3.2237 \(\int \frac{(f+g x) \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=231 \[ -\frac{(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{2 e^2 (d+e x)^{7/2} (2 c d-b e)}-\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-4 b e g+7 c d g+c e f)}{4 e^2 (d+e x)^{3/2} (2 c d-b e)}+\frac{c (-4 b e g+7 c d g+c e f) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{4 e^2 (2 c d-b e)^{3/2}} \]

[Out]

-((c*e*f + 7*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)*(d + e*x)^(3/2))
- ((e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(2*e^2*(2*c*d - b*e)*(d + e*x)^(7/2)) + (c*(c*e*f
+ 7*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*(2*c*d - b*e)^(3/2))

________________________________________________________________________________________

Rubi [A]  time = 0.368409, antiderivative size = 231, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {792, 662, 660, 208} \[ -\frac{(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{2 e^2 (d+e x)^{7/2} (2 c d-b e)}-\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-4 b e g+7 c d g+c e f)}{4 e^2 (d+e x)^{3/2} (2 c d-b e)}+\frac{c (-4 b e g+7 c d g+c e f) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{4 e^2 (2 c d-b e)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[((f + g*x)*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2])/(d + e*x)^(7/2),x]

[Out]

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

Mathematica [A]  time = 0.674621, size = 222, normalized size = 0.96 \[ \frac{\sqrt{(d+e x) (c (d-e x)-b e)} \left (-2 e (e f-d g) (b e-c d+c e x)^2-\frac{e (d+e x) (-4 b e g+7 c d g+c e f) \left (c \sqrt{e} (d+e x) \sqrt{c (d-e x)-b e} \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{c (d-e x)-b e}}{\sqrt{e (b e-2 c d)}}\right )-\sqrt{e (b e-2 c d)} (b e-c d+c e x)\right )}{\sqrt{e (b e-2 c d)}}\right )}{4 e^3 (d+e x)^{5/2} (b e-2 c d) (b e-c d+c e x)} \]

Antiderivative was successfully verified.

[In]

Integrate[((f + g*x)*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2])/(d + e*x)^(7/2),x]

[Out]

(Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(-2*e*(e*f - d*g)*(-(c*d) + b*e + c*e*x)^2 - (e*(c*e*f + 7*c*d*g - 4*b
*e*g)*(d + e*x)*(-(Sqrt[e*(-2*c*d + b*e)]*(-(c*d) + b*e + c*e*x)) + c*Sqrt[e]*(d + e*x)*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)]))/(4*e^3*(-2
*c*d + b*e)*(d + e*x)^(5/2)*(-(c*d) + b*e + c*e*x))

________________________________________________________________________________________

Maple [B]  time = 0.023, size = 630, normalized size = 2.7 \begin{align*} -{\frac{1}{4\,{e}^{2}} \left ( 4\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ){x}^{2}bc{e}^{3}g-7\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ){x}^{2}{c}^{2}d{e}^{2}g-\arctan \left ({\sqrt{-cex-be+cd}{\frac{1}{\sqrt{be-2\,cd}}}} \right ){x}^{2}{c}^{2}{e}^{3}f+8\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) xbcd{e}^{2}g-14\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) x{c}^{2}{d}^{2}eg-2\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) x{c}^{2}d{e}^{2}f+4\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) bc{d}^{2}eg-7\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ){c}^{2}{d}^{3}g-\arctan \left ({\sqrt{-cex-be+cd}{\frac{1}{\sqrt{be-2\,cd}}}} \right ){c}^{2}{d}^{2}ef+4\,xb{e}^{2}g\sqrt{-cex-be+cd}\sqrt{be-2\,cd}-9\,xcdeg\sqrt{-cex-be+cd}\sqrt{be-2\,cd}+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}-5\,c{d}^{2}g\sqrt{-cex-be+cd}\sqrt{be-2\,cd}-3\,cdef\sqrt{-cex-be+cd}\sqrt{be-2\,cd} \right ) \sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}} \left ( be-2\,cd \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{-cex-be+cd}}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \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)^(1/2)/(e*x+d)^(7/2),x)

[Out]

