3.23.18 \(\int \frac {(d+e x)^2 (f+g x)}{(c d^2-b d e-b e^2 x-c e^2 x^2)^{3/2}} \, dx\) [2218]

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

[Out]

-1/2*(-3*b*e*g+4*c*d*g+2*c*e*f)*arctan(1/2*e*(2*c*x+b)/c^(1/2)/(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2))/c^(5/2)
/e^2+2*(-b*e*g+c*d*g+c*e*f)*(e*x+d)^2/c/e^2/(-b*e+2*c*d)/(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)+(-3*b*e*g+4*c*
d*g+2*c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)/c^2/e^2/(-b*e+2*c*d)

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Rubi [A]
time = 0.19, 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 = {802, 654, 635, 210} \begin {gather*} -\frac {\text {ArcTan}\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 ) (-3 b e g+4 c d g+2 c e f)}{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}} \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 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 635

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 654

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 802

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) \text {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 = 0.27, size = 142, normalized size = 0.67 \begin {gather*} \frac {\sqrt {c} (d+e x) (-3 b e g+c (2 e f+3 d g-e g x))+(2 c e f+4 c d g-3 b e g) \sqrt {d+e x} \sqrt {-b e+c (d-e x)} \tan ^{-1}\left (\frac {\sqrt {c d-b e-c e x}}{\sqrt {c} \sqrt {d+e x}}\right )}{c^{5/2} e^2 \sqrt {(d+e x) (-b e+c (d-e x))}} \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]*(d + e*x)*(-3*b*e*g + c*(2*e*f + 3*d*g - e*g*x)) + (2*c*e*f + 4*c*d*g - 3*b*e*g)*Sqrt[d + e*x]*Sqrt[-
(b*e) + c*(d - e*x)]*ArcTan[Sqrt[c*d - b*e - c*e*x]/(Sqrt[c]*Sqrt[d + e*x])])/(c^(5/2)*e^2*Sqrt[(d + e*x)*(-(b
*e) + c*(d - e*x))])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(839\) vs. \(2(199)=398\).
time = 0.04, size = 840, normalized size = 3.94

method	result	size
default	\(e^{2} g \left (-\frac {x^{2}}{c \,e^{2} \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}-\frac {3 b \left (\frac {x}{c \,e^{2} \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}-\frac {b \left (\frac {1}{c \,e^{2} \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}-\frac {b \left (-2 c \,e^{2} x -b \,e^{2}\right )}{c \left (-4 c \,e^{2} \left (-b d e +c \,d^{2}\right )-b^{2} e^{4}\right ) \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}\right )}{2 c}-\frac {\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 )}{c \,e^{2} \sqrt {c \,e^{2}}}\right )}{2 c}+\frac {2 \left (-b d e +c \,d^{2}\right ) \left (\frac {1}{c \,e^{2} \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}-\frac {b \left (-2 c \,e^{2} x -b \,e^{2}\right )}{c \left (-4 c \,e^{2} \left (-b d e +c \,d^{2}\right )-b^{2} e^{4}\right ) \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}\right )}{c \,e^{2}}\right )+\left (2 d e g +e^{2} f \right ) \left (\frac {x}{c \,e^{2} \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}-\frac {b \left (\frac {1}{c \,e^{2} \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}-\frac {b \left (-2 c \,e^{2} x -b \,e^{2}\right )}{c \left (-4 c \,e^{2} \left (-b d e +c \,d^{2}\right )-b^{2} e^{4}\right ) \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}\right )}{2 c}-\frac {\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 )}{c \,e^{2} \sqrt {c \,e^{2}}}\right )+\left (d^{2} g +2 d e f \right ) \left (\frac {1}{c \,e^{2} \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}-\frac {b \left (-2 c \,e^{2} x -b \,e^{2}\right )}{c \left (-4 c \,e^{2} \left (-b d e +c \,d^{2}\right )-b^{2} e^{4}\right ) \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}\right )+\frac {2 d^{2} f \left (-2 c \,e^{2} x -b \,e^{2}\right )}{\left (-4 c \,e^{2} \left (-b d e +c \,d^{2}\right )-b^{2} e^{4}\right ) \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}\)	\(840\)

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,method=_RETURNVERBOSE)

[Out]

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

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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((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(2*c*d-%e*b>0)', see `assume?`
for more det

