3.20.47 \(\int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^3} \, dx\)
Optimal. Leaf size=168 \[ -\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^3 (2 c d-b e)}-\frac {2 g \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 (d+e x)}-\frac {\sqrt {c} g \tan ^{-1}\left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{e^2} \]
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Rubi [A] time = 0.32, antiderivative size = 168, normalized size of antiderivative = 1.00,
number of steps used = 4, number of rules used = 4, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used =
{792, 662, 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 )^{3/2}}{3 e^2 (d+e x)^3 (2 c d-b e)}-\frac {2 g \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 (d+e x)}-\frac {\sqrt {c} 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 )}{e^2} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[((f + g*x)*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2])/(d + e*x)^3,x]
[Out]
(-2*g*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(e^2*(d + e*x)) - (2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c
*e^2*x^2)^(3/2))/(3*e^2*(2*c*d - b*e)*(d + e*x)^3) - (Sqrt[c]*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])])/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 662
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 + p + 1)), x] - Dist[(c*p)/(e^2*(m + p + 1)), Int[(d + e*x)^(m + 2)*(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] && (LtQ[m, -2] || EqQ[m + 2*p + 1, 0]) && NeQ[m + 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) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^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 )^{3/2}}{3 e^2 (2 c d-b e) (d+e x)^3}+\frac {g \int \frac {\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^2} \, dx}{e}\\ &=-\frac {2 g \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 (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 )^{3/2}}{3 e^2 (2 c d-b e) (d+e x)^3}-\frac {(c g) \int \frac {1}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{e}\\ &=-\frac {2 g \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 (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 )^{3/2}}{3 e^2 (2 c d-b e) (d+e x)^3}-\frac {(2 c 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 )}{e}\\ &=-\frac {2 g \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 (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 )^{3/2}}{3 e^2 (2 c d-b e) (d+e x)^3}-\frac {\sqrt {c} 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 )}{e^2}\\ \end {align*}
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Mathematica [C] time = 0.22, size = 146, normalized size = 0.87 \begin {gather*} \frac {2 \sqrt {(d+e x) (c (d-e x)-b e)} \left ((b e-c d+c e x) (-b e g+c d g+c e f)+\frac {g (b e-2 c d)^2 \, _2F_1\left (-\frac {3}{2},-\frac {3}{2};-\frac {1}{2};\frac {c (d+e x)}{2 c d-b e}\right )}{\sqrt {\frac {b e-c d+c e x}{b e-2 c d}}}\right )}{3 c e^2 (d+e x)^2 (2 c d-b e)} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[((f + g*x)*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2])/(d + e*x)^3,x]
[Out]
(2*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*((c*e*f + c*d*g - b*e*g)*(-(c*d) + b*e + c*e*x) + ((-2*c*d + b*e)^2*
g*Hypergeometric2F1[-3/2, -3/2, -1/2, (c*(d + e*x))/(2*c*d - b*e)])/Sqrt[(-(c*d) + b*e + c*e*x)/(-2*c*d + b*e)
]))/(3*c*e^2*(2*c*d - b*e)*(d + e*x)^2)
