3.2190 \(\int \frac{(f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{3/2}}{(d+e x)^5} \, dx\)
Optimal. Leaf size=214 \[ \frac{c^{3/2} g \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{e^2}-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e^2 (d+e x)^5 (2 c d-b e)}-\frac{2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^3}+\frac{2 c g \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 (d+e x)} \]
[Out]
(2*c*g*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(e^2*(d + e*x)) - (2*g*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)
^(3/2))/(3*e^2*(d + e*x)^3) - (2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(5*e^2*(2*c*d - b*e)
*(d + e*x)^5) + (c^(3/2)*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])])/e^2
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Rubi [A] time = 0.443403, antiderivative size = 214, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 4, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used =
{792, 662, 621, 204} \[ \frac{c^{3/2} g \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{e^2}-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e^2 (d+e x)^5 (2 c d-b e)}-\frac{2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^3}+\frac{2 c g \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 (d+e x)} \]
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)^5,x]
[Out]
(2*c*g*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(e^2*(d + e*x)) - (2*g*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)
^(3/2))/(3*e^2*(d + e*x)^3) - (2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(5*e^2*(2*c*d - b*e)
*(d + e*x)^5) + (c^(3/2)*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])])/e^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 621
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 204
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])
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)^5} \, dx &=-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e^2 (2 c d-b e) (d+e x)^5}+\frac{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)^4} \, dx}{e}\\ &=-\frac{2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^3}-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e^2 (2 c d-b e) (d+e x)^5}-\frac{(c g) \int \frac{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^2} \, dx}{e}\\ &=\frac{2 c g \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 (d+e x)}-\frac{2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^3}-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e^2 (2 c d-b e) (d+e x)^5}+\frac{\left (c^2 g\right ) \int \frac{1}{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{e}\\ &=\frac{2 c g \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 (d+e x)}-\frac{2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^3}-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e^2 (2 c d-b e) (d+e x)^5}+\frac{\left (2 c^2 g\right ) \operatorname{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 )}{e}\\ &=\frac{2 c g \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 (d+e x)}-\frac{2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^3}-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e^2 (2 c d-b e) (d+e x)^5}+\frac{c^{3/2} 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 )}{e^2}\\ \end{align*}
Mathematica [C] time = 0.319863, size = 150, normalized size = 0.7 \[ \frac{2 \sqrt{(d+e x) (c (d-e x)-b e)} \left (\frac{g (2 c d-b e)^3 \, _2F_1\left (-\frac{5}{2},-\frac{5}{2};-\frac{3}{2};\frac{c (d+e x)}{2 c d-b e}\right )}{\sqrt{\frac{b e-c d+c e x}{b e-2 c d}}}-(b e-c d+c e x)^2 (-b e g+c d g+c e f)\right )}{5 c e^2 (d+e x)^3 (2 c d-b e)} \]
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)^5,x]
[Out]
(2*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(-((c*e*f + c*d*g - b*e*g)*(-(c*d) + b*e + c*e*x)^2) + ((2*c*d - b*e
)^3*g*Hypergeometric2F1[-5/2, -5/2, -3/2, (c*(d + e*x))/(2*c*d - b*e)])/Sqrt[(-(c*d) + b*e + c*e*x)/(-2*c*d +
b*e)]))/(5*c*e^2*(2*c*d - b*e)*(d + e*x)^3)
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Maple [B] time = 0.015, size = 1023, normalized size = 4.8 \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)^5,x)
[Out]
-2/3*g/e^5/(-b*e^2+2*c*d*e)/(x+d/e)^4*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(5/2)+4/3*g/e^3*c/(-b*e^2+2*
c*d*e)^2/(x+d/e)^3*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(5/2)+16/3*g/e*c^2/(-b*e^2+2*c*d*e)^3/(x+d/e)^2
*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(5/2)+16/3*g*e*c^3/(-b*e^2+2*c*d*e)^3*(-(x+d/e)^2*c*e^2+(-b*e^2+2
*c*d*e)*(x+d/e))^(3/2)-4*g*e^3*c^3/(-b*e^2+2*c*d*e)^3*b*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*x-2*
g*e^3*c^2/(-b*e^2+2*c*d*e)^3*b^2*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)-g*e^5*c^2/(-b*e^2+2*c*d*e)^
3*b^3/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)
*(x+d/e))^(1/2))+6*g*e^4*c^3/(-b*e^2+2*c*d*e)^3*b^2/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*
d*e)/c/e^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2))*d-12*g*e^3*c^4/(-b*e^2+2*c*d*e)^3*b/(c*e^2)^(1/
2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2))*
d^2+8*g*e^2*c^4/(-b*e^2+2*c*d*e)^3*d*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*x+4*g*e^2*c^3/(-b*e^2+2
*c*d*e)^3*d*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)*b+8*g*e^2*c^5/(-b*e^2+2*c*d*e)^3*d^3/(c*e^2)^(1/
2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2))-
2/5*(-d*g+e*f)/e^6/(-b*e^2+2*c*d*e)/(x+d/e)^5*(-(x+d/e)^2*c*e^2+(-b*e^2+2*c*d*e)*(x+d/e))^(5/2)
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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)^5,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 88.4453, size = 1804, normalized size = 8.43 \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)^5,x, algorithm="fricas")
[Out]
[1/30*(15*((2*c^2*d*e^3 - b*c*e^4)*g*x^3 + 3*(2*c^2*d^2*e^2 - b*c*d*e^3)*g*x^2 + 3*(2*c^2*d^3*e - b*c*d^2*e^2)
*g*x + (2*c^2*d^4 - b*c*d^3*e)*g)*sqrt(-c)*log(8*c^2*e^2*x^2 + 8*b*c*e^2*x - 4*c^2*d^2 + 4*b*c*d*e + b^2*e^2 +
4*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*c*e*x + b*e)*sqrt(-c)) - 4*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2
- b*d*e)*((3*c^2*e^3*f - (43*c^2*d*e^2 - 20*b*c*e^3)*g)*x^2 + 3*(c^2*d^2*e - 2*b*c*d*e^2 + b^2*e^3)*f - (23*c^
2*d^3 - 6*b*c*d^2*e - 2*b^2*d*e^2)*g - (6*(c^2*d*e^2 - b*c*e^3)*f + (54*c^2*d^2*e - 14*b*c*d*e^2 - 5*b^2*e^3)*
g)*x))/(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), -1/15*(15*((2*c^2*d*e^3 - b*c*e^4)*g*x^3 + 3*(2*c^2*d^2*e^2 - b*c*d*e^3)*g*x^2 + 3*(2*c^2*d^3*e
- b*c*d^2*e^2)*g*x + (2*c^2*d^4 - b*c*d^3*e)*g)*sqrt(c)*arctan(1/2*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*
(2*c*e*x + b*e)*sqrt(c)/(c^2*e^2*x^2 + b*c*e^2*x - c^2*d^2 + b*c*d*e)) + 2*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 -
b*d*e)*((3*c^2*e^3*f - (43*c^2*d*e^2 - 20*b*c*e^3)*g)*x^2 + 3*(c^2*d^2*e - 2*b*c*d*e^2 + b^2*e^3)*f - (23*c^2
*d^3 - 6*b*c*d^2*e - 2*b^2*d*e^2)*g - (6*(c^2*d*e^2 - b*c*e^3)*f + (54*c^2*d^2*e - 14*b*c*d*e^2 - 5*b^2*e^3)*g
)*x))/(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)]
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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)**5,x)
[Out]
Timed out
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \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)^5,x, algorithm="giac")
[Out]
Exception raised: TypeError