^(1/2)*x*c*f-3/2/(d*g-e*f)/g*ln(((x+f/g)*c+1/2*(b*g-2*c*f)/g)/c^(1/2)+((x+f/g)^2*c+(b*g-2*c*f)*(x+f/g)/g+(
a*g^2-b*f*g+c*f^2)/g^2)^(1/2))*c^(1/2)*f*a-3/8/(d*g-e*f)/g*ln(((x+f/g)*c+1/2*(b*g-2*c*f)/g)/c^(1/2)+((x+f/g)^2
*c+(b*g-2*c*f)*(x+f/g)/g+(a*g^2-b*f*g+c*f^2)/g^2)^(1/2))/c^(1/2)*b^2*f-1/(d*g-e*f)*((x+d/e)^2*c+(b*e-2*c*d)*(x
+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*a+1/(d*g-e*f)*((x+f/g)^2*c+(b*g-2*c*f)*(x+f/g)/g+(a*g^2-b*f*g+c*f^2)/g^
2)^(1/2)*a+2/(d*g-e*f)/g^3/((a*g^2-b*f*g+c*f^2)/g^2)^(1/2)*ln(((b*g-2*c*f)*(x+f/g)/g+2*(a*g^2-b*f*g+c*f^2)/g^2
+2*((a*g^2-b*f*g+c*f^2)/g^2)^(1/2)*((x+f/g)^2*c+(b*g-2*c*f)*(x+f/g)/g+(a*g^2-b*f*g+c*f^2)/g^2)^(1/2))/(x+f/g))
*b*f^3*c-2/(d*g-e*f)/e/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln(((b*e-2*c*d)*(x+d/e)/e+2*(a*e^2-b*d*e+c*d^2)/e^2+2*(
(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e))*a*b
*d-2/(d*g-e*f)/e^3/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln(((b*e-2*c*d)*(x+d/e)/e+2*(a*e^2-b*d*e+c*d^2)/e^2+2*((a*e
^2-b*d*e+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e))*b*d^3*c
-2/(d*g-e*f)/g^2/((a*g^2-b*f*g+c*f^2)/g^2)^(1/2)*ln(((b*g-2*c*f)*(x+f/g)/g+2*(a*g^2-b*f*g+c*f^2)/g^2+2*((a*g^2
-b*f*g+c*f^2)/g^2)^(1/2)*((x+f/g)^2*c+(b*g-2*c*f)*(x+f/g)/g+(a*g^2-b*f*g+c*f^2)/g^2)^(1/2))/(x+f/g))*a*c*f^2+3
/2/(d*g-e*f)/g^2*ln(((x+f/g)*c+1/2*(b*g-2*c*f)/g)/c^(1/2)+((x+f/g)^2*c+(b*g-2*c*f)*(x+f/g)/g+(a*g^2-b*f*g+c*f^
2)/g^2)^(1/2))*c^(1/2)*f^2*b-1/(d*g-e*f)/g^2/((a*g^2-b*f*g+c*f^2)/g^2)^(1/2)*ln(((b*g-2*c*f)*(x+f/g)/g+2*(a*g^
2-b*f*g+c*f^2)/g^2+2*((a*g^2-b*f*g+c*f^2)/g^2)^(1/2)*((x+f/g)^2*c+(b*g-2*c*f)*(x+f/g)/g+(a*g^2-b*f*g+c*f^2)/g^
2)^(1/2))/(x+f/g))*b^2*f^2-1/(d*g-e*f)/g^4/((a*g^2-b*f*g+c*f^2)/g^2)^(1/2)*ln(((b*g-2*c*f)*(x+f/g)/g+2*(a*g^2-
b*f*g+c*f^2)/g^2+2*((a*g^2-b*f*g+c*f^2)/g^2)^(1/2)*((x+f/g)^2*c+(b*g-2*c*f)*(x+f/g)/g+(a*g^2-b*f*g+c*f^2)/g^2)
^(1/2))/(x+f/g))*c^2*f^4-5/4/(d*g-e*f)/g*((x+f/g)^2*c+(b*g-2*c*f)*(x+f/g)/g+(a*g^2-b*f*g+c*f^2)/g^2)^(1/2)*b*f
+3/4/(d*g-e*f)/c^(1/2)*ln(((x+f/g)*c+1/2*(b*g-2*c*f)/g)/c^(1/2)+((x+f/g)^2*c+(b*g-2*c*f)*(x+f/g)/g+(a*g^2-b*f*
g+c*f^2)/g^2)^(1/2))*a*b+2/(d*g-e*f)/g/((a*g^2-b*f*g+c*f^2)/g^2)^(1/2)*ln(((b*g-2*c*f)*(x+f/g)/g+2*(a*g^2-b*f*
g+c*f^2)/g^2+2*((a*g^2-b*f*g+c*f^2)/g^2)^(1/2)*((x+f/g)^2*c+(b*g-2*c*f)*(x+f/g)/g+(a*g^2-b*f*g+c*f^2)/g^2)^(1/
2))/(x+f/g))*a*b*f+2/(d*g-e*f)/e^2/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln(((b*e-2*c*d)*(x+d/e)/e+2*(a*e^2-b*d*e+c*
d^2)/e^2+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/
(x+d/e))*a*c*d^2+1/(d*g-e*f)/g^2*((x+f/g)^2*c+(b*g-2*c*f)*(x+f/g)/g+(a*g^2-b*f*g+c*f^2)/g^2)^(1/2)*c*f^2-1/(d*
g-e*f)/g^3*ln(((x+f/g)*c+1/2*(b*g-2*c*f)/g)/c^(1/2)+((x+f/g)^2*c+(b*g-2*c*f)*(x+f/g)/g+(a*g^2-b*f*g+c*f^2)/g^2
)^(1/2))*c^(3/2)*f^3+5/4/(d*g-e*f)/e*((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b*d-3/4
/(d*g-e*f)/c^(1/2)*ln(((x+d/e)*c+1/2*(b*e-2*c*d)/e)/c^(1/2)+((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*
d^2)/e^2)^(1/2))*a*b-1/(d*g-e*f)/e^2*((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*c*d^2+1
/(d*g-e*f)/e^3*ln(((x+d/e)*c+1/2*(b*e-2*c*d)/e)/c^(1/2)+((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)
/e^2)^(1/2))*c^(3/2)*d^3

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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