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IntegrateAlgebraic [A] time = 2.41, size = 290, normalized size = 1.73 \begin {gather*} -\frac {g \sqrt {-c e^2} \log \left (b^2 e^2-8 c x \sqrt {-c e^2} \sqrt {-b d e-b e^2 x+c d^2-c e^2 x^2}-4 b c d e-4 b c e^2 x+4 c^2 d^2-8 c^2 e^2 x^2\right )}{2 e^3}-\frac {2 \sqrt {-b d e-b e^2 x+c d^2-c e^2 x^2} \left (2 b d e g+b e^2 f+3 b e^2 g x-5 c d^2 g-c d e f-7 c d e g x+c e^2 f x\right )}{3 e^2 (d+e x)^2 (b e-2 c d)}-\frac {\sqrt {c} g \tan ^{-1}\left (\frac {2 \sqrt {c} x \sqrt {-c e^2}}{b e}-\frac {2 \sqrt {c} \sqrt {-b d e-b e^2 x+c d^2-c e^2 x^2}}{b e}\right )}{e^2} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[((f + g*x)*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2])/(d + e*x)^3,x]
[Out]
(-2*(-(c*d*e*f) + b*e^2*f - 5*c*d^2*g + 2*b*d*e*g + c*e^2*f*x - 7*c*d*e*g*x + 3*b*e^2*g*x)*Sqrt[c*d^2 - b*d*e
- b*e^2*x - c*e^2*x^2])/(3*e^2*(-2*c*d + b*e)*(d + e*x)^2) - (Sqrt[c]*g*ArcTan[(2*Sqrt[c]*Sqrt[-(c*e^2)]*x)/(b
*e) - (2*Sqrt[c]*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2])/(b*e)])/e^2 - (Sqrt[-(c*e^2)]*g*Log[4*c^2*d^2 - 4*
b*c*d*e + b^2*e^2 - 4*b*c*e^2*x - 8*c^2*e^2*x^2 - 8*c*Sqrt[-(c*e^2)]*x*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^
2]])/(2*e^3)
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fricas [A] time = 1.50, size = 579, normalized size = 3.45 \begin {gather*} \left [\frac {3 \, {\left ({\left (2 \, c d e^{2} - b e^{3}\right )} g x^{2} + 2 \, {\left (2 \, c d^{2} e - b d e^{2}\right )} g x + {\left (2 \, c d^{3} - b d^{2} e\right )} g\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 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left ({\left (c d e - b e^{2}\right )} f + {\left (5 \, c d^{2} - 2 \, b d e\right )} g - {\left (c e^{2} f - {\left (7 \, c d e - 3 \, b e^{2}\right )} g\right )} x\right )}}{6 \, {\left (2 \, c d^{3} e^{2} - b d^{2} e^{3} + {\left (2 \, c d e^{4} - b e^{5}\right )} x^{2} + 2 \, {\left (2 \, c d^{2} e^{3} - b d e^{4}\right )} x\right )}}, \frac {3 \, {\left ({\left (2 \, c d e^{2} - b e^{3}\right )} g x^{2} + 2 \, {\left (2 \, c d^{2} e - b d e^{2}\right )} g x + {\left (2 \, c d^{3} - b d^{2} e\right )} g\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 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left ({\left (c d e - b e^{2}\right )} f + {\left (5 \, c d^{2} - 2 \, b d e\right )} g - {\left (c e^{2} f - {\left (7 \, c d e - 3 \, b e^{2}\right )} g\right )} x\right )}}{3 \, {\left (2 \, c d^{3} e^{2} - b d^{2} e^{3} + {\left (2 \, c d e^{4} - b e^{5}\right )} x^{2} + 2 \, {\left (2 \, c d^{2} e^{3} - b d e^{4}\right )} x\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)^(1/2)/(e*x+d)^3,x, algorithm="fricas")
[Out]
[1/6*(3*((2*c*d*e^2 - b*e^3)*g*x^2 + 2*(2*c*d^2*e - b*d*e^2)*g*x + (2*c*d^3 - b*d^2*e)*g)*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*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*((c*d*e - b*e^2)*f + (5*c*d^2 - 2*b*d*e)*g -
(c*e^2*f - (7*c*d*e - 3*b*e^2)*g)*x))/(2*c*d^3*e^2 - b*d^2*e^3 + (2*c*d*e^4 - b*e^5)*x^2 + 2*(2*c*d^2*e^3 - b*
d*e^4)*x), 1/3*(3*((2*c*d*e^2 - b*e^3)*g*x^2 + 2*(2*c*d^2*e - b*d*e^2)*g*x + (2*c*d^3 - b*d^2*e)*g)*sqrt(c)*ar
ctan(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*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*((c*d*e - b*e^2)*f + (5*c*d^2 - 2*b*d*e)*g - (c*e^
2*f - (7*c*d*e - 3*b*e^2)*g)*x))/(2*c*d^3*e^2 - b*d^2*e^3 + (2*c*d*e^4 - b*e^5)*x^2 + 2*(2*c*d^2*e^3 - b*d*e^4
)*x)]
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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)^(1/2)/(e*x+d)^3,x, algorithm="giac")
[Out]
Exception raised: NotImplementedError >> Unable to parse Giac output: 2*(-c*sqrt(-c*exp(2))*g/2/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))+(4*exp(2)
*g*exp(1)^4*b^2*d-16*c*exp(2)*g*exp(1)^3*b*d^2-3*exp(2)^2*g*exp(1)^2*b^2*d+12*c^2*exp(2)*g*exp(1)^2*d^3+12*c*e
