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

Optimal. Leaf size=271 \[ -\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{e^2 (d+e x)^3 (2 c d-b e)}-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (b e g-6 c d g+4 c e f)}{2 e^2 (d+e x) (2 c d-b e)}-\frac {3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (b e g-6 c d g+4 c e f)}{4 e^2}-\frac {3 (2 c d-b e) (b e g-6 c d g+4 c e f) \tan ^{-1}\left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{8 \sqrt {c} e^2} \]

________________________________________________________________________________________

Rubi [A]  time = 0.47, antiderivative size = 271, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {792, 664, 621, 204} \begin {gather*} -\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{e^2 (d+e x)^3 (2 c d-b e)}-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (b e g-6 c d g+4 c e f)}{2 e^2 (d+e x) (2 c d-b e)}-\frac {3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (b e g-6 c d g+4 c e f)}{4 e^2}-\frac {3 (2 c d-b e) (b e g-6 c d g+4 c e f) \tan ^{-1}\left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{8 \sqrt {c} e^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[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 664

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*(2*c*d - b*e))/(e^2*(m + 2*p + 1)), Int[(d + e*x)^(m + 1)*
(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a
*e^2, 0] && GtQ[p, 0] && (LeQ[-2, m, 0] || EqQ[m + p + 1, 0]) && NeQ[m + 2*p + 1, 0] && IntegerQ[2*p]

Rule 792

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((d*g - e*f)*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/((2*c*d - b*e)*(m + p + 1)), x] + Dist[(m*(g*(c*d - b*e)
+ c*e*f) + e*(p + 1)*(2*c*f - b*g))/(e*(2*c*d - b*e)*(m + p + 1)), Int[(d + e*x)^(m + 1)*(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] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ((L
tQ[m, -1] &&  !IGtQ[m + p + 1, 0]) || (LtQ[m, 0] && LtQ[p, -1]) || EqQ[m + 2*p + 2, 0]) && NeQ[m + p + 1, 0]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.85, size = 208, normalized size = 0.77 \begin {gather*} \frac {\sqrt {(d+e x) (c (d-e x)-b e)} \left (-\frac {\sqrt {e} \left (b e (13 d g-8 e f+5 e g x)+c \left (-28 d^2 g+10 d e (2 f-g x)+2 e^2 x (2 f+g x)\right )\right )}{d+e x}-\frac {3 \sqrt {e (2 c d-b e)} (b e g-6 c d g+4 c e f) \sin ^{-1}\left (\frac {\sqrt {c} \sqrt {e} \sqrt {d+e x}}{\sqrt {e (2 c d-b e)}}\right )}{\sqrt {c} \sqrt {d+e x} \sqrt {\frac {b e-c d+c e x}{b e-2 c d}}}\right )}{4 e^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(-((Sqrt[e]*(b*e*(-8*e*f + 13*d*g + 5*e*g*x) + c*(-28*d^2*g + 10*d*e*(
2*f - g*x) + 2*e^2*x*(2*f + g*x))))/(d + e*x)) - (3*Sqrt[e*(2*c*d - b*e)]*(4*c*e*f - 6*c*d*g + b*e*g)*ArcSin[(
Sqrt[c]*Sqrt[e]*Sqrt[d + e*x])/Sqrt[e*(2*c*d - b*e)]])/(Sqrt[c]*Sqrt[d + e*x]*Sqrt[(-(c*d) + b*e + c*e*x)/(-2*
c*d + b*e)])))/(4*e^(5/2))

________________________________________________________________________________________

IntegrateAlgebraic [B]  time = 53.95, size = 14673, normalized size = 54.14 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Result too large to show

