3.2248 \(\int \frac{(f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{3/2}}{(d+e x)^{11/2}} \, dx\)

Optimal. Leaf size=307 \[ -\frac{c^2 (-6 b e g+11 c d g+c e f) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{8 e^2 (2 c d-b e)^{3/2}}-\frac{(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{3 e^2 (d+e x)^{11/2} (2 c d-b e)}-\frac{\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-6 b e g+11 c d g+c e f)}{12 e^2 (d+e x)^{7/2} (2 c d-b e)}+\frac{c \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-6 b e g+11 c d g+c e f)}{8 e^2 (d+e x)^{3/2} (2 c d-b e)} \]

[Out]

(c*(c*e*f + 11*c*d*g - 6*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(8*e^2*(2*c*d - b*e)*(d + e*x)^(3/2
)) - ((c*e*f + 11*c*d*g - 6*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(12*e^2*(2*c*d - b*e)*(d + e*x
)^(7/2)) - ((e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(3*e^2*(2*c*d - b*e)*(d + e*x)^(11/2)) -
(c^2*(c*e*f + 11*c*d*g - 6*b*e*g)*ArcTanh[Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]/(Sqrt[2*c*d - b*e]*Sqrt[d
+ e*x])])/(8*e^2*(2*c*d - b*e)^(3/2))

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Rubi [A]  time = 0.482011, antiderivative size = 307, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {792, 662, 660, 208} \[ -\frac{c^2 (-6 b e g+11 c d g+c e f) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{8 e^2 (2 c d-b e)^{3/2}}-\frac{(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{3 e^2 (d+e x)^{11/2} (2 c d-b e)}-\frac{\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-6 b e g+11 c d g+c e f)}{12 e^2 (d+e x)^{7/2} (2 c d-b e)}+\frac{c \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-6 b e g+11 c d g+c e f)}{8 e^2 (d+e x)^{3/2} (2 c d-b e)} \]

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)^(11/2),x]

[Out]

(c*(c*e*f + 11*c*d*g - 6*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(8*e^2*(2*c*d - b*e)*(d + e*x)^(3/2
)) - ((c*e*f + 11*c*d*g - 6*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(12*e^2*(2*c*d - b*e)*(d + e*x
)^(7/2)) - ((e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(3*e^2*(2*c*d - b*e)*(d + e*x)^(11/2)) -
(c^2*(c*e*f + 11*c*d*g - 6*b*e*g)*ArcTanh[Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]/(Sqrt[2*c*d - b*e]*Sqrt[d
+ e*x])])/(8*e^2*(2*c*d - b*e)^(3/2))

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]

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 660

Int[1/(Sqrt[(d_.) + (e_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[2*e, Subst[Int[1/(
2*c*d - b*e + e^2*x^2), x], x, Sqrt[a + b*x + c*x^2]/Sqrt[d + e*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]

Rule 208

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

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)^{11/2}} \, dx &=-\frac{(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{3 e^2 (2 c d-b e) (d+e x)^{11/2}}+\frac{(c e f+11 c d g-6 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)^{9/2}} \, dx}{6 e (2 c d-b e)}\\ &=-\frac{(c e f+11 c d g-6 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{12 e^2 (2 c d-b e) (d+e x)^{7/2}}-\frac{(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{3 e^2 (2 c d-b e) (d+e x)^{11/2}}-\frac{(c (c e f+11 c d g-6 b e g)) \int \frac{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{5/2}} \, dx}{8 e (2 c d-b e)}\\ &=\frac{c (c e f+11 c d g-6 b e g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{8 e^2 (2 c d-b e) (d+e x)^{3/2}}-\frac{(c e f+11 c d g-6 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{12 e^2 (2 c d-b e) (d+e x)^{7/2}}-\frac{(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{3 e^2 (2 c d-b e) (d+e x)^{11/2}}+\frac{\left (c^2 (c e f+11 c d g-6 b e g)\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{16 e (2 c d-b e)}\\ &=\frac{c (c e f+11 c d g-6 b e g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{8 e^2 (2 c d-b e) (d+e x)^{3/2}}-\frac{(c e f+11 c d g-6 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{12 e^2 (2 c d-b e) (d+e x)^{7/2}}-\frac{(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{3 e^2 (2 c d-b e) (d+e x)^{11/2}}+\frac{\left (c^2 (c e f+11 c d g-6 b e g)\right ) \operatorname{Subst}\left (\int \frac{1}{-2 c d e^2+b e^3+e^2 x^2} \, dx,x,\frac{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}}{\sqrt{d+e x}}\right )}{8 (2 c d-b e)}\\ &=\frac{c (c e f+11 c d g-6 b e g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{8 e^2 (2 c d-b e) (d+e x)^{3/2}}-\frac{(c e f+11 c d g-6 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{12 e^2 (2 c d-b e) (d+e x)^{7/2}}-\frac{(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{3 e^2 (2 c d-b e) (d+e x)^{11/2}}-\frac{c^2 (c e f+11 c d g-6 b e g) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{2 c d-b e} \sqrt{d+e x}}\right )}{8 e^2 (2 c d-b e)^{3/2}}\\ \end{align*}

