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

Optimal. Leaf size=266 \[ \frac {(2 c d-b e)^3 (-3 b e g-2 c d g+8 c e f) \tan ^{-1}\left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{128 c^{5/2} e^2}+\frac {(b+2 c x) (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-3 b e g-2 c d g+8 c e f)}{64 c^2 e}+\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-3 b e g-2 c d g+8 c e f)}{24 c e^2}-\frac {g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{4 c e^2 (d+e x)} \]

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Rubi [A]  time = 0.33, antiderivative size = 266, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {794, 664, 612, 621, 204} \begin {gather*} \frac {(b+2 c x) (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-3 b e g-2 c d g+8 c e f)}{64 c^2 e}+\frac {(2 c d-b e)^3 (-3 b e g-2 c d g+8 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 )}{128 c^{5/2} e^2}+\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-3 b e g-2 c d g+8 c e f)}{24 c e^2}-\frac {g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{4 c e^2 (d+e x)} \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),x]

[Out]

((2*c*d - b*e)*(8*c*e*f - 2*c*d*g - 3*b*e*g)*(b + 2*c*x)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(64*c^2*e)
+ ((8*c*e*f - 2*c*d*g - 3*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(24*c*e^2) - (g*(d*(c*d - b*e)
- b*e^2*x - c*e^2*x^2)^(5/2))/(4*c*e^2*(d + e*x)) + ((2*c*d - b*e)^3*(8*c*e*f - 2*c*d*g - 3*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])])/(128*c^(5/2)*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 612

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^p)/(2*c*(2*p +
1)), x] - Dist[(p*(b^2 - 4*a*c))/(2*c*(2*p + 1)), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

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 794

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[(g*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] + Dist[(m*(g*(c*d - b*e) + c*e*f) + e*(p + 1)
*(2*c*f - b*g))/(c*e*(m + 2*p + 2)), 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] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[m + 2*p + 2, 0] && (NeQ[m, 2] || Eq
Q[d, 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} \, dx &=-\frac {g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{4 c e^2 (d+e x)}-\frac {\left (c e^3 f-\left (-c d e^2+b e^3\right ) g+\frac {5}{2} e \left (-2 c e^2 f+b e^2 g\right )\right ) \int \frac {\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{d+e x} \, dx}{4 c e^3}\\ &=\frac {(8 c e f-2 c d g-3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{24 c e^2}-\frac {g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{4 c e^2 (d+e x)}+\frac {((2 c d-b e) (8 c e f-2 c d g-3 b e g)) \int \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx}{16 c e}\\ &=\frac {(2 c d-b e) (8 c e f-2 c d g-3 b e g) (b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{64 c^2 e}+\frac {(8 c e f-2 c d g-3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{24 c e^2}-\frac {g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{4 c e^2 (d+e x)}+\frac {\left ((2 c d-b e)^3 (8 c e f-2 c d g-3 b e g)\right ) \int \frac {1}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{128 c^2 e}\\ &=\frac {(2 c d-b e) (8 c e f-2 c d g-3 b e g) (b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{64 c^2 e}+\frac {(8 c e f-2 c d g-3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{24 c e^2}-\frac {g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{4 c e^2 (d+e x)}+\frac {\left ((2 c d-b e)^3 (8 c e f-2 c d g-3 b e g)\right ) \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 )}{64 c^2 e}\\ &=\frac {(2 c d-b e) (8 c e f-2 c d g-3 b e g) (b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{64 c^2 e}+\frac {(8 c e f-2 c d g-3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{24 c e^2}-\frac {g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{4 c e^2 (d+e x)}+\frac {(2 c d-b e)^3 (8 c e f-2 c d g-3 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 )}{128 c^{5/2} e^2}\\ \end {align*}

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Mathematica [A]  time = 2.57, size = 361, normalized size = 1.36 \begin {gather*} \frac {(d+e x) \sqrt {(d+e x) (c (d-e x)-b e)} \left (\frac {\sqrt {e (2 c d-b e)} \left (\frac {b e-c d+c e x}{b e-2 c d}\right )^{3/2} (-3 b e g-2 c d g+8 c e f) \left (2 c^2 e^8 (d+e x)^2 \sqrt {e (2 c d-b e)} (b e-2 c d) \sqrt {\frac {b e-c d+c e x}{b e-2 c d}} (7 b e-10 c d+4 c e x)-3 e^6 (2 c d-b e)^3 \left (\sqrt {c} e^{5/2} \sqrt {d+e x} (b e-2 c d) \sin ^{-1}\left (\frac {\sqrt {c} \sqrt {e} \sqrt {d+e x}}{\sqrt {e (2 c d-b e)}}\right )+c e^2 (d+e x) \sqrt {e (2 c d-b e)} \sqrt {\frac {b e-c d+c e x}{b e-2 c d}}\right )\right )}{16 c^2 e^7 (d+e x)^2 (b e-c d+c e x)^2}-3 e^2 g (b e-c d+c e x)^2\right )}{12 c e^4} \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),x]

