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

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

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Rubi [A]  time = 0.85, antiderivative size = 441, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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} \end {gather*}

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 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]

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*}

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Mathematica [A]  time = 1.13, size = 420, normalized size = 0.95 \begin {gather*} \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} \end {gather*}

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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IntegrateAlgebraic [A]  time = 3.47, size = 687, normalized size = 1.56 \begin {gather*} \frac {\log \left (-2 \sqrt {c} \sqrt {a+b x+c x^2}+b+2 c x\right ) \left (-48 a^2 c^2 e^4 g+24 a b^2 c e^4 g+96 a b c^2 d e^3 g-96 a b c^2 e^4 f-192 a c^3 d^2 e^2 g+192 a c^3 d e^3 f-3 b^4 e^4 g-8 b^3 c d e^3 g+8 b^3 c e^4 f-48 b^2 c^2 d^2 e^2 g+48 b^2 c^2 d e^3 f+192 b c^3 d^3 e g-192 b c^3 d^2 e^2 f-128 c^4 d^4 g+128 c^4 d^3 e f\right )}{128 c^{5/2} e^5}+\frac {\sqrt {a+b x+c x^2} \left (60 a b c e^3 g-256 a c^2 d e^2 g+256 a c^2 e^3 f+120 a c^2 e^3 g x-9 b^3 e^3 g-24 b^2 c d e^2 g+24 b^2 c e^3 f+6 b^2 c e^3 g x+240 b c^2 d^2 e g-240 b c^2 d e^2 f-112 b c^2 d e^2 g x+112 b c^2 e^3 f x+72 b c^2 e^3 g x^2-192 c^3 d^3 g+192 c^3 d^2 e f+96 c^3 d^2 e g x-96 c^3 d e^2 f x-64 c^3 d e^2 g x^2+64 c^3 e^3 f x^2+48 c^3 e^3 g x^3\right )}{192 c^2 e^4}-\frac {2 \left (-c d^2 e f \sqrt {-a e^2+b d e-c d^2}+b d e^2 f \sqrt {-a e^2+b d e-c d^2}-b d^2 e g \sqrt {-a e^2+b d e-c d^2}+a d e^2 g \sqrt {-a e^2+b d e-c d^2}-a e^3 f \sqrt {-a e^2+b d e-c d^2}+c d^3 g \sqrt {-a e^2+b d e-c d^2}\right ) \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)*(a + b*x + c*x^2)^(3/2))/(d + e*x),x]

[Out]

(Sqrt[a + b*x + c*x^2]*(192*c^3*d^2*e*f - 240*b*c^2*d*e^2*f + 24*b^2*c*e^3*f + 256*a*c^2*e^3*f - 192*c^3*d^3*g
+ 240*b*c^2*d^2*e*g - 24*b^2*c*d*e^2*g - 256*a*c^2*d*e^2*g - 9*b^3*e^3*g + 60*a*b*c*e^3*g - 96*c^3*d*e^2*f*x
+ 112*b*c^2*e^3*f*x + 96*c^3*d^2*e*g*x - 112*b*c^2*d*e^2*g*x + 6*b^2*c*e^3*g*x + 120*a*c^2*e^3*g*x + 64*c^3*e^
3*f*x^2 - 64*c^3*d*e^2*g*x^2 + 72*b*c^2*e^3*g*x^2 + 48*c^3*e^3*g*x^3))/(192*c^2*e^4) - (2*(-(c*d^2*e*Sqrt[-(c*
d^2) + b*d*e - a*e^2]*f) + b*d*e^2*Sqrt[-(c*d^2) + b*d*e - a*e^2]*f - a*e^3*Sqrt[-(c*d^2) + b*d*e - a*e^2]*f +
c*d^3*Sqrt[-(c*d^2) + b*d*e - a*e^2]*g - b*d^2*e*Sqrt[-(c*d^2) + b*d*e - a*e^2]*g + a*d*e^2*Sqrt[-(c*d^2) + b
*d*e - a*e^2]*g)*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^3*e*f - 192*b*c^3*d^2*e^2*f + 48*b^2*c^2*d*e^3*f + 192*a*c^3*d*e^3*f + 8*b^3*c*e^4*f - 96*a*b
*c^2*e^4*f - 128*c^4*d^4*g + 192*b*c^3*d^3*e*g - 48*b^2*c^2*d^2*e^2*g - 192*a*c^3*d^2*e^2*g - 8*b^3*c*d*e^3*g
+ 96*a*b*c^2*d*e^3*g - 3*b^4*e^4*g + 24*a*b^2*c*e^4*g - 48*a^2*c^2*e^4*g)*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[a + b
*x + c*x^2]])/(128*c^(5/2)*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)*(c*x^2+b*x+a)^(3/2)/(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)*(c*x^2+b*x+a)^(3/2)/(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 = 0.01, size = 4188, normalized size = 9.50 \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)*(c*x^2+b*x+a)^(3/2)/(e*x+d),x)

[Out]

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

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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)*(c*x^2+b*x+a)^(3/2)/(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 )\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{d+e\,x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(((f + g*x)*(a + b*x + c*x^2)^(3/2))/(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 ) \left (a + b x + c x^{2}\right )^{\frac {3}{2}}}{d + e x}\, dx \end {gather*}

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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