3.2202 \(\int \frac{(f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2}}{(d+e x)^5} \, dx\)

Optimal. Leaf size=350 \[ -\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (d+e x)^5 (2 c d-b e)}+\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (3 b e g-10 c d g+4 c e f)}{3 e^2 (d+e x)^3 (2 c d-b e)}+\frac{5 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (3 b e g-10 c d g+4 c e f)}{6 e^2 (d+e x) (2 c d-b e)}+\frac{5 c \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (3 b e g-10 c d g+4 c e f)}{4 e^2}+\frac{5 \sqrt{c} (2 c d-b e) (3 b e g-10 c d g+4 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{8 e^2} \]

[Out]

(5*c*(4*c*e*f - 10*c*d*g + 3*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(4*e^2) + (5*c*(4*c*e*f - 10*c*
d*g + 3*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(6*e^2*(2*c*d - b*e)*(d + e*x)) + (2*(4*c*e*f - 10
*c*d*g + 3*b*e*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)^3) - (2*(e*f - d
*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(3*e^2*(2*c*d - b*e)*(d + e*x)^5) + (5*Sqrt[c]*(2*c*d - b*e)*
(4*c*e*f - 10*c*d*g + 3*b*e*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])])/
(8*e^2)

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Rubi [A]  time = 0.627347, antiderivative size = 350, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.114, Rules used = {792, 662, 664, 621, 204} \[ -\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (d+e x)^5 (2 c d-b e)}+\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (3 b e g-10 c d g+4 c e f)}{3 e^2 (d+e x)^3 (2 c d-b e)}+\frac{5 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (3 b e g-10 c d g+4 c e f)}{6 e^2 (d+e x) (2 c d-b e)}+\frac{5 c \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (3 b e g-10 c d g+4 c e f)}{4 e^2}+\frac{5 \sqrt{c} (2 c d-b e) (3 b e g-10 c d g+4 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{8 e^2} \]

Antiderivative was successfully verified.

[In]

Int[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2))/(d + e*x)^5,x]

[Out]

(5*c*(4*c*e*f - 10*c*d*g + 3*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(4*e^2) + (5*c*(4*c*e*f - 10*c*
d*g + 3*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(6*e^2*(2*c*d - b*e)*(d + e*x)) + (2*(4*c*e*f - 10
*c*d*g + 3*b*e*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)^3) - (2*(e*f - d
*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(3*e^2*(2*c*d - b*e)*(d + e*x)^5) + (5*Sqrt[c]*(2*c*d - b*e)*
(4*c*e*f - 10*c*d*g + 3*b*e*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])])/
(8*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 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 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 )^{5/2}}{(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 )^{7/2}}{3 e^2 (2 c d-b e) (d+e x)^5}-\frac{(4 c e f-10 c d g+3 b e g) \int \frac{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx}{3 e (2 c d-b e)}\\ &=\frac{2 (4 c e f-10 c d g+3 b e 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)^3}-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (2 c d-b e) (d+e x)^5}+\frac{(5 c (4 c e f-10 c d g+3 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)^2} \, dx}{3 e (2 c d-b e)}\\ &=\frac{5 c (4 c e f-10 c d g+3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{6 e^2 (2 c d-b e) (d+e x)}+\frac{2 (4 c e f-10 c d g+3 b e 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)^3}-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (2 c d-b e) (d+e x)^5}+\frac{(5 c (4 c e f-10 c d g+3 b e g)) \int \frac{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}}{d+e x} \, dx}{4 e}\\ &=\frac{5 c (4 c e f-10 c d g+3 b e g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{4 e^2}+\frac{5 c (4 c e f-10 c d g+3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{6 e^2 (2 c d-b e) (d+e x)}+\frac{2 (4 c e f-10 c d g+3 b e 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)^3}-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (2 c d-b e) (d+e x)^5}+\frac{(5 c (2 c d-b e) (4 c e f-10 c d g+3 b e g)) \int \frac{1}{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{8 e}\\ &=\frac{5 c (4 c e f-10 c d g+3 b e g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{4 e^2}+\frac{5 c (4 c e f-10 c d g+3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{6 e^2 (2 c d-b e) (d+e x)}+\frac{2 (4 c e f-10 c d g+3 b e 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)^3}-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (2 c d-b e) (d+e x)^5}+\frac{(5 c (2 c d-b e) (4 c e f-10 c d g+3 b e g)) \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 )}{4 e}\\ &=\frac{5 c (4 c e f-10 c d g+3 b e g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{4 e^2}+\frac{5 c (4 c e f-10 c d g+3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{6 e^2 (2 c d-b e) (d+e x)}+\frac{2 (4 c e f-10 c d g+3 b e 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)^3}-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (2 c d-b e) (d+e x)^5}+\frac{5 \sqrt{c} (2 c d-b e) (4 c e f-10 c d g+3 b e 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 )}{8 e^2}\\ \end{align*}

