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

Optimal. Leaf size=441 \[ \frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (48 c^2 e^2 \left (a^2 e^2 g+2 a b e (e f-d g)+b^2 (-d) (e f-d g)\right )-8 b^2 c e^3 (3 a e g-b d g+b e f)+192 c^3 d e (b d-a e) (e f-d g)+3 b^4 e^4 g-128 c^4 d^3 (e f-d g)\right )}{128 c^{5/2} e^5}-\frac{\sqrt{a+b x+c x^2} \left (2 c e x \left (-4 c e (3 a e g-2 b d g+2 b e f)+3 b^2 e^2 g+16 c^2 d (e f-d g)\right )+16 c^2 e (5 b d-4 a e) (e f-d g)-4 b c e^2 (3 a e g-2 b d g+2 b e f)+3 b^3 e^3 g-64 c^3 d^2 (e f-d g)\right )}{64 c^2 e^4}+\frac{(e f-d g) \left (a e^2-b d e+c d^2\right )^{3/2} \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{e^5}+\frac{\left (a+b x+c x^2\right )^{3/2} (3 b e g-8 c d g+8 c e f+6 c e g x)}{24 c e^2} \]

[Out]

-((3*b^3*e^3*g - 64*c^3*d^2*(e*f - d*g) + 16*c^2*e*(5*b*d - 4*a*e)*(e*f - d*g) - 4*b*c*e^2*(2*b*e*f - 2*b*d*g
+ 3*a*e*g) + 2*c*e*(3*b^2*e^2*g + 16*c^2*d*(e*f - d*g) - 4*c*e*(2*b*e*f - 2*b*d*g + 3*a*e*g))*x)*Sqrt[a + b*x
+ c*x^2])/(64*c^2*e^4) + ((8*c*e*f - 8*c*d*g + 3*b*e*g + 6*c*e*g*x)*(a + b*x + c*x^2)^(3/2))/(24*c*e^2) + ((3*
b^4*e^4*g - 128*c^4*d^3*(e*f - d*g) + 192*c^3*d*e*(b*d - a*e)*(e*f - d*g) - 8*b^2*c*e^3*(b*e*f - b*d*g + 3*a*e
*g) + 48*c^2*e^2*(a^2*e^2*g - b^2*d*(e*f - d*g) + 2*a*b*e*(e*f - d*g)))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a
+ b*x + c*x^2])])/(128*c^(5/2)*e^5) + ((c*d^2 - b*d*e + a*e^2)^(3/2)*(e*f - d*g)*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

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Rubi [A]  time = 0.853214, antiderivative size = 441, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {814, 843, 621, 206, 724} \[ \frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (48 c^2 e^2 \left (a^2 e^2 g+2 a b e (e f-d g)+b^2 (-d) (e f-d g)\right )-8 b^2 c e^3 (3 a e g-b d g+b e f)+192 c^3 d e (b d-a e) (e f-d g)+3 b^4 e^4 g-128 c^4 d^3 (e f-d g)\right )}{128 c^{5/2} e^5}-\frac{\sqrt{a+b x+c x^2} \left (2 c e x \left (-4 c e (3 a e g-2 b d g+2 b e f)+3 b^2 e^2 g+16 c^2 d (e f-d g)\right )+16 c^2 e (5 b d-4 a e) (e f-d g)-4 b c e^2 (3 a e g-2 b d g+2 b e f)+3 b^3 e^3 g-64 c^3 d^2 (e f-d g)\right )}{64 c^2 e^4}+\frac{(e f-d g) \left (a e^2-b d e+c d^2\right )^{3/2} \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{e^5}+\frac{\left (a+b x+c x^2\right )^{3/2} (3 b e g-8 c d g+8 c e f+6 c e g x)}{24 c e^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((3*b^3*e^3*g - 64*c^3*d^2*(e*f - d*g) + 16*c^2*e*(5*b*d - 4*a*e)*(e*f - d*g) - 4*b*c*e^2*(2*b*e*f - 2*b*d*g
+ 3*a*e*g) + 2*c*e*(3*b^2*e^2*g + 16*c^2*d*(e*f - d*g) - 4*c*e*(2*b*e*f - 2*b*d*g + 3*a*e*g))*x)*Sqrt[a + b*x
+ c*x^2])/(64*c^2*e^4) + ((8*c*e*f - 8*c*d*g + 3*b*e*g + 6*c*e*g*x)*(a + b*x + c*x^2)^(3/2))/(24*c*e^2) + ((3*
b^4*e^4*g - 128*c^4*d^3*(e*f - d*g) + 192*c^3*d*e*(b*d - a*e)*(e*f - d*g) - 8*b^2*c*e^3*(b*e*f - b*d*g + 3*a*e
*g) + 48*c^2*e^2*(a^2*e^2*g - b^2*d*(e*f - d*g) + 2*a*b*e*(e*f - d*g)))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a
+ b*x + c*x^2])])/(128*c^(5/2)*e^5) + ((c*d^2 - b*d*e + a*e^2)^(3/2)*(e*f - d*g)*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 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 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 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 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]