________________________________________________________________________________________

fricas [A]  time = 1.28, size = 635, normalized size = 2.34 \begin {gather*} \left [\frac {3 \, {\left (4 \, {\left (2 \, c^{2} d^{2} e - b c d e^{2}\right )} f - {\left (12 \, c^{2} d^{3} - 8 \, b c d^{2} e + b^{2} d e^{2}\right )} g + {\left (4 \, {\left (2 \, c^{2} d e^{2} - b c e^{3}\right )} f - {\left (12 \, c^{2} d^{2} e - 8 \, b c d e^{2} + b^{2} e^{3}\right )} g\right )} x\right )} \sqrt {-c} \log \left (8 \, c^{2} e^{2} x^{2} + 8 \, b c e^{2} x - 4 \, c^{2} d^{2} + 4 \, b c d e + b^{2} e^{2} - 4 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (2 \, c e x + b e\right )} \sqrt {-c}\right ) - 4 \, {\left (2 \, c^{2} e^{2} g x^{2} + 4 \, {\left (5 \, c^{2} d e - 2 \, b c e^{2}\right )} f - {\left (28 \, c^{2} d^{2} - 13 \, b c d e\right )} g + {\left (4 \, c^{2} e^{2} f - 5 \, {\left (2 \, c^{2} d e - b c e^{2}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}{16 \, {\left (c e^{3} x + c d e^{2}\right )}}, \frac {3 \, {\left (4 \, {\left (2 \, c^{2} d^{2} e - b c d e^{2}\right )} f - {\left (12 \, c^{2} d^{3} - 8 \, b c d^{2} e + b^{2} d e^{2}\right )} g + {\left (4 \, {\left (2 \, c^{2} d e^{2} - b c e^{3}\right )} f - {\left (12 \, c^{2} d^{2} e - 8 \, b c d e^{2} + b^{2} e^{3}\right )} g\right )} x\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (2 \, c e x + b e\right )} \sqrt {c}}{2 \, {\left (c^{2} e^{2} x^{2} + b c e^{2} x - c^{2} d^{2} + b c d e\right )}}\right ) - 2 \, {\left (2 \, c^{2} e^{2} g x^{2} + 4 \, {\left (5 \, c^{2} d e - 2 \, b c e^{2}\right )} f - {\left (28 \, c^{2} d^{2} - 13 \, b c d e\right )} g + {\left (4 \, c^{2} e^{2} f - 5 \, {\left (2 \, c^{2} d e - b c e^{2}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}{8 \, {\left (c e^{3} x + c d e^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/16*(3*(4*(2*c^2*d^2*e - b*c*d*e^2)*f - (12*c^2*d^3 - 8*b*c*d^2*e + b^2*d*e^2)*g + (4*(2*c^2*d*e^2 - b*c*e^3
)*f - (12*c^2*d^2*e - 8*b*c*d*e^2 + b^2*e^3)*g)*x)*sqrt(-c)*log(8*c^2*e^2*x^2 + 8*b*c*e^2*x - 4*c^2*d^2 + 4*b*
c*d*e + b^2*e^2 - 4*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*c*e*x + b*e)*sqrt(-c)) - 4*(2*c^2*e^2*g*x^2
+ 4*(5*c^2*d*e - 2*b*c*e^2)*f - (28*c^2*d^2 - 13*b*c*d*e)*g + (4*c^2*e^2*f - 5*(2*c^2*d*e - b*c*e^2)*g)*x)*sqr
t(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))/(c*e^3*x + c*d*e^2), 1/8*(3*(4*(2*c^2*d^2*e - b*c*d*e^2)*f - (12*c^2*
d^3 - 8*b*c*d^2*e + b^2*d*e^2)*g + (4*(2*c^2*d*e^2 - b*c*e^3)*f - (12*c^2*d^2*e - 8*b*c*d*e^2 + b^2*e^3)*g)*x)
*sqrt(c)*arctan(1/2*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*c*e*x + b*e)*sqrt(c)/(c^2*e^2*x^2 + b*c*e^2*
x - c^2*d^2 + b*c*d*e)) - 2*(2*c^2*e^2*g*x^2 + 4*(5*c^2*d*e - 2*b*c*e^2)*f - (28*c^2*d^2 - 13*b*c*d*e)*g + (4*
c^2*e^2*f - 5*(2*c^2*d*e - b*c*e^2)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))/(c*e^3*x + c*d*e^2)]