xp(2)^2*g*exp(1)*b*d^2-8*c^2*exp(2)^2*g*d^3+4*c*exp(2)*exp(1)^4*b*d*f-exp(2)^2*exp(1)^3*b^2*f-4*c^2*exp(2)*exp
(1)^3*d^2*f)/2/(-4*exp(1)^6*b*d+4*c*exp(1)^5*d^2+4*exp(2)*exp(1)^4*b*d-4*c*exp(2)*exp(1)^3*d^2)/sqrt(b*d*exp(1
)^3-c*d^2*exp(1)^2+c*d^2*exp(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*g*exp(1)^5*b^2*d-16*c*exp(2)*(sq
rt(-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)^4*b*d^2-5*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)^3*b^2*d+12*c^2*exp(2)*(sqrt(-b*d*exp(1)-b*x*ex
p(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)^3*g*exp(1)^3*d^3+20*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*g*exp(1)^2*b*d^2-16*c^2*exp(2)^2*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*e
xp(2))-sqrt(-c*exp(2))*x)^3*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*e
xp(2))*x)^3*exp(1)^5*b*d*f+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)^4*b^2*f-4*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*exp(1)^4*d^2*f-8
*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*exp(1)^3*b*d*f+8*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*g*exp(1)^6*b^2*d^2+16*c*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)^5*b*d^3+12*exp(2)*sqrt(-c*exp(2))*(s
qrt(-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*b^2*d^2-8*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^4-8*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*g*exp(1)^3*b*d^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*g*exp(1)^2*b^2*d^2-4*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*g*exp(1)^2*d^4-20*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*g*exp(1)*b*d^3+24*c^2*ex
p(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^4-8*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*exp(1)^5*b^2*d*f+12*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*exp(1)^4*b*d^2*f+5*ex
p(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)^3*b^2*d*f-
4*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-8*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*exp(1
)*d^3*f+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)*g*exp(1)^6*b^3*d^2-28*c*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)*g*exp(1)^5*b^2*d^3-7*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)^4*b^3*d^2+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)*g*exp(1)^4*b*d^4+55*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)*g*exp(1)^3*b^2*d^3-20*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+3*exp(2)^3*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sq
rt(-c*exp(2))*x)*g*exp(1)^2*b^3*d^2-80*c^2*exp(2)^2*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*e
xp(2))*x)*g*exp(1)^2*b*d^4-24*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)*g
*exp(1)*b^2*d^3+32*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+24*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)*g*b*d^4-4*c*exp(2)*(sqr
t(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqrt(-c*exp(2))*x)*exp(1)^6*b^2*d^2*f-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)*exp(1)^5*b^3*d*f+8*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)*exp(1)^5*b*d^3*f-3*c*exp(2)^2*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*e
xp(2))-sqrt(-c*exp(2))*x)*exp(1)^4*b^2*d^2*f-4*c^3*exp(2)*(sqrt(-b*d*exp(1)-b*x*exp(2)+c*d^2-c*x^2*exp(2))-sqr
t(-c*exp(2))*x)*exp(1)^4*d^4*f+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)*ex
p(1)^3*b^3*d*f+12*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)*exp(1)^3*b*
d^3*f+4*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)*exp(1)^2*b^2*d^2*f-8*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)*exp(1)^2*d^4*f-8*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)*exp(1)*b*d^3*f-8*sqrt(-c*exp(2))*g*exp(1)^7