Mathematica [A]  time = 1.40268, size = 254, normalized size = 0.83 \[ \frac{((d+e x) (c (d-e x)-b e))^{3/2} \left (\frac{e (d+e x) (-6 b e g+11 c d g+c e f) \left (\sqrt{e (b e-2 c d)} \left (-2 b^2 e^2+b c e (d-7 e x)+c^2 \left (d^2+4 d e x-5 e^2 x^2\right )\right )+3 c^2 \sqrt{e} (d+e x)^2 \sqrt{c (d-e x)-b e} \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{c (d-e x)-b e}}{\sqrt{e (b e-2 c d)}}\right )\right )}{\sqrt{e (b e-2 c d)}}+8 e (e f-d g) (b e-c d+c e x)^3\right )}{24 e^3 (d+e x)^{9/2} (2 c d-b e) (b e-c d+c e x)^2} \]

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)^(11/2),x]

[Out]

(((d + e*x)*(-(b*e) + c*(d - e*x)))^(3/2)*(8*e*(e*f - d*g)*(-(c*d) + b*e + c*e*x)^3 + (e*(c*e*f + 11*c*d*g - 6
*b*e*g)*(d + e*x)*(Sqrt[e*(-2*c*d + b*e)]*(-2*b^2*e^2 + b*c*e*(d - 7*e*x) + c^2*(d^2 + 4*d*e*x - 5*e^2*x^2)) +
3*c^2*Sqrt[e]*(d + e*x)^2*Sqrt[-(b*e) + c*(d - e*x)]*ArcTan[(Sqrt[e]*Sqrt[-(b*e) + c*(d - e*x)])/Sqrt[e*(-2*c
*d + b*e)]]))/Sqrt[e*(-2*c*d + b*e)]))/(24*e^3*(2*c*d - b*e)*(d + e*x)^(9/2)*(-(c*d) + b*e + c*e*x)^2)

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Maple [B]  time = 0.03, size = 999, normalized size = 3.3 \begin{align*} \text{result too large to display} \end{align*}

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)^(11/2),x)

[Out]

1/24*(-9*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x*c^3*d^2*e^2*f+18*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*
e-2*c*d)^(1/2))*b*c^2*d^3*e*g+3*x^2*c^2*e^3*f*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+12*x*b^2*e^3*g*(-c*e*x-
b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-99*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x^2*c^3*d^2*e^2*g-9*arcta
n((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x^2*c^3*d*e^3*f-99*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2)
)*x*c^3*d^3*e*g+8*b^2*e^3*f*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+4*b^2*d*e^2*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e
-2*c*d)^(1/2)+7*c^2*d^2*e*f*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+18*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c
*d)^(1/2))*x^3*b*c^2*e^4*g-33*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x^3*c^3*d*e^3*g-2*x*b*c*d*e^2*g
*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-33*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^3*d^4*g-19*c^2
*d^3*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-3*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x^3*c^3*e^4
*f-3*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^3*d^3*e*f+30*x^2*b*c*e^3*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e
-2*c*d)^(1/2)-63*x^2*c^2*d*e^2*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+14*x*b*c*e^3*f*(-c*e*x-b*e+c*d)^(1/2
)*(b*e-2*c*d)^(1/2)-50*x*c^2*d^2*e*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-22*x*c^2*d*e^2*f*(-c*e*x-b*e+c*d
)^(1/2)*(b*e-2*c*d)^(1/2)-18*b*c*d*e^2*f*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+54*arctan((-c*e*x-b*e+c*d)^(
1/2)/(b*e-2*c*d)^(1/2))*x^2*b*c^2*d*e^3*g+54*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x*b*c^2*d^2*e^2*
g)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(b*e-2*c*d)^(3/2)/e^2/(-c*e*x-b*e+c*d)^(1/2)/(e*x+d)^(7/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e\right )}^{\frac{3}{2}}{\left (g x + f\right )}}{{\left (e x + d\right )}^{\frac{11}{2}}}\,{d x} \end{align*}

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)^(11/2),x, algorithm="maxima")

[Out]

integrate((-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(3/2)*(g*x + f)/(e*x + d)^(11/2), x)

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Fricas [B]  time = 1.5994, size = 3004, normalized size = 9.79 \begin{align*} \text{result too large to display} \end{align*}

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)^(11/2),x, algorithm="fricas")

[Out]