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: 2*(-4*exp(1)^3*c^2*g*1/16/exp(1)^4/c*x+(
-8*exp(1)^3*c^2*f-10*exp(1)^3*c*g*b+24*exp(1)^2*c^2*g*d)*1/16/exp(1)^4/c)*sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*
x^2*exp(2))+2*((3*sqrt(-c*exp(2))*b^2*g*exp(1)^2-24*c*sqrt(-c*exp(2))*b*g*exp(1)*d+12*c*sqrt(-c*exp(2))*b*exp(
1)^2*f+36*c^2*sqrt(-c*exp(2))*g*d^2-24*c^2*sqrt(-c*exp(2))*exp(1)*d*f)/16/c/exp(2)/exp(1)*ln(abs(2*c*(sqrt(-b*
d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)-sqrt(-c*exp(2))*b))+(12*exp(2)*b^2*g*exp(1)^4*d-15*
exp(2)^2*b^2*g*exp(1)^2*d+3*exp(2)^2*b^2*exp(1)^3*f-48*c*exp(2)*b*g*exp(1)^3*d^2+60*c*exp(2)^2*b*g*exp(1)*d^2+
12*c*exp(2)*b*exp(1)^4*d*f-24*c*exp(2)^2*b*exp(1)^2*d*f+36*c^2*exp(2)*g*exp(1)^2*d^3-48*c^2*exp(2)^2*g*d^3-12*
c^2*exp(2)*exp(1)^3*d^2*f+24*c^2*exp(2)^2*exp(1)*d^2*f)/4/exp(1)^5/2/sqrt(b*d*exp(1)^3-c*d^2*exp(1)^2+c*d^2*ex
p(2)-b*d*exp(1)*exp(2))*atan((-d*sqrt(-c*exp(2))+(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(
2))*x)*exp(1))/sqrt(b*d*exp(1)^3-c*d^2*exp(1)^2+c*d^2*exp(2)-b*d*exp(1)*exp(2)))-(4*exp(2)*(sqrt(-b*d*exp(1)-b
*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^3*b^2*g*exp(1)^5*d-9*exp(2)^2*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*
d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^3*b^2*g*exp(1)^3*d+5*exp(2)^2*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*ex
p(2))-sqrt(-c*exp(2))*x)^3*b^2*exp(1)^4*f-16*c*exp(2)*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c
*exp(2))*x)^3*b*g*exp(1)^4*d^2+36*c*exp(2)^2*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*
x)^3*b*g*exp(1)^2*d^2+4*c*exp(2)*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^3*b*exp(1
)^5*d*f-24*c*exp(2)^2*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^3*b*exp(1)^3*d*f+12*
c^2*exp(2)*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^3*g*exp(1)^3*d^3-32*c^2*exp(2)^
2*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^3*g*exp(1)*d^3-4*c^2*exp(2)*(sqrt(-b*d*e
xp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^3*exp(1)^4*d^2*f+24*c^2*exp(2)^2*(sqrt(-b*d*exp(1)-b*x
*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^3*exp(1)^2*d^2*f-8*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)
+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^2*b^2*g*exp(1)^6*d^2+20*exp(2)*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*e
xp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^2*b^2*g*exp(1)^4*d^2+3*exp(2)^2*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)
-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^2*b^2*g*exp(1)^2*d^2-16*exp(2)*sqrt(-c*exp(2))*(sqrt(-b*d*e
xp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^2*b^2*exp(1)^5*d*f+exp(2)^2*sqrt(-c*exp(2))*(sqrt(-b*d
*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^2*b^2*exp(1)^3*d*f+16*c*sqrt(-c*exp(2))*(sqrt(-b*d*e
xp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^2*b*g*exp(1)^5*d^3-32*c*exp(2)*sqrt(-c*exp(2))*(sqrt(-
b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^2*b*g*exp(1)^3*d^3-44*c*exp(2)^2*sqrt(-c*exp(2))*
(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^2*b*g*exp(1)*d^3+36*c*exp(2)*sqrt(-c*exp(2
))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^2*b*exp(1)^4*d^2*f+24*c*exp(2)^2*sqrt(-
c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^2*b*exp(1)^2*d^2*f-8*c^2*sqrt(-c
*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^2*g*exp(1)^4*d^4+12*c^2*exp(2)*sq
rt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^2*g*exp(1)^2*d^4+56*c^2*exp(