*b^3*d^3+24*c*sqrt(-c*exp(2))*g*exp(1)^6*b^2*d^4+20*exp(2)*sqrt(-c*exp(2))*g*exp(1)^5*b^3*d^3-24*c^2*sqrt(-c*e
xp(2))*g*exp(1)^5*b*d^5-52*c*exp(2)*sqrt(-c*exp(2))*g*exp(1)^4*b^2*d^4+8*c^3*sqrt(-c*exp(2))*g*exp(1)^4*d^6-17
*exp(2)^2*sqrt(-c*exp(2))*g*exp(1)^3*b^3*d^3+44*c^2*exp(2)*sqrt(-c*exp(2))*g*exp(1)^3*b*d^5+33*c*exp(2)^2*sqrt
(-c*exp(2))*g*exp(1)^2*b^2*d^4-12*c^3*exp(2)*sqrt(-c*exp(2))*g*exp(1)^2*d^6+5*exp(2)^3*sqrt(-c*exp(2))*g*exp(1
)*b^3*d^3-16*c^2*exp(2)^2*sqrt(-c*exp(2))*g*exp(1)*b*d^5-6*c*exp(2)^3*sqrt(-c*exp(2))*g*b^2*d^4+4*c*exp(2)*sqr
t(-c*exp(2))*exp(1)^5*b^2*d^3*f+exp(2)^2*sqrt(-c*exp(2))*exp(1)^4*b^3*d^2*f-8*c^2*exp(2)*sqrt(-c*exp(2))*exp(1
)^4*b*d^4*f-5*c*exp(2)^2*sqrt(-c*exp(2))*exp(1)^3*b^2*d^3*f+4*c^3*exp(2)*sqrt(-c*exp(2))*exp(1)^3*d^5*f-exp(2)
^3*sqrt(-c*exp(2))*exp(1)^2*b^3*d^2*f+4*c^2*exp(2)^2*sqrt(-c*exp(2))*exp(1)^2*b*d^4*f+2*c*exp(2)^3*sqrt(-c*exp
(2))*exp(1)*b^2*d^3*f)/(8*exp(1)^6*b*d-8*c*exp(1)^5*d^2-8*exp(2)*exp(1)^4*b*d+8*c*exp(2)*exp(1)^3*d^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)-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+exp(1)^2*b*d-c*exp(1)*d^2-exp(2)*b*d)^2)
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maple [B] time = 0.07, size = 404, normalized size = 2.40 \begin {gather*} \frac {b c e g \arctan \left (\frac {\sqrt {c \,e^{2}}\, \left (x +\frac {d}{e}-\frac {-b \,e^{2}+2 c d e}{2 c \,e^{2}}\right )}{\sqrt {-\left (x +\frac {d}{e}\right )^{2} c \,e^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )}}\right )}{\left (-b \,e^{2}+2 c d e \right ) \sqrt {c \,e^{2}}}-\frac {2 c^{2} d g \arctan \left (\frac {\sqrt {c \,e^{2}}\, \left (x +\frac {d}{e}-\frac {-b \,e^{2}+2 c d e}{2 c \,e^{2}}\right )}{\sqrt {-\left (x +\frac {d}{e}\right )^{2} c \,e^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )}}\right )}{\left (-b \,e^{2}+2 c d e \right ) \sqrt {c \,e^{2}}}-\frac {2 \sqrt {-\left (x +\frac {d}{e}\right )^{2} c \,e^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )}\, c g}{\left (-b \,e^{2}+2 c d e \right ) e}-\frac {2 \left (-\left (x +\frac {d}{e}\right )^{2} c \,e^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}} g}{\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )^{2} e^{3}}-\frac {2 \left (-d g +e f \right ) \left (-\left (x +\frac {d}{e}\right )^{2} c \,e^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{3 \left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )^{3} e^{4}} \end {gather*}
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)^(1/2)/(e*x+d)^3,x)
[Out]
-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))^(3/2)-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))^(1/2)+g*e*c/(-b*e^2+2*c*d*e)/(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))*b-2*g*c^2/(-b*e^2+2*c*d*e
)/(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-2/3*(-d*g+e*f)/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))^(3/2)
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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*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(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?
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (f+g\,x\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{{\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)^(1/2))/(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)^(1/2))/(d + e*x)^3, x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {- \left (d + e x\right ) \left (b e - c d + c e x\right )} \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)**(1/2)/(e*x+d)**3,x)
[Out]
Integral(sqrt(-(d + e*x)*(b*e - c*d + c*e*x))*(f + g*x)/(d + e*x)**3, x)
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