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

Optimal. Leaf size=213 \[ -\frac {(-3 b e g+4 c d g+2 c e f) \tan ^{-1}\left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{2 c^{5/2} e^2}+\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-3 b e g+4 c d g+2 c e f)}{c^2 e^2 (2 c d-b e)}+\frac {2 (d+e x)^2 (-b e g+c d g+c e f)}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}} \]

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Rubi [A]  time = 0.28, antiderivative size = 213, 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 = {788, 640, 621, 204} \begin {gather*} \frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-3 b e g+4 c d g+2 c e f)}{c^2 e^2 (2 c d-b e)}-\frac {(-3 b e g+4 c d g+2 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 )}{2 c^{5/2} e^2}+\frac {2 (d+e x)^2 (-b e g+c d g+c e f)}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(c*e*f + c*d*g - b*e*g)*(d + e*x)^2)/(c*e^2*(2*c*d - b*e)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) + ((2*
c*e*f + 4*c*d*g - 3*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(c^2*e^2*(2*c*d - b*e)) - ((2*c*e*f + 4*
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])])/(2*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 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 640

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

Rule 788

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((g*(c*d - b*e) + c*e*f)*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/(c*(p + 1)*(2*c*d - b*e)), x] - Dist[(e*(m*(g
*(c*d - b*e) + c*e*f) + e*(p + 1)*(2*c*f - b*g)))/(c*(p + 1)*(2*c*d - b*e)), Int[(d + e*x)^(m - 1)*(a + b*x +
c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2,
0] && LtQ[p, -1] && GtQ[m, 0]

Rubi steps

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

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Mathematica [A]  time = 0.58, size = 201, normalized size = 0.94 \begin {gather*} \frac {\sqrt {c} \sqrt {e} (d+e x) \sqrt {e (2 c d-b e)} (c (3 d g+2 e f-e g x)-3 b e g)+e \sqrt {d+e x} (b e-2 c d) \sqrt {\frac {b e-c d+c e x}{b e-2 c d}} (-3 b e g+4 c d g+2 c e f) \sin ^{-1}\left (\frac {\sqrt {c} \sqrt {e} \sqrt {d+e x}}{\sqrt {e (2 c d-b e)}}\right )}{c^{5/2} e^{5/2} \sqrt {e (2 c d-b e)} \sqrt {(d+e x) (c (d-e x)-b e)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[c]*Sqrt[e]*Sqrt[e*(2*c*d - b*e)]*(d + e*x)*(-3*b*e*g + c*(2*e*f + 3*d*g - e*g*x)) + e*(-2*c*d + b*e)*(2*
c*e*f + 4*c*d*g - 3*b*e*g)*Sqrt[d + e*x]*Sqrt[(-(c*d) + b*e + c*e*x)/(-2*c*d + b*e)]*ArcSin[(Sqrt[c]*Sqrt[e]*S
qrt[d + e*x])/Sqrt[e*(2*c*d - b*e)]])/(c^(5/2)*e^(5/2)*Sqrt[e*(2*c*d - b*e)]*Sqrt[(d + e*x)*(-(b*e) + c*(d - e
*x))])

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IntegrateAlgebraic [A]  time = 6.38, size = 285, normalized size = 1.34 \begin {gather*} -\frac {\sqrt {-c e^2} (-3 b e g+4 c d g+2 c e f) \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 )}{4 c^3 e^3}+\frac {(-3 b e g+4 c d g+2 c e f) \tan ^{-1}\left (\frac {\sqrt {c} \left (2 \sqrt {-b d e-b e^2 x+c d^2-c e^2 x^2}-2 x \sqrt {-c e^2}\right )}{b e}\right )}{2 c^{5/2} e^2}+\frac {\sqrt {-b d e-b e^2 x+c d^2-c e^2 x^2} (3 b e g-3 c d g-2 c e f+c e g x)}{c^2 e^2 (b e-c d+c e x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.95, size = 503, normalized size = 2.36 \begin {gather*} \left [-\frac {{\left (2 \, {\left (c^{2} d e - b c e^{2}\right )} f + {\left (4 \, c^{2} d^{2} - 7 \, b c d e + 3 \, b^{2} e^{2}\right )} g - {\left (2 \, c^{2} e^{2} f + {\left (4 \, c^{2} d e - 3 \, b c e^{2}\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 (c^{2} e g x - 2 \, c^{2} e f - 3 \, {\left (c^{2} d - b c e\right )} g\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}{4 \, {\left (c^{4} e^{3} x - c^{4} d e^{2} + b c^{3} e^{3}\right )}}, -\frac {{\left (2 \, {\left (c^{2} d e - b c e^{2}\right )} f + {\left (4 \, c^{2} d^{2} - 7 \, b c d e + 3 \, b^{2} e^{2}\right )} g - {\left (2 \, c^{2} e^{2} f + {\left (4 \, c^{2} d e - 3 \, b c e^{2}\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 (c^{2} e g x - 2 \, c^{2} e f - 3 \, {\left (c^{2} d - b c e\right )} g\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}{2 \, {\left (c^{4} e^{3} x - c^{4} d e^{2} + b c^{3} e^{3}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/4*((2*(c^2*d*e - b*c*e^2)*f + (4*c^2*d^2 - 7*b*c*d*e + 3*b^2*e^2)*g - (2*c^2*e^2*f + (4*c^2*d*e - 3*b*c*e^
2)*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*(c^2*e*g*x - 2*c^2*e*f - 3*(c^2*d - b*c*e)*g)*sqrt(-c*e^2*x
^2 - b*e^2*x + c*d^2 - b*d*e))/(c^4*e^3*x - c^4*d*e^2 + b*c^3*e^3), -1/2*((2*(c^2*d*e - b*c*e^2)*f + (4*c^2*d^
2 - 7*b*c*d*e + 3*b^2*e^2)*g - (2*c^2*e^2*f + (4*c^2*d*e - 3*b*c*e^2)*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*(c^2*e*
g*x - 2*c^2*e*f - 3*(c^2*d - b*c*e)*g)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))/(c^4*e^3*x - c^4*d*e^2 + b*
c^3*e^3)]