2)^2*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^2*g*d^4-20*c^2*exp(2)
*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^2*exp(1)^3*d^3*f-40*c^2*e
xp(2)^2*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^2*exp(1)*d^3*f+4*e
xp(2)*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b^3*g*exp(1)^6*d^2-11*exp(2)^2*(sqrt
(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b^3*g*exp(1)^4*d^2+7*exp(2)^3*(sqrt(-b*d*exp(1)
-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b^3*g*exp(1)^2*d^2+3*exp(2)^2*(sqrt(-b*d*exp(1)-b*x*exp(2)+
c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b^3*exp(1)^5*d*f-3*exp(2)^3*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*ex
p(2))-sqrt(-c*exp(2))*x)*b^3*exp(1)^3*d*f-28*c*exp(2)*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c
*exp(2))*x)*b^2*g*exp(1)^5*d^3+91*c*exp(2)^2*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*
x)*b^2*g*exp(1)^3*d^3-48*c*exp(2)^3*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b^2*g*
exp(1)*d^3-4*c*exp(2)*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b^2*exp(1)^6*d^2*f-3
9*c*exp(2)^2*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b^2*exp(1)^4*d^2*f+28*c*exp(2
)^3*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b^2*exp(1)^2*d^2*f+44*c^2*exp(2)*(sqrt
(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b*g*exp(1)^4*d^4-160*c^2*exp(2)^2*(sqrt(-b*d*ex
p(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b*g*exp(1)^2*d^4+56*c^2*exp(2)^3*(sqrt(-b*d*exp(1)-b*x*
exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*b*g*d^4+8*c^2*exp(2)*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*ex
p(2))-sqrt(-c*exp(2))*x)*b*exp(1)^5*d^3*f+92*c^2*exp(2)^2*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqr
t(-c*exp(2))*x)*b*exp(1)^3*d^3*f-40*c^2*exp(2)^3*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(
2))*x)*b*exp(1)*d^3*f-20*c^3*exp(2)*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*g*exp(
1)^3*d^5+80*c^3*exp(2)^2*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*g*exp(1)*d^5-4*c^
3*exp(2)*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*exp(1)^4*d^4*f-56*c^3*exp(2)^2*(s
qrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*exp(1)^2*d^4*f-8*sqrt(-c*exp(2))*b^3*g*exp(1
)^7*d^3+28*exp(2)*sqrt(-c*exp(2))*b^3*g*exp(1)^5*d^3-29*exp(2)^2*sqrt(-c*exp(2))*b^3*g*exp(1)^3*d^3+9*exp(2)^3
*sqrt(-c*exp(2))*b^3*g*exp(1)*d^3-8*exp(2)*sqrt(-c*exp(2))*b^3*exp(1)^6*d^2*f+13*exp(2)^2*sqrt(-c*exp(2))*b^3*
exp(1)^4*d^2*f-5*exp(2)^3*sqrt(-c*exp(2))*b^3*exp(1)^2*d^2*f+24*c*sqrt(-c*exp(2))*b^2*g*exp(1)^6*d^4-84*c*exp(
2)*sqrt(-c*exp(2))*b^2*g*exp(1)^4*d^4+69*c*exp(2)^2*sqrt(-c*exp(2))*b^2*g*exp(1)^2*d^4-14*c*exp(2)^3*sqrt(-c*e
xp(2))*b^2*g*d^4+36*c*exp(2)*sqrt(-c*exp(2))*b^2*exp(1)^5*d^3*f-41*c*exp(2)^2*sqrt(-c*exp(2))*b^2*exp(1)^3*d^3
*f+10*c*exp(2)^3*sqrt(-c*exp(2))*b^2*exp(1)*d^3*f-24*c^2*sqrt(-c*exp(2))*b*g*exp(1)^5*d^5+84*c^2*exp(2)*sqrt(-
c*exp(2))*b*g*exp(1)^3*d^5-40*c^2*exp(2)^2*sqrt(-c*exp(2))*b*g*exp(1)*d^5-48*c^2*exp(2)*sqrt(-c*exp(2))*b*exp(
1)^4*d^4*f+28*c^2*exp(2)^2*sqrt(-c*exp(2))*b*exp(1)^2*d^4*f+8*c^3*sqrt(-c*exp(2))*g*exp(1)^4*d^6-28*c^3*exp(2)
*sqrt(-c*exp(2))*g*exp(1)^2*d^6+20*c^3*exp(2)*sqrt(-c*exp(2))*exp(1)^3*d^5*f)/8/exp(1)^5/((sqrt(-b*d*exp(1)-b*
x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^2*exp(1)-2*sqrt(-c*exp(2))*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-
c*x^2*exp(2))-sqrt(-c*exp(2))*x)*d+b*exp(1)^2*d-exp(2)*b*d-c*exp(1)*d^2)^2)