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Fricas [A]
time = 5.63, size = 470, normalized size = 2.21 \begin {gather*} \left [\frac {{\left (4 \, c^{2} d^{2} g - {\left (2 \, b c f - 3 \, b^{2} g + {\left (2 \, c^{2} f - 3 \, b c g\right )} x\right )} e^{2} - {\left (4 \, c^{2} d g x - 2 \, c^{2} d f + 7 \, b c d g\right )} e\right )} \sqrt {-c} \log \left (-4 \, c^{2} d^{2} + 4 \, b c d e - 4 \, \sqrt {c d^{2} - b d e - {\left (c x^{2} + b x\right )} e^{2}} {\left (2 \, c x + b\right )} \sqrt {-c} e + {\left (8 \, c^{2} x^{2} + 8 \, b c x + b^{2}\right )} e^{2}\right ) + 4 \, {\left (3 \, c^{2} d g - {\left (c^{2} g x - 2 \, c^{2} f + 3 \, b c g\right )} e\right )} \sqrt {c d^{2} - b d e - {\left (c x^{2} + b x\right )} e^{2}}}{4 \, {\left (c^{4} d e^{2} - {\left (c^{4} x + b c^{3}\right )} e^{3}\right )}}, \frac {{\left (4 \, c^{2} d^{2} g - {\left (2 \, b c f - 3 \, b^{2} g + {\left (2 \, c^{2} f - 3 \, b c g\right )} x\right )} e^{2} - {\left (4 \, c^{2} d g x - 2 \, c^{2} d f + 7 \, b c d g\right )} e\right )} \sqrt {c} \arctan \left (-\frac {\sqrt {c d^{2} - b d e - {\left (c x^{2} + b x\right )} e^{2}} {\left (2 \, c x + b\right )} \sqrt {c} e}{2 \, {\left (c^{2} d^{2} - b c d e - {\left (c^{2} x^{2} + b c x\right )} e^{2}\right )}}\right ) + 2 \, {\left (3 \, c^{2} d g - {\left (c^{2} g x - 2 \, c^{2} f + 3 \, b c g\right )} e\right )} \sqrt {c d^{2} - b d e - {\left (c x^{2} + b x\right )} e^{2}}}{2 \, {\left (c^{4} d e^{2} - {\left (c^{4} x + b c^{3}\right )} 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*((4*c^2*d^2*g - (2*b*c*f - 3*b^2*g + (2*c^2*f - 3*b*c*g)*x)*e^2 - (4*c^2*d*g*x - 2*c^2*d*f + 7*b*c*d*g)*e
)*sqrt(-c)*log(-4*c^2*d^2 + 4*b*c*d*e - 4*sqrt(c*d^2 - b*d*e - (c*x^2 + b*x)*e^2)*(2*c*x + b)*sqrt(-c)*e + (8*
c^2*x^2 + 8*b*c*x + b^2)*e^2) + 4*(3*c^2*d*g - (c^2*g*x - 2*c^2*f + 3*b*c*g)*e)*sqrt(c*d^2 - b*d*e - (c*x^2 +
b*x)*e^2))/(c^4*d*e^2 - (c^4*x + b*c^3)*e^3), 1/2*((4*c^2*d^2*g - (2*b*c*f - 3*b^2*g + (2*c^2*f - 3*b*c*g)*x)*
e^2 - (4*c^2*d*g*x - 2*c^2*d*f + 7*b*c*d*g)*e)*sqrt(c)*arctan(-1/2*sqrt(c*d^2 - b*d*e - (c*x^2 + b*x)*e^2)*(2*
c*x + b)*sqrt(c)*e/(c^2*d^2 - b*c*d*e - (c^2*x^2 + b*c*x)*e^2)) + 2*(3*c^2*d*g - (c^2*g*x - 2*c^2*f + 3*b*c*g)
*e)*sqrt(c*d^2 - b*d*e - (c*x^2 + b*x)*e^2))/(c^4*d*e^2 - (c^4*x + b*c^3)*e^3)]

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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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Giac [A]
time = 1.10, size = 231, normalized size = 1.08 \begin {gather*} \frac {\sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e} g e^{\left (-2\right )}}{c^{2}} + \frac {{\left (4 \, c d g + 2 \, c f e - 3 \, b g e\right )} e^{\left (-2\right )} \log \left ({\left | -b e + 2 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )} \sqrt {-c} \right |}\right )}{2 \, \sqrt {-c} c^{2}} + \frac {2 \, {\left (2 \, c^{2} d^{2} g + 2 \, c^{2} d f e - 3 \, b c d g e - b c f e^{2} + b^{2} g e^{2}\right )} e^{\left (-2\right )}}{{\left (c d - b e + {\left (\sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )} \sqrt {-c}\right )} \sqrt {-c} c^{2}} \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)*g*e^(-2)/c^2 + 1/2*(4*c*d*g + 2*c*f*e - 3*b*g*e)*e^(-2)*log(abs(-b*
e + 2*(sqrt(-c*e^2)*x - sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e))*sqrt(-c)))/(sqrt(-c)*c^2) + 2*(2*c^2*d^2*g
+ 2*c^2*d*f*e - 3*b*c*d*g*e - b*c*f*e^2 + b^2*g*e^2)*e^(-2)/((c*d - b*e + (sqrt(-c*e^2)*x - sqrt(-c*x^2*e^2 +
c*d^2 - b*x*e^2 - b*d*e))*sqrt(-c))*sqrt(-c)*c^2)

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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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