[Out]

((d + e*x)*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(-3*e^2*g*(-(c*d) + b*e + c*e*x)^2 + (Sqrt[e*(2*c*d - b*e)]*
(8*c*e*f - 2*c*d*g - 3*b*e*g)*((-(c*d) + b*e + c*e*x)/(-2*c*d + b*e))^(3/2)*(2*c^2*e^8*Sqrt[e*(2*c*d - b*e)]*(
-2*c*d + b*e)*(d + e*x)^2*Sqrt[(-(c*d) + b*e + c*e*x)/(-2*c*d + b*e)]*(-10*c*d + 7*b*e + 4*c*e*x) - 3*e^6*(2*c
*d - b*e)^3*(c*e^2*Sqrt[e*(2*c*d - b*e)]*(d + e*x)*Sqrt[(-(c*d) + b*e + c*e*x)/(-2*c*d + b*e)] + Sqrt[c]*e^(5/
2)*(-2*c*d + b*e)*Sqrt[d + e*x]*ArcSin[(Sqrt[c]*Sqrt[e]*Sqrt[d + e*x])/Sqrt[e*(2*c*d - b*e)]])))/(16*c^2*e^7*(
d + e*x)^2*(-(c*d) + b*e + c*e*x)^2)))/(12*c*e^4)

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IntegrateAlgebraic [F]  time = 180.26, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[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),x]

[Out]

$Aborted

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fricas [A]  time = 0.59, size = 809, normalized size = 3.04 \begin {gather*} \left [-\frac {3 \, {\left (8 \, {\left (8 \, c^{4} d^{3} e - 12 \, b c^{3} d^{2} e^{2} + 6 \, b^{2} c^{2} d e^{3} - b^{3} c e^{4}\right )} f - {\left (16 \, c^{4} d^{4} - 24 \, b^{2} c^{2} d^{2} e^{2} + 16 \, b^{3} c d e^{3} - 3 \, b^{4} e^{4}\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 \, {\left (48 \, c^{4} e^{3} g x^{3} + 8 \, {\left (8 \, c^{4} e^{3} f - {\left (8 \, c^{4} d e^{2} - 9 \, b c^{3} e^{3}\right )} g\right )} x^{2} - 8 \, {\left (8 \, c^{4} d^{2} e - 2 \, b c^{3} d e^{2} - 3 \, b^{2} c^{2} e^{3}\right )} f + {\left (64 \, c^{4} d^{3} - 76 \, b c^{3} d^{2} e + 36 \, b^{2} c^{2} d e^{2} - 9 \, b^{3} c e^{3}\right )} g - 2 \, {\left (8 \, {\left (6 \, c^{4} d e^{2} - 7 \, b c^{3} e^{3}\right )} f + {\left (12 \, c^{4} d^{2} e - 4 \, b c^{3} d e^{2} - 3 \, b^{2} c^{2} e^{3}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}{768 \, c^{3} e^{2}}, -\frac {3 \, {\left (8 \, {\left (8 \, c^{4} d^{3} e - 12 \, b c^{3} d^{2} e^{2} + 6 \, b^{2} c^{2} d e^{3} - b^{3} c e^{4}\right )} f - {\left (16 \, c^{4} d^{4} - 24 \, b^{2} c^{2} d^{2} e^{2} + 16 \, b^{3} c d e^{3} - 3 \, b^{4} e^{4}\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 \, {\left (48 \, c^{4} e^{3} g x^{3} + 8 \, {\left (8 \, c^{4} e^{3} f - {\left (8 \, c^{4} d e^{2} - 9 \, b c^{3} e^{3}\right )} g\right )} x^{2} - 8 \, {\left (8 \, c^{4} d^{2} e - 2 \, b c^{3} d e^{2} - 3 \, b^{2} c^{2} e^{3}\right )} f + {\left (64 \, c^{4} d^{3} - 76 \, b c^{3} d^{2} e + 36 \, b^{2} c^{2} d e^{2} - 9 \, b^{3} c e^{3}\right )} g - 2 \, {\left (8 \, {\left (6 \, c^{4} d e^{2} - 7 \, b c^{3} e^{3}\right )} f + {\left (12 \, c^{4} d^{2} e - 4 \, b c^{3} d e^{2} - 3 \, b^{2} c^{2} e^{3}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}{384 \, c^{3} e^{2}}\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),x, algorithm="fricas")