________________________________________________________________________________________

maple [B]  time = 0.07, size = 2535, normalized size = 9.35

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9*e^4*c^2/(-b*e^2+2*c*d*e)^2*b^2/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-(x+d
/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2))*d*f-18*e^2*c^3/(-b*e^2+2*c*d*e)^2*b/(c*e^2)^(1/2)*arctan((c*e^2)^
(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2))*d^3*g-6*e^2*c^2/(-
b*e^2+2*c*d*e)^2*b*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*x*d*g+18*e^3*c^3/(-b*e^2+2*c*d*e)^2*b/(c*
e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))
^(1/2))*d^2*f-3/2*e^4*c/(-b*e^2+2*c*d*e)^2*b^3/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/
c/e^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2))*d*g+9*e^3*c^2/(-b*e^2+2*c*d*e)^2*b^2/(c*e^2)^(1/2)*a
rctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2))*d^2*
g-9/2*g*e*c^2/(-b*e^2+2*c*d*e)*b/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-(x+d/
e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2))*d^2+9/4*g*e^2*c/(-b*e^2+2*c*d*e)*b^2/(c*e^2)^(1/2)*arctan((c*e^2)^
(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2))*d+3*g*c^3/(-b*e^2+
2*c*d*e)*d^3/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2
*c*d*e)*(x+d/e))^(1/2))+3*g*c^2/(-b*e^2+2*c*d*e)*d*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*x+3/2*g*c
/(-b*e^2+2*c*d*e)*d*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*b-3/8*g*e^3/(-b*e^2+2*c*d*e)*b^3/(c*e^2)
^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/
2))+2/e^4/(-b*e^2+2*c*d*e)/(x+d/e)^3*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(5/2)*d*g-8/e*c/(-b*e^2+2*c*d
*e)^2/(x+d/e)^2*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(5/2)*f+3*e^3*c/(-b*e^2+2*c*d*e)^2*b^2*(-(x+d/e)^2
*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*f+2*g/e^3/(-b*e^2+2*c*d*e)/(x+d/e)^2*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)
*(x+d/e))^(5/2)+2*g/e*c/(-b*e^2+2*c*d*e)*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(3/2)-3/4*g*e/(-b*e^2+2*c
*d*e)*b^2*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)-2/e^3/(-b*e^2+2*c*d*e)/(x+d/e)^3*(-(x+d/e)^2*c*e^2
+(-b*e^2+2*c*d*e)*(x+d/e))^(5/2)*f+8*c^2/(-b*e^2+2*c*d*e)^2*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(3/2)*
d*g-8*e*c^2/(-b*e^2+2*c*d*e)^2*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(3/2)*f+6*e*c^2/(-b*e^2+2*c*d*e)^2*
d^2*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*b*g-6*e^2*c^2/(-b*e^2+2*c*d*e)^2*d*(-(x+d/e)^2*c*e^2+(-b
*e^2+2*c*d*e)*(x+d/e))^(1/2)*b*f+12*e*c^4/(-b*e^2+2*c*d*e)^2*d^4/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2
*(-b*e^2+2*c*d*e)/c/e^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2))*g-12*e^2*c^4/(-b*e^2+2*c*d*e)^2*d^
3/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+
d/e))^(1/2))*f+12*e*c^3/(-b*e^2+2*c*d*e)^2*d^2*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*x*g-3/2*g*e*c
/(-b*e^2+2*c*d*e)*b*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*x-12*e^2*c^3/(-b*e^2+2*c*d*e)^2*d*(-(x+d
/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*x*f+8/e^2*c/(-b*e^2+2*c*d*e)^2/(x+d/e)^2*(-(x+d/e)^2*c*e^2+(-b*e^2
+2*c*d*e)*(x+d/e))^(5/2)*d*g+6*e^3*c^2/(-b*e^2+2*c*d*e)^2*b*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*
x*f-3*e^2*c/(-b*e^2+2*c*d*e)^2*b^2*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*d*g+3/2*e^5*c/(-b*e^2+2*c
*d*e)^2*b^3/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*
c*d*e)*(x+d/e))^(1/2))*f

________________________________________________________________________________________

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*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/(e*x+d)^3,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?

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________