[1/48*(3*(c^3*d^4*e*f + (c^3*e^5*f + (11*c^3*d*e^4 - 6*b*c^2*e^5)*g)*x^4 + 4*(c^3*d*e^4*f + (11*c^3*d^2*e^3 -
6*b*c^2*d*e^4)*g)*x^3 + 6*(c^3*d^2*e^3*f + (11*c^3*d^3*e^2 - 6*b*c^2*d^2*e^3)*g)*x^2 + (11*c^3*d^5 - 6*b*c^2*d
^4*e)*g + 4*(c^3*d^3*e^2*f + (11*c^3*d^4*e - 6*b*c^2*d^3*e^2)*g)*x)*sqrt(2*c*d - b*e)*log(-(c*e^2*x^2 - 3*c*d^
2 + 2*b*d*e - 2*(c*d*e - b*e^2)*x + 2*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(2*c*d - b*e)*sqrt(e*x +
d))/(e^2*x^2 + 2*d*e*x + d^2)) - 2*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(3*((2*c^3*d*e^3 - b*c^2*e^4)*f
- (42*c^3*d^2*e^2 - 41*b*c^2*d*e^3 + 10*b^2*c*e^4)*g)*x^2 + (14*c^3*d^3*e - 43*b*c^2*d^2*e^2 + 34*b^2*c*d*e^3
- 8*b^3*e^4)*f - (38*c^3*d^4 - 19*b*c^2*d^3*e - 8*b^2*c*d^2*e^2 + 4*b^3*d*e^3)*g - 2*((22*c^3*d^2*e^2 - 25*b*c
^2*d*e^3 + 7*b^2*c*e^4)*f + (50*c^3*d^3*e - 23*b*c^2*d^2*e^2 - 13*b^2*c*d*e^3 + 6*b^3*e^4)*g)*x)*sqrt(e*x + d)
)/(4*c^2*d^6*e^2 - 4*b*c*d^5*e^3 + b^2*d^4*e^4 + (4*c^2*d^2*e^6 - 4*b*c*d*e^7 + b^2*e^8)*x^4 + 4*(4*c^2*d^3*e^
5 - 4*b*c*d^2*e^6 + b^2*d*e^7)*x^3 + 6*(4*c^2*d^4*e^4 - 4*b*c*d^3*e^5 + b^2*d^2*e^6)*x^2 + 4*(4*c^2*d^5*e^3 -
4*b*c*d^4*e^4 + b^2*d^3*e^5)*x), -1/24*(3*(c^3*d^4*e*f + (c^3*e^5*f + (11*c^3*d*e^4 - 6*b*c^2*e^5)*g)*x^4 + 4*
(c^3*d*e^4*f + (11*c^3*d^2*e^3 - 6*b*c^2*d*e^4)*g)*x^3 + 6*(c^3*d^2*e^3*f + (11*c^3*d^3*e^2 - 6*b*c^2*d^2*e^3)
*g)*x^2 + (11*c^3*d^5 - 6*b*c^2*d^4*e)*g + 4*(c^3*d^3*e^2*f + (11*c^3*d^4*e - 6*b*c^2*d^3*e^2)*g)*x)*sqrt(-2*c
*d + b*e)*arctan(sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(-2*c*d + b*e)*sqrt(e*x + d)/(c*e^2*x^2 + b*e^
2*x - c*d^2 + b*d*e)) + sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(3*((2*c^3*d*e^3 - b*c^2*e^4)*f - (42*c^3*d
^2*e^2 - 41*b*c^2*d*e^3 + 10*b^2*c*e^4)*g)*x^2 + (14*c^3*d^3*e - 43*b*c^2*d^2*e^2 + 34*b^2*c*d*e^3 - 8*b^3*e^4
)*f - (38*c^3*d^4 - 19*b*c^2*d^3*e - 8*b^2*c*d^2*e^2 + 4*b^3*d*e^3)*g - 2*((22*c^3*d^2*e^2 - 25*b*c^2*d*e^3 +
7*b^2*c*e^4)*f + (50*c^3*d^3*e - 23*b*c^2*d^2*e^2 - 13*b^2*c*d*e^3 + 6*b^3*e^4)*g)*x)*sqrt(e*x + d))/(4*c^2*d^
6*e^2 - 4*b*c*d^5*e^3 + b^2*d^4*e^4 + (4*c^2*d^2*e^6 - 4*b*c*d*e^7 + b^2*e^8)*x^4 + 4*(4*c^2*d^3*e^5 - 4*b*c*d
^2*e^6 + b^2*d*e^7)*x^3 + 6*(4*c^2*d^4*e^4 - 4*b*c*d^3*e^5 + b^2*d^2*e^6)*x^2 + 4*(4*c^2*d^5*e^3 - 4*b*c*d^4*e
^4 + b^2*d^3*e^5)*x)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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)**(11/2),x)

[Out]

Timed out

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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)^(11/2),x, algorithm="giac")

[Out]

Timed out