[Out]

[-1/768*(3*(8*(8*c^4*d^3*e - 12*b*c^3*d^2*e^2 + 6*b^2*c^2*d*e^3 - b^3*c*e^4)*f - (16*c^4*d^4 - 24*b^2*c^2*d^2*
e^2 + 16*b^3*c*d*e^3 - 3*b^4*e^4)*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*(48*c^4*e^3*g*x^3 + 8*(8*c^4*e^
3*f - (8*c^4*d*e^2 - 9*b*c^3*e^3)*g)*x^2 - 8*(8*c^4*d^2*e - 2*b*c^3*d*e^2 - 3*b^2*c^2*e^3)*f + (64*c^4*d^3 - 7
6*b*c^3*d^2*e + 36*b^2*c^2*d*e^2 - 9*b^3*c*e^3)*g - 2*(8*(6*c^4*d*e^2 - 7*b*c^3*e^3)*f + (12*c^4*d^2*e - 4*b*c
^3*d*e^2 - 3*b^2*c^2*e^3)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))/(c^3*e^2), -1/384*(3*(8*(8*c^4*d^3
*e - 12*b*c^3*d^2*e^2 + 6*b^2*c^2*d*e^3 - b^3*c*e^4)*f - (16*c^4*d^4 - 24*b^2*c^2*d^2*e^2 + 16*b^3*c*d*e^3 - 3
*b^4*e^4)*g)*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*(48*c^4*e^3*g*x^3 + 8*(8*c^4*e^3*f - (8*c^4*d*e^2 - 9*b*c^3*e^3)*g)*x^
2 - 8*(8*c^4*d^2*e - 2*b*c^3*d*e^2 - 3*b^2*c^2*e^3)*f + (64*c^4*d^3 - 76*b*c^3*d^2*e + 36*b^2*c^2*d*e^2 - 9*b^
3*c*e^3)*g - 2*(8*(6*c^4*d*e^2 - 7*b*c^3*e^3)*f + (12*c^4*d^2*e - 4*b*c^3*d*e^2 - 3*b^2*c^2*e^3)*g)*x)*sqrt(-c
*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))/(c^3*e^2)]

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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*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(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 = 0.06, size = 1817, normalized size = 6.83

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),x)

[Out]

3/4*b*c/(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-1/2/e*c*d^2*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*x*g-1/2/e*c^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-1/16*e^3*b^3/c/(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-3/8*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^2*g+3/8*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+3/8*g/e*c^2/(c*e^2)^
(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*d^4-1/3/e^2*(-(x+d/e)^2*c*e^2+(
-b*e^2+2*c*d*e)*(x+d/e))^(3/2)*d*g+1/4*d*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*b*f+1/4*g/e*(-c*e^2
*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x-1/8*e*b^2/c*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*f+1/2*c*d*(-(x
+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*x*f+1/16*e^2*b^3/c/(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-3/4*e*b*c/(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+3/6
4*g*e/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*b^3-1/4*e*b*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)
*x*f+1/8*b^2/c*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*d*g+3/16*g/e*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)
^(1/2)*b*d^2-1/4/e*d^2*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*b*g+1/2*c^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+1/8*g/e/
c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*b-3/8*g*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*b*d+1/4*b*(-(x+d/e)^
2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*x*d*g-3/16*g/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*b^2*d-3/16*g*e^2
/c/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*b^3*d+1/3/e*(-(x+d/e
)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(3/2)*f-3/4*g*c/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2
-b*e^2*x-b*d*e+c*d^2)^(1/2))*b*d^3+3/128*g*e^3/c^2/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-
b*e^2*x-b*d*e+c*d^2)^(1/2))*b^4+9/16*g*e/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*
d*e+c*d^2)^(1/2))*b^2*d^2+3/32*g*e/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*b^2+3/8*g/e*c*(-c*e^2*x^2-b*e^2*
x-b*d*e+c*d^2)^(1/2)*x*d^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)^(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\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x\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)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(3/2))/(d + e*x),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), x)

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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 )}{d + e x}\, 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),x)

[Out]

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

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