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giac [B]  time = 0.62, size = 427, normalized size = 2.00 \begin {gather*} \frac {\sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e} {\left ({\left (\frac {{\left (4 \, c^{3} d^{2} g e^{3} - 4 \, b c^{2} d g e^{4} + b^{2} c g e^{5}\right )} x}{4 \, c^{4} d^{2} e^{3} - 4 \, b c^{3} d e^{4} + b^{2} c^{2} e^{5}} - \frac {8 \, c^{3} d^{3} g e^{2} + 8 \, c^{3} d^{2} f e^{3} - 20 \, b c^{2} d^{2} g e^{3} - 8 \, b c^{2} d f e^{4} + 14 \, b^{2} c d g e^{4} + 2 \, b^{2} c f e^{5} - 3 \, b^{3} g e^{5}}{4 \, c^{4} d^{2} e^{3} - 4 \, b c^{3} d e^{4} + b^{2} c^{2} e^{5}}\right )} x - \frac {12 \, c^{3} d^{4} g e + 8 \, c^{3} d^{3} f e^{2} - 24 \, b c^{2} d^{3} g e^{2} - 8 \, b c^{2} d^{2} f e^{3} + 15 \, b^{2} c d^{2} g e^{3} + 2 \, b^{2} c d f e^{4} - 3 \, b^{3} d g e^{4}}{4 \, c^{4} d^{2} e^{3} - 4 \, b c^{3} d e^{4} + b^{2} c^{2} e^{5}}\right )}}{c x^{2} e^{2} - c d^{2} + b x e^{2} + b d e} - \frac {{\left (4 \, c d g + 2 \, c f e - 3 \, b g e\right )} \sqrt {-c e^{2}} e^{\left (-3\right )} \log \left ({\left | -2 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )} c - \sqrt {-c e^{2}} b \right |}\right )}{2 \, c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e)*(((4*c^3*d^2*g*e^3 - 4*b*c^2*d*g*e^4 + b^2*c*g*e^5)*x/(4*c^4*d^2*e^
3 - 4*b*c^3*d*e^4 + b^2*c^2*e^5) - (8*c^3*d^3*g*e^2 + 8*c^3*d^2*f*e^3 - 20*b*c^2*d^2*g*e^3 - 8*b*c^2*d*f*e^4 +
14*b^2*c*d*g*e^4 + 2*b^2*c*f*e^5 - 3*b^3*g*e^5)/(4*c^4*d^2*e^3 - 4*b*c^3*d*e^4 + b^2*c^2*e^5))*x - (12*c^3*d^
4*g*e + 8*c^3*d^3*f*e^2 - 24*b*c^2*d^3*g*e^2 - 8*b*c^2*d^2*f*e^3 + 15*b^2*c*d^2*g*e^3 + 2*b^2*c*d*f*e^4 - 3*b^
3*d*g*e^4)/(4*c^4*d^2*e^3 - 4*b*c^3*d*e^4 + b^2*c^2*e^5))/(c*x^2*e^2 - c*d^2 + b*x*e^2 + b*d*e) - 1/2*(4*c*d*g
+ 2*c*f*e - 3*b*g*e)*sqrt(-c*e^2)*e^(-3)*log(abs(-2*(sqrt(-c*e^2)*x - sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d
*e))*c - sqrt(-c*e^2)*b))/c^3