Rubi steps

\begin{align*} \int \frac{(f+g x) \left (a+b x+c x^2\right )^{3/2}}{d+e x} \, dx &=\frac{(8 c e f-8 c d g+3 b e g+6 c e g x) \left (a+b x+c x^2\right )^{3/2}}{24 c e^2}-\frac{\int \frac{\left (\frac{1}{2} \left (8 c e (b d-2 a e) f+4 a c d e g-2 b d \left (4 c d-\frac{3 b e}{2}\right ) g\right )+\frac{1}{2} \left (3 b^2 e^2 g+16 c^2 d (e f-d g)-4 c e (2 b e f-2 b d g+3 a e g)\right ) x\right ) \sqrt{a+b x+c x^2}}{d+e x} \, dx}{8 c e^2}\\ &=-\frac{\left (3 b^3 e^3 g-64 c^3 d^2 (e f-d g)+16 c^2 e (5 b d-4 a e) (e f-d g)-4 b c e^2 (2 b e f-2 b d g+3 a e g)+2 c e \left (3 b^2 e^2 g+16 c^2 d (e f-d g)-4 c e (2 b e f-2 b d g+3 a e g)\right ) x\right ) \sqrt{a+b x+c x^2}}{64 c^2 e^4}+\frac{(8 c e f-8 c d g+3 b e g+6 c e g x) \left (a+b x+c x^2\right )^{3/2}}{24 c e^2}+\frac{\int \frac{\frac{1}{4} \left (4 c e (b d-2 a e) (8 c e (b d-2 a e) f+4 a c d e g-b d (8 c d-3 b e) g)-d \left (4 b c d-b^2 e-4 a c e\right ) \left (3 b^2 e^2 g+16 c^2 d (e f-d g)-4 c e (2 b e f-2 b d g+3 a e g)\right )\right )+\frac{1}{4} \left (3 b^4 e^4 g-128 c^4 d^3 (e f-d g)+192 c^3 d e (b d-a e) (e f-d g)-8 b^2 c e^3 (b e f-b d g+3 a e g)+48 c^2 e^2 \left (a^2 e^2 g-b^2 d (e f-d g)+2 a b e (e f-d g)\right )\right ) x}{(d+e x) \sqrt{a+b x+c x^2}} \, dx}{32 c^2 e^4}\\ &=-\frac{\left (3 b^3 e^3 g-64 c^3 d^2 (e f-d g)+16 c^2 e (5 b d-4 a e) (e f-d g)-4 b c e^2 (2 b e f-2 b d g+3 a e g)+2 c e \left (3 b^2 e^2 g+16 c^2 d (e f-d g)-4 c e (2 b e f-2 b d g+3 a e g)\right ) x\right ) \sqrt{a+b x+c x^2}}{64 c^2 e^4}+\frac{(8 c e f-8 c d g+3 b e g+6 c e g x) \left (a+b x+c x^2\right )^{3/2}}{24 c e^2}+\frac{\left (\left (c d^2-b d e+a e^2\right )^2 (e f-d g)\right ) \int \frac{1}{(d+e x) \sqrt{a+b x+c x^2}} \, dx}{e^5}+\frac{\left (3 b^4 e^4 g-128 c^4 d^3 (e f-d g)+192 c^3 d e (b d-a e) (e f-d g)-8 b^2 c e^3 (b e f-b d g+3 a e g)+48 c^2 e^2 \left (a^2 e^2 g-b^2 d (e f-d g)+2 a b e (e f-d g)\right )\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{128 c^2 e^5}\\ &=-\frac{\left (3 b^3 e^3 g-64 c^3 d^2 (e f-d g)+16 c^2 e (5 b d-4 a e) (e f-d g)-4 b c e^2 (2 b e f-2 b d g+3 a e g)+2 c e \left (3 b^2 e^2 g+16 c^2 d (e f-d g)-4 c e (2 b e f-2 b d g+3 a e g)\right ) x\right ) \sqrt{a+b x+c x^2}}{64 c^2 e^4}+\frac{(8 c e f-8 c d g+3 b e g+6 c e g x) \left (a+b x+c x^2\right )^{3/2}}{24 c e^2}-\frac{\left (2 \left (c d^2-b d e+a e^2\right )^2 (e f-d g)\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 (3 b^4 e^4 g-128 c^4 d^3 (e f-d g)+192 c^3 d e (b d-a e) (e f-d g)-8 b^2 c e^3 (b e f-b d g+3 a e g)+48 c^2 e^2 \left (a^2 e^2 g-b^2 d (e f-d g)+2 a b e (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 )}{64 c^2 e^5}\\ &=-\frac{\left (3 b^3 e^3 g-64 c^3 d^2 (e f-d g)+16 c^2 e (5 b d-4 a e) (e f-d g)-4 b c e^2 (2 b e f-2 b d g+3 a e g)+2 c e \left (3 b^2 e^2 g+16 c^2 d (e f-d g)-4 c e (2 b e f-2 b d g+3 a e g)\right ) x\right ) \sqrt{a+b x+c x^2}}{64 c^2 e^4}+\frac{(8 c e f-8 c d g+3 b e g+6 c e g x) \left (a+b x+c x^2\right )^{3/2}}{24 c e^2}+\frac{\left (3 b^4 e^4 g-128 c^4 d^3 (e f-d g)+192 c^3 d e (b d-a e) (e f-d g)-8 b^2 c e^3 (b e f-b d g+3 a e g)+48 c^2 e^2 \left (a^2 e^2 g-b^2 d (e f-d g)+2 a b e (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 )}{128 c^{5/2} e^5}+\frac{\left (c d^2-b d e+a e^2\right )^{3/2} (e f-d g) \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*}

