3.2267 \(\int \frac{f+g x}{(d+e x)^{5/2} \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx\)
Optimal. Leaf size=233 \[ -\frac{(e f-d g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{2 e^2 (d+e x)^{5/2} (2 c d-b e)}-\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-4 b e g+5 c d g+3 c e f)}{4 e^2 (d+e x)^{3/2} (2 c d-b e)^2}-\frac{c (-4 b e g+5 c d g+3 c e f) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{4 e^2 (2 c d-b e)^{5/2}} \]
[Out]
-((e*f - d*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(2*e^2*(2*c*d - b*e)*(d + e*x)^(5/2)) - ((3*c*e*f + 5
*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)^2*(d + e*x)^(3/2)) - (c*(3*c
*e*f + 5*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)^(5/2))
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Rubi [A] time = 0.373967, antiderivative size = 233, 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, 672, 660, 208} \[ -\frac{(e f-d g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{2 e^2 (d+e x)^{5/2} (2 c d-b e)}-\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-4 b e g+5 c d g+3 c e f)}{4 e^2 (d+e x)^{3/2} (2 c d-b e)^2}-\frac{c (-4 b e g+5 c d g+3 c e f) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{4 e^2 (2 c d-b e)^{5/2}} \]
Antiderivative was successfully verified.
[In]
Int[(f + g*x)/((d + e*x)^(5/2)*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2]),x]
[Out]
-((e*f - d*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(2*e^2*(2*c*d - b*e)*(d + e*x)^(5/2)) - ((3*c*e*f + 5
*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)^2*(d + e*x)^(3/2)) - (c*(3*c
*e*f + 5*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)^(5/2))
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 672
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[(e*(d + e*x)^m*(a +
b*x + c*x^2)^(p + 1))/((m + p + 1)*(2*c*d - b*e)), x] + Dist[(c*(m + 2*p + 2))/((m + p + 1)*(2*c*d - b*e)), I
nt[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ
[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, 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}{(d+e x)^{5/2} \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx &=-\frac{(e f-d g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{2 e^2 (2 c d-b e) (d+e x)^{5/2}}+\frac{(3 c e f+5 c d g-4 b e g) \int \frac{1}{(d+e x)^{3/2} \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{4 e (2 c d-b e)}\\ &=-\frac{(e f-d g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{2 e^2 (2 c d-b e) (d+e x)^{5/2}}-\frac{(3 c e f+5 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)^2 (d+e x)^{3/2}}+\frac{(c (3 c e f+5 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)^2}\\ &=-\frac{(e f-d g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{2 e^2 (2 c d-b e) (d+e x)^{5/2}}-\frac{(3 c e f+5 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)^2 (d+e x)^{3/2}}+\frac{(c (3 c e f+5 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)^2}\\ &=-\frac{(e f-d g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{2 e^2 (2 c d-b e) (d+e x)^{5/2}}-\frac{(3 c e f+5 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)^2 (d+e x)^{3/2}}-\frac{c (3 c e f+5 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)^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.62371, size = 209, normalized size = 0.9 \[ \frac{\sqrt{(d+e x) (c (d-e x)-b e)} \left (-\frac{(d+e x) (-4 b e g+5 c d g+3 c e f) \left (\sqrt{e} (2 c d-b e) \sqrt{c (d-e x)-b e}+c (d+e x) \sqrt{e (b e-2 c d)} \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{c (d-e x)-b e}}{\sqrt{e (b e-2 c d)}}\right )\right )}{2 e^{3/2} (b e-2 c d)^2 \sqrt{c (d-e x)-b e}}+\frac{d g}{e}-f\right )}{2 e (d+e x)^{5/2} (2 c d-b e)} \]
Antiderivative was successfully verified.
[In]
Integrate[(f + g*x)/((d + e*x)^(5/2)*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2]),x]
[Out]
(Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(-f + (d*g)/e - ((3*c*e*f + 5*c*d*g - 4*b*e*g)*(d + e*x)*(Sqrt[e]*(2*c
*d - b*e)*Sqrt[-(b*e) + c*(d - e*x)] + c*Sqrt[e*(-2*c*d + b*e)]*(d + e*x)*ArcTan[(Sqrt[e]*Sqrt[-(b*e) + c*(d -
e*x)])/Sqrt[e*(-2*c*d + b*e)]]))/(2*e^(3/2)*(-2*c*d + b*e)^2*Sqrt[-(b*e) + c*(d - e*x)])))/(2*e*(2*c*d - b*e)
*(d + e*x)^(5/2))
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Maple [B] time = 0.03, size = 630, normalized size = 2.7 \begin{align*} -{\frac{1}{4\,{e}^{2}}\sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}} \left ( 4\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ){x}^{2}bc{e}^{3}g-5\,\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+8\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) xbcd{e}^{2}g-10\,\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+4\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) bc{d}^{2}eg-5\,\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}+5\,xcdeg\sqrt{-cex-be+cd}\sqrt{be-2\,cd}+3\,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}+c{d}^{2}g\sqrt{-cex-be+cd}\sqrt{be-2\,cd}+7\,cdef\sqrt{-cex-be+cd}\sqrt{be-2\,cd} \right ) \left ( ex+d \right ) ^{-{\frac{5}{2}}} \left ( be-2\,cd \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt{-cex-be+cd}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((g*x+f)/(e*x+d)^(5/2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x)
[Out]
-1/4*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*(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-
5*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+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-10*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+4*
arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c*d^2*e*g-5*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)+5*x*c*d*e*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+3*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)+c*d^2*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+7*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)/(b*e-2*c*d)^(5/2)/e^2/(-c*e*x-b*e+c*d)^(1/2)
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{g x + f}{\sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}{\left (e x + d\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((g*x+f)/(e*x+d)^(5/2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, algorithm="maxima")
[Out]
integrate((g*x + f)/(sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(e*x + d)^(5/2)), x)
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Fricas [B] time = 1.41195, size = 2414, normalized size = 10.36 \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)/(e*x+d)^(5/2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, algorithm="fricas")
[Out]
[-1/8*((3*c^2*d^3*e*f + (3*c^2*e^4*f + (5*c^2*d*e^3 - 4*b*c*e^4)*g)*x^3 + 3*(3*c^2*d*e^3*f + (5*c^2*d^2*e^2 -
4*b*c*d*e^3)*g)*x^2 + (5*c^2*d^4 - 4*b*c*d^3*e)*g + 3*(3*c^2*d^2*e^2*f + (5*c^2*d^3*e - 4*b*c*d^2*e^2)*g)*x)*s
qrt(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)*((14*c^2*d^2*e - 11*b*c*d*e^2 + 2*b^2*e^3)*f + (2*c^2*d^3 - 5*b*c*d^2*e + 2*b^2*d*e^2)*g + (3*(2*c^2*
d*e^2 - b*c*e^3)*f + (10*c^2*d^2*e - 13*b*c*d*e^2 + 4*b^2*e^3)*g)*x)*sqrt(e*x + d))/(8*c^3*d^6*e^2 - 12*b*c^2*
d^5*e^3 + 6*b^2*c*d^4*e^4 - b^3*d^3*e^5 + (8*c^3*d^3*e^5 - 12*b*c^2*d^2*e^6 + 6*b^2*c*d*e^7 - b^3*e^8)*x^3 + 3
*(8*c^3*d^4*e^4 - 12*b*c^2*d^3*e^5 + 6*b^2*c*d^2*e^6 - b^3*d*e^7)*x^2 + 3*(8*c^3*d^5*e^3 - 12*b*c^2*d^4*e^4 +
6*b^2*c*d^3*e^5 - b^3*d^2*e^6)*x), -1/4*((3*c^2*d^3*e*f + (3*c^2*e^4*f + (5*c^2*d*e^3 - 4*b*c*e^4)*g)*x^3 + 3*
(3*c^2*d*e^3*f + (5*c^2*d^2*e^2 - 4*b*c*d*e^3)*g)*x^2 + (5*c^2*d^4 - 4*b*c*d^3*e)*g + 3*(3*c^2*d^2*e^2*f + (5*
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)*(
(14*c^2*d^2*e - 11*b*c*d*e^2 + 2*b^2*e^3)*f + (2*c^2*d^3 - 5*b*c*d^2*e + 2*b^2*d*e^2)*g + (3*(2*c^2*d*e^2 - b*
c*e^3)*f + (10*c^2*d^2*e - 13*b*c*d*e^2 + 4*b^2*e^3)*g)*x)*sqrt(e*x + d))/(8*c^3*d^6*e^2 - 12*b*c^2*d^5*e^3 +
6*b^2*c*d^4*e^4 - b^3*d^3*e^5 + (8*c^3*d^3*e^5 - 12*b*c^2*d^2*e^6 + 6*b^2*c*d*e^7 - b^3*e^8)*x^3 + 3*(8*c^3*d^
4*e^4 - 12*b*c^2*d^3*e^5 + 6*b^2*c*d^2*e^6 - b^3*d*e^7)*x^2 + 3*(8*c^3*d^5*e^3 - 12*b*c^2*d^4*e^4 + 6*b^2*c*d^
3*e^5 - b^3*d^2*e^6)*x)]
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f + g x}{\sqrt{- \left (d + e x\right ) \left (b e - c d + c e x\right )} \left (d + e x\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((g*x+f)/(e*x+d)**(5/2)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(1/2),x)
[Out]
Integral((f + g*x)/(sqrt(-(d + e*x)*(b*e - c*d + c*e*x))*(d + e*x)**(5/2)), x)
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((g*x+f)/(e*x+d)^(5/2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, algorithm="giac")
[Out]
sage0*x