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maple [B]  time = 0.06, size = 1320, normalized size = 6.20 \begin {gather*} \frac {3 b^{3} e^{4} g x}{2 \left (-b^{2} e^{4}+4 b c d \,e^{3}-4 d^{2} e^{2} c^{2}\right ) \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, c^{2}}-\frac {6 b^{2} d \,e^{3} g x}{\left (-b^{2} e^{4}+4 b c d \,e^{3}-4 d^{2} e^{2} c^{2}\right ) \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, c}-\frac {b^{2} e^{4} f x}{\left (-b^{2} e^{4}+4 b c d \,e^{3}-4 d^{2} e^{2} c^{2}\right ) \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, c}+\frac {6 b \,d^{2} e^{2} g x}{\left (-b^{2} e^{4}+4 b c d \,e^{3}-4 d^{2} e^{2} c^{2}\right ) \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}+\frac {4 b d \,e^{3} f x}{\left (-b^{2} e^{4}+4 b c d \,e^{3}-4 d^{2} e^{2} c^{2}\right ) \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}+\frac {3 b^{4} e^{4} g}{4 \left (-b^{2} e^{4}+4 b c d \,e^{3}-4 d^{2} e^{2} c^{2}\right ) \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, c^{3}}-\frac {3 b^{3} d \,e^{3} g}{\left (-b^{2} e^{4}+4 b c d \,e^{3}-4 d^{2} e^{2} c^{2}\right ) \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, c^{2}}-\frac {b^{3} e^{4} f}{2 \left (-b^{2} e^{4}+4 b c d \,e^{3}-4 d^{2} e^{2} c^{2}\right ) \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, c^{2}}+\frac {3 b^{2} d^{2} e^{2} g}{\left (-b^{2} e^{4}+4 b c d \,e^{3}-4 d^{2} e^{2} c^{2}\right ) \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, c}+\frac {2 b^{2} d \,e^{3} f}{\left (-b^{2} e^{4}+4 b c d \,e^{3}-4 d^{2} e^{2} c^{2}\right ) \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, c}+\frac {2 \left (-2 c \,e^{2} x -b \,e^{2}\right ) d^{2} f}{\left (-b^{2} e^{4}-4 \left (-b d e +c \,d^{2}\right ) c \,e^{2}\right ) \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}-\frac {g \,x^{2}}{\sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, c}-\frac {3 b g x}{2 \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, c^{2}}+\frac {3 b g \arctan \left (\frac {\sqrt {c \,e^{2}}\, \left (x +\frac {b}{2 c}\right )}{\sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}\right )}{2 \sqrt {c \,e^{2}}\, c^{2}}+\frac {2 d g x}{\sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, c e}-\frac {2 d g \arctan \left (\frac {\sqrt {c \,e^{2}}\, \left (x +\frac {b}{2 c}\right )}{\sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}\right )}{\sqrt {c \,e^{2}}\, c e}+\frac {f x}{\sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, c}-\frac {f \arctan \left (\frac {\sqrt {c \,e^{2}}\, \left (x +\frac {b}{2 c}\right )}{\sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}\right )}{\sqrt {c \,e^{2}}\, c}+\frac {3 b^{2} g}{4 \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, c^{3}}-\frac {3 b d g}{\sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, c^{2} e}-\frac {b f}{2 \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, c^{2}}+\frac {3 d^{2} g}{\sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, c \,e^{2}}+\frac {2 d f}{\sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, c e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x/c/e/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d*g-1/2*b^3/c^2/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)/(-c*e^2*x^
2-b*e^2*x-b*d*e+c*d^2)^(1/2)*e^4*f-2/c/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))*d*g-3/e*g/c^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*b*d+3/4*e^4*g*b^4/c^3/(-b^2*e^4+4*b*c*d
*e^3-4*c^2*d^2*e^2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)-6*e^3*g/c*b^2/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)/
(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d+3/2*e^4*g*b^3/c^2/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)/(-c*e^2*x^2-
b*e^2*x-b*d*e+c*d^2)^(1/2)*x+6*e^2*g*b/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(
1/2)*x*d^2-g*x^2/c/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)+3/4*g*b^2/c^3/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)
+2/c/e/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d*f-3/2*g*b/c^2*x/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)+3/2*g*b
/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))+3/e^2*g/c/(-c*e^2*
x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d^2+2*b^2/c/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^
2)^(1/2)*e^3*d*f+2*d^2*f*(-2*c*e^2*x-b*e^2)/(-b^2*e^4-4*(-b*d*e+c*d^2)*c*e^2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)
^(1/2)-3*e^3*g/c^2*b^3/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d+3*e^2*g/c
*b^2/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d^2-b^2/c/(-b^2*e^4+4*b*c*d*e
^3-4*c^2*d^2*e^2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*e^4*x*f+4*b/(-b^2*e^4+4*b*c*d*e^3-4*c^2*d^2*e^2)/(-c*
e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*e^3*x*d*f+x/c/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*f-1/2*b/c^2/(-c*e^2*x^
2-b*e^2*x-b*d*e+c*d^2)^(1/2)*f-1/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))*f

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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