-1/4*(4*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x^2*b*c*e^3*g-7*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-arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x^2*c^2*e^3*f+8*arctan((-c*e*x-b*
e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x*b*c*d*e^2*g-14*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x*c^2*d^2*e*
g-2*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x*c^2*d*e^2*f+4*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)
^(1/2))*b*c*d^2*e*g-7*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^2*d^3*g-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)-9*x*c*d*e*g*(-c*e*x-b*e+c
*d)^(1/2)*(b*e-2*c*d)^(1/2)+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)-5*c*d^2*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2
*c*d)^(1/2)-3*c*d*e*f*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2))*(-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)^(5/2)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}{\left (g x + f\right )}}{{\left (e x + d\right )}^{\frac{7}{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)^(1/2)/(e*x+d)^(7/2),x, algorithm="maxima")

[Out]

integrate(sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(g*x + f)/(e*x + d)^(7/2), x)

________________________________________________________________________________________

Fricas [B]  time = 1.52585, size = 2155, normalized size = 9.33 \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)^(1/2)/(e*x+d)^(7/2),x, algorithm="fricas")

[Out]

[1/8*((c^2*d^3*e*f + (c^2*e^4*f + (7*c^2*d*e^3 - 4*b*c*e^4)*g)*x^3 + 3*(c^2*d*e^3*f + (7*c^2*d^2*e^2 - 4*b*c*d
*e^3)*g)*x^2 + (7*c^2*d^4 - 4*b*c*d^3*e)*g + 3*(c^2*d^2*e^2*f + (7*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)*
((6*c^2*d^2*e - 7*b*c*d*e^2 + 2*b^2*e^3)*f + (10*c^2*d^3 - 9*b*c*d^2*e + 2*b^2*d*e^2)*g - ((2*c^2*d*e^2 - b*c*
e^3)*f - (18*c^2*d^2*e - 17*b*c*d*e^2 + 4*b^2*e^3)*g)*x)*sqrt(e*x + d))/(4*c^2*d^5*e^2 - 4*b*c*d^4*e^3 + b^2*d
^3*e^4 + (4*c^2*d^2*e^5 - 4*b*c*d*e^6 + b^2*e^7)*x^3 + 3*(4*c^2*d^3*e^4 - 4*b*c*d^2*e^5 + b^2*d*e^6)*x^2 + 3*(
4*c^2*d^4*e^3 - 4*b*c*d^3*e^4 + b^2*d^2*e^5)*x), 1/4*((c^2*d^3*e*f + (c^2*e^4*f + (7*c^2*d*e^3 - 4*b*c*e^4)*g)
*x^3 + 3*(c^2*d*e^3*f + (7*c^2*d^2*e^2 - 4*b*c*d*e^3)*g)*x^2 + (7*c^2*d^4 - 4*b*c*d^3*e)*g + 3*(c^2*d^2*e^2*f
+ (7*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)*sqr
t(-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)*((6*c^2*d^2*e - 7*b*c*d*e^2 + 2*b^2*e^3)*f + (10*c^2*d^3 - 9*b*c*d^2*e + 2*b^2*d*e^2)*g - ((2*c^2*d*e^2 -
b*c*e^3)*f - (18*c^2*d^2*e - 17*b*c*d*e^2 + 4*b^2*e^3)*g)*x)*sqrt(e*x + d))/(4*c^2*d^5*e^2 - 4*b*c*d^4*e^3 + b
^2*d^3*e^4 + (4*c^2*d^2*e^5 - 4*b*c*d*e^6 + b^2*e^7)*x^3 + 3*(4*c^2*d^3*e^4 - 4*b*c*d^2*e^5 + b^2*d*e^6)*x^2 +
3*(4*c^2*d^4*e^3 - 4*b*c*d^3*e^4 + b^2*d^2*e^5)*x)]

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{- \left (d + e x\right ) \left (b e - c d + c e x\right )} \left (f + g x\right )}{\left (d + e x\right )^{\frac{7}{2}}}\, dx \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)**(1/2)/(e*x+d)**(7/2),x)

[Out]

Integral(sqrt(-(d + e*x)*(b*e - c*d + c*e*x))*(f + g*x)/(d + e*x)**(7/2), x)

________________________________________________________________________________________

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)^(1/2)/(e*x+d)^(7/2),x, algorithm="giac")

[Out]

Timed out