Mathematica [C]  time = 0.22416, size = 173, normalized size = 0.49 \[ \frac{2 ((d+e x) (c (d-e x)-b e))^{5/2} \left (\frac{(d+e x) (b e-2 c d)^2 (3 b e g-10 c d g+4 c e f) \, _2F_1\left (-\frac{5}{2},-\frac{1}{2};\frac{1}{2};\frac{c (d+e x)}{2 c d-b e}\right )}{\sqrt{\frac{b e-c d+c e x}{b e-2 c d}}}+(e f-d g) (b e-c d+c e x)^3\right )}{3 e^2 (d+e x)^4 (2 c d-b e) (b e-c d+c e x)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2))/(d + e*x)^5,x]

[Out]

(2*((d + e*x)*(-(b*e) + c*(d - e*x)))^(5/2)*((e*f - d*g)*(-(c*d) + b*e + c*e*x)^3 + ((-2*c*d + b*e)^2*(4*c*e*f
- 10*c*d*g + 3*b*e*g)*(d + e*x)*Hypergeometric2F1[-5/2, -1/2, 1/2, (c*(d + e*x))/(2*c*d - b*e)])/Sqrt[(-(c*d)
+ b*e + c*e*x)/(-2*c*d + b*e)]))/(3*e^2*(2*c*d - b*e)*(d + e*x)^4*(-(c*d) + b*e + c*e*x)^2)

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Maple [B]  time = 0.016, size = 5227, normalized size = 14.9 \begin{align*} \text{output 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)^(5/2)/(e*x+d)^5,x)

[Out]

result too large to display

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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)^(5/2)/(e*x+d)^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 22.2778, size = 2010, normalized size = 5.74 \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)^(5/2)/(e*x+d)^5,x, algorithm="fricas")

[Out]

[-1/48*(15*((4*(2*c^2*d*e^3 - b*c*e^4)*f - (20*c^2*d^2*e^2 - 16*b*c*d*e^3 + 3*b^2*e^4)*g)*x^2 + 4*(2*c^2*d^3*e
- b*c*d^2*e^2)*f - (20*c^2*d^4 - 16*b*c*d^3*e + 3*b^2*d^2*e^2)*g + 2*(4*(2*c^2*d^2*e^2 - b*c*d*e^3)*f - (20*c
^2*d^3*e - 16*b*c*d^2*e^2 + 3*b^2*d*e^3)*g)*x)*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*(6*c^2*e^3*g*x^3 + 3*
(4*c^2*e^3*f - (16*c^2*d*e^2 - 9*b*c*e^3)*g)*x^2 + 4*(23*c^2*d^2*e - 6*b*c*d*e^2 - 2*b^2*e^3)*f - (236*c^2*d^3
- 147*b*c*d^2*e + 16*b^2*d*e^2)*g + 2*(4*(17*c^2*d*e^2 - 7*b*c*e^3)*f - (161*c^2*d^2*e - 103*b*c*d*e^2 + 12*b
^2*e^3)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))/(e^4*x^2 + 2*d*e^3*x + d^2*e^2), -1/24*(15*((4*(2*c^
2*d*e^3 - b*c*e^4)*f - (20*c^2*d^2*e^2 - 16*b*c*d*e^3 + 3*b^2*e^4)*g)*x^2 + 4*(2*c^2*d^3*e - b*c*d^2*e^2)*f -
(20*c^2*d^4 - 16*b*c*d^3*e + 3*b^2*d^2*e^2)*g + 2*(4*(2*c^2*d^2*e^2 - b*c*d*e^3)*f - (20*c^2*d^3*e - 16*b*c*d^
2*e^2 + 3*b^2*d*e^3)*g)*x)*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*(6*c^2*e^3*g*x^3 + 3*(4*c^2*e^3*f - (16*c^2*d*e^2 - 9*b*
c*e^3)*g)*x^2 + 4*(23*c^2*d^2*e - 6*b*c*d*e^2 - 2*b^2*e^3)*f - (236*c^2*d^3 - 147*b*c*d^2*e + 16*b^2*d*e^2)*g
+ 2*(4*(17*c^2*d*e^2 - 7*b*c*e^3)*f - (161*c^2*d^2*e - 103*b*c*d*e^2 + 12*b^2*e^3)*g)*x)*sqrt(-c*e^2*x^2 - b*e
^2*x + c*d^2 - b*d*e))/(e^4*x^2 + 2*d*e^3*x + d^2*e^2)]

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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)**(5/2)/(e*x+d)**5,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)^(5/2)/(e*x+d)^5,x, algorithm="giac")

[Out]

Timed out