Mathematica [A]  time = 1.17622, size = 420, normalized size = 0.95 \[ \frac{\frac{3 \left (\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right ) \left (48 c^2 e^2 \left (a^2 e^2 g+2 a b e (e f-d g)+b^2 d (d g-e f)\right )-8 b^2 c e^3 (3 a e g-b d g+b e f)-192 c^3 d e (b d-a e) (d g-e f)+3 b^4 e^4 g+128 c^4 d^3 (d g-e f)\right )+2 \sqrt{c} e \sqrt{a+x (b+c x)} \left (8 c^2 e (a e (-8 d g+8 e f+3 e g x)+2 b (e x-5 d) (e f-d g))+2 b c e^2 (6 a e g+b (-4 d g+4 e f-3 e g x))-3 b^3 e^3 g-32 c^3 d (e x-2 d) (e f-d g)\right )+128 c^{5/2} (d g-e f) \left (e (a e-b d)+c d^2\right )^{3/2} \tanh ^{-1}\left (\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right )\right )}{16 c^{3/2} e^3}+(a+x (b+c x))^{3/2} (3 b e g+c (-8 d g+8 e f+6 e g x))}{24 c e^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + x*(b + c*x))^(3/2)*(3*b*e*g + c*(8*e*f - 8*d*g + 6*e*g*x)) + (3*(2*Sqrt[c]*e*Sqrt[a + x*(b + c*x)]*(-3*b
^3*e^3*g - 32*c^3*d*(e*f - d*g)*(-2*d + e*x) + 2*b*c*e^2*(6*a*e*g + b*(4*e*f - 4*d*g - 3*e*g*x)) + 8*c^2*e*(2*
b*(e*f - d*g)*(-5*d + e*x) + a*e*(8*e*f - 8*d*g + 3*e*g*x))) + (3*b^4*e^4*g + 128*c^4*d^3*(-(e*f) + d*g) - 192
*c^3*d*e*(b*d - a*e)*(-(e*f) + d*g) - 8*b^2*c*e^3*(b*e*f - b*d*g + 3*a*e*g) + 48*c^2*e^2*(a^2*e^2*g + 2*a*b*e*
(e*f - d*g) + b^2*d*(-(e*f) + d*g)))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])] + 128*c^(5/2)*(c*d
^2 + e*(-(b*d) + a*e))^(3/2)*(-(e*f) + d*g)*ArcTanh[(-(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)])]))/(16*c^(3/2)*e^3))/(24*c*e^2)

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Maple [B]  time = 0.276, size = 4188, normalized size = 9.5 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError