3.2247 \(\int \frac{(f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{3/2}}{(d+e x)^{9/2}} \, dx\)

Optimal. Leaf size=305 \[ -\frac{(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{2 e^2 (d+e x)^{9/2} (2 c d-b e)}+\frac{\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (4 b e g-9 c d g+c e f)}{4 e^2 (d+e x)^{5/2} (2 c d-b e)}+\frac{3 c \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (4 b e g-9 c d g+c e f)}{4 e^2 \sqrt{d+e x} (2 c d-b e)}-\frac{3 c (4 b e g-9 c d g+c e f) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{4 e^2 \sqrt{2 c d-b e}} \]

[Out]

(3*c*(c*e*f - 9*c*d*g + 4*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(4*e^2*(2*c*d - b*e)*Sqrt[d + e*x]
) + ((c*e*f - 9*c*d*g + 4*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(4*e^2*(2*c*d - b*e)*(d + e*x)^(
5/2)) - ((e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(2*e^2*(2*c*d - b*e)*(d + e*x)^(9/2)) - (3*c
*(c*e*f - 9*c*d*g + 4*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
])])/(4*e^2*Sqrt[2*c*d - b*e])

________________________________________________________________________________________

Rubi [A]  time = 0.483125, antiderivative size = 305, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.109, Rules used = {792, 662, 664, 660, 208} \[ -\frac{(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{2 e^2 (d+e x)^{9/2} (2 c d-b e)}+\frac{\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (4 b e g-9 c d g+c e f)}{4 e^2 (d+e x)^{5/2} (2 c d-b e)}+\frac{3 c \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (4 b e g-9 c d g+c e f)}{4 e^2 \sqrt{d+e x} (2 c d-b e)}-\frac{3 c (4 b e g-9 c d g+c e f) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{4 e^2 \sqrt{2 c d-b e}} \]

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)^(9/2),x]

[Out]

(3*c*(c*e*f - 9*c*d*g + 4*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(4*e^2*(2*c*d - b*e)*Sqrt[d + e*x]
) + ((c*e*f - 9*c*d*g + 4*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(4*e^2*(2*c*d - b*e)*(d + e*x)^(
5/2)) - ((e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(2*e^2*(2*c*d - b*e)*(d + e*x)^(9/2)) - (3*c
*(c*e*f - 9*c*d*g + 4*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
])])/(4*e^2*Sqrt[2*c*d - b*e])

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 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 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)^{9/2}} \, dx &=-\frac{(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{2 e^2 (2 c d-b e) (d+e x)^{9/2}}-\frac{(c e f-9 c d g+4 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)^{7/2}} \, dx}{4 e (2 c d-b e)}\\ &=\frac{(c e f-9 c d g+4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{4 e^2 (2 c d-b e) (d+e x)^{5/2}}-\frac{(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{2 e^2 (2 c d-b e) (d+e x)^{9/2}}+\frac{(3 c (c e f-9 c d g+4 b e g)) \int \frac{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{3/2}} \, dx}{8 e (2 c d-b e)}\\ &=\frac{3 c (c e f-9 c d g+4 b e g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{4 e^2 (2 c d-b e) \sqrt{d+e x}}+\frac{(c e f-9 c d g+4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{4 e^2 (2 c d-b e) (d+e x)^{5/2}}-\frac{(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{2 e^2 (2 c d-b e) (d+e x)^{9/2}}+\frac{(3 c (c e f-9 c d g+4 b e g)) \int \frac{1}{\sqrt{d+e x} \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{8 e}\\ &=\frac{3 c (c e f-9 c d g+4 b e g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{4 e^2 (2 c d-b e) \sqrt{d+e x}}+\frac{(c e f-9 c d g+4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{4 e^2 (2 c d-b e) (d+e x)^{5/2}}-\frac{(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{2 e^2 (2 c d-b e) (d+e x)^{9/2}}+\frac{1}{4} (3 c (c e f-9 c d g+4 b e g)) \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 )\\ &=\frac{3 c (c e f-9 c d g+4 b e g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{4 e^2 (2 c d-b e) \sqrt{d+e x}}+\frac{(c e f-9 c d g+4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{4 e^2 (2 c d-b e) (d+e x)^{5/2}}-\frac{(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{2 e^2 (2 c d-b e) (d+e x)^{9/2}}-\frac{3 c (c e f-9 c d g+4 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 )}{4 e^2 \sqrt{2 c d-b e}}\\ \end{align*}

Mathematica [C]  time = 0.209185, size = 129, normalized size = 0.42 \[ \frac{((d+e x) (c (d-e x)-b e))^{5/2} \left (\frac{c (d+e x)^2 (4 b e g-9 c d g+c e f) \, _2F_1\left (2,\frac{5}{2};\frac{7}{2};\frac{-c d+b e+c e x}{b e-2 c d}\right )}{e (b e-2 c d)^2}+\frac{5 d g}{e}-5 f\right )}{10 e (d+e x)^{9/2} (2 c d-b e)} \]

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)^(9/2),x]

[Out]

(((d + e*x)*(-(b*e) + c*(d - e*x)))^(5/2)*(-5*f + (5*d*g)/e + (c*(c*e*f - 9*c*d*g + 4*b*e*g)*(d + e*x)^2*Hyper
geometric2F1[2, 5/2, 7/2, (-(c*d) + b*e + c*e*x)/(-2*c*d + b*e)])/(e*(-2*c*d + b*e)^2)))/(10*e*(2*c*d - b*e)*(
d + e*x)^(9/2))

________________________________________________________________________________________

Maple [B]  time = 0.025, size = 665, normalized size = 2.2 \begin{align*}{\frac{1}{4\,{e}^{2}}\sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}} \left ( 12\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ){x}^{2}bc{e}^{3}g-27\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ){x}^{2}{c}^{2}d{e}^{2}g+3\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ){x}^{2}{c}^{2}{e}^{3}f+24\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) xbcd{e}^{2}g-54\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) x{c}^{2}{d}^{2}eg+6\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) x{c}^{2}d{e}^{2}f-8\,{x}^{2}c{e}^{2}g\sqrt{-cex-be+cd}\sqrt{be-2\,cd}+12\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) bc{d}^{2}eg-27\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ){c}^{2}{d}^{3}g+3\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ){c}^{2}{d}^{2}ef+4\,xb{e}^{2}g\sqrt{-cex-be+cd}\sqrt{be-2\,cd}-29\,xcdeg\sqrt{-cex-be+cd}\sqrt{be-2\,cd}+5\,xc{e}^{2}f\sqrt{-cex-be+cd}\sqrt{be-2\,cd}+2\,bdeg\sqrt{-cex-be+cd}\sqrt{be-2\,cd}+2\,b{e}^{2}f\sqrt{-cex-be+cd}\sqrt{be-2\,cd}-17\,c{d}^{2}g\sqrt{-cex-be+cd}\sqrt{be-2\,cd}+cdef\sqrt{-cex-be+cd}\sqrt{be-2\,cd} \right ) \left ( ex+d \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt{-cex-be+cd}}}{\frac{1}{\sqrt{be-2\,cd}}}} \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)^(9/2),x)

[Out]

1/4*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*(12*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x^2*b*c*e^3*g-
27*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x^2*c^2*d*e^2*g+3*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d
)^(1/2))*x^2*c^2*e^3*f+24*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x*b*c*d*e^2*g-54*arctan((-c*e*x-b*e
+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x*c^2*d^2*e*g+6*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x*c^2*d*e^2*f-
8*x^2*c*e^2*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+12*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c
*d^2*e*g-27*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^2*d^3*g+3*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*
c*d)^(1/2))*c^2*d^2*e*f+4*x*b*e^2*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-29*x*c*d*e*g*(-c*e*x-b*e+c*d)^(1/
2)*(b*e-2*c*d)^(1/2)+5*x*c*e^2*f*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+2*b*d*e*g*(-c*e*x-b*e+c*d)^(1/2)*(b*
e-2*c*d)^(1/2)+2*b*e^2*f*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-17*c*d^2*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d
)^(1/2)+c*d*e*f*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2))/(e*x+d)^(5/2)/(-c*e*x-b*e+c*d)^(1/2)/e^2/(b*e-2*c*d)
^(1/2)

________________________________________________________________________________________

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{9}{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)^(9/2),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)^(9/2), x)

________________________________________________________________________________________

Fricas [A]  time = 1.72167, size = 2045, normalized size = 6.7 \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)^(9/2),x, algorithm="fricas")

[Out]

[1/8*(3*(c^2*d^3*e*f + (c^2*e^4*f - (9*c^2*d*e^3 - 4*b*c*e^4)*g)*x^3 + 3*(c^2*d*e^3*f - (9*c^2*d^2*e^2 - 4*b*c
*d*e^3)*g)*x^2 - (9*c^2*d^4 - 4*b*c*d^3*e)*g + 3*(c^2*d^2*e^2*f - (9*c^2*d^3*e - 4*b*c*d^2*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
)*(8*(2*c^2*d*e^2 - b*c*e^3)*g*x^2 - (2*c^2*d^2*e + 3*b*c*d*e^2 - 2*b^2*e^3)*f + (34*c^2*d^3 - 21*b*c*d^2*e +
2*b^2*d*e^2)*g - (5*(2*c^2*d*e^2 - b*c*e^3)*f - (58*c^2*d^2*e - 37*b*c*d*e^2 + 4*b^2*e^3)*g)*x)*sqrt(e*x + d))
/(2*c*d^4*e^2 - b*d^3*e^3 + (2*c*d*e^5 - b*e^6)*x^3 + 3*(2*c*d^2*e^4 - b*d*e^5)*x^2 + 3*(2*c*d^3*e^3 - b*d^2*e
^4)*x), -1/4*(3*(c^2*d^3*e*f + (c^2*e^4*f - (9*c^2*d*e^3 - 4*b*c*e^4)*g)*x^3 + 3*(c^2*d*e^3*f - (9*c^2*d^2*e^2
- 4*b*c*d*e^3)*g)*x^2 - (9*c^2*d^4 - 4*b*c*d^3*e)*g + 3*(c^2*d^2*e^2*f - (9*c^2*d^3*e - 4*b*c*d^2*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)*(8*(2*c^2*d*e^2 - b*c*e^3)*g*x^2 -
(2*c^2*d^2*e + 3*b*c*d*e^2 - 2*b^2*e^3)*f + (34*c^2*d^3 - 21*b*c*d^2*e + 2*b^2*d*e^2)*g - (5*(2*c^2*d*e^2 - b
*c*e^3)*f - (58*c^2*d^2*e - 37*b*c*d*e^2 + 4*b^2*e^3)*g)*x)*sqrt(e*x + d))/(2*c*d^4*e^2 - b*d^3*e^3 + (2*c*d*e
^5 - b*e^6)*x^3 + 3*(2*c*d^2*e^4 - b*d*e^5)*x^2 + 3*(2*c*d^3*e^3 - b*d^2*e^4)*x)]

________________________________________________________________________________________

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)**(9/2),x)

[Out]

Timed out

________________________________________________________________________________________

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)^(9/2),x, algorithm="giac")

[Out]

Timed out