3.2261 \(\int \frac{(f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2}}{(d+e x)^{17/2}} \, dx\)

Optimal. Leaf size=464 \[ \frac{c^3 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-10 b e g+17 c d g+3 c e f)}{128 e^2 (d+e x)^{3/2} (2 c d-b e)^2}-\frac{c^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-10 b e g+17 c d g+3 c e f)}{64 e^2 (d+e x)^{5/2} (2 c d-b e)}+\frac{c^4 (-10 b e g+17 c d g+3 c e f) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{128 e^2 (2 c d-b e)^{5/2}}+\frac{c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-10 b e g+17 c d g+3 c e f)}{48 e^2 (d+e x)^{9/2} (2 c d-b e)}-\frac{(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{5 e^2 (d+e x)^{17/2} (2 c d-b e)}-\frac{\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-10 b e g+17 c d g+3 c e f)}{40 e^2 (d+e x)^{13/2} (2 c d-b e)} \]

[Out]

-(c^2*(3*c*e*f + 17*c*d*g - 10*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(64*e^2*(2*c*d - b*e)*(d + e*
x)^(5/2)) + (c^3*(3*c*e*f + 17*c*d*g - 10*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(128*e^2*(2*c*d -
b*e)^2*(d + e*x)^(3/2)) + (c*(3*c*e*f + 17*c*d*g - 10*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(48*
e^2*(2*c*d - b*e)*(d + e*x)^(9/2)) - ((3*c*e*f + 17*c*d*g - 10*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5
/2))/(40*e^2*(2*c*d - b*e)*(d + e*x)^(13/2)) - ((e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(5*e^
2*(2*c*d - b*e)*(d + e*x)^(17/2)) + (c^4*(3*c*e*f + 17*c*d*g - 10*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])])/(128*e^2*(2*c*d - b*e)^(5/2))

________________________________________________________________________________________

Rubi [A]  time = 0.751239, antiderivative size = 464, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.109, Rules used = {792, 662, 672, 660, 208} \[ \frac{c^3 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-10 b e g+17 c d g+3 c e f)}{128 e^2 (d+e x)^{3/2} (2 c d-b e)^2}-\frac{c^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-10 b e g+17 c d g+3 c e f)}{64 e^2 (d+e x)^{5/2} (2 c d-b e)}+\frac{c^4 (-10 b e g+17 c d g+3 c e f) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{128 e^2 (2 c d-b e)^{5/2}}+\frac{c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-10 b e g+17 c d g+3 c e f)}{48 e^2 (d+e x)^{9/2} (2 c d-b e)}-\frac{(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{5 e^2 (d+e x)^{17/2} (2 c d-b e)}-\frac{\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-10 b e g+17 c d g+3 c e f)}{40 e^2 (d+e x)^{13/2} (2 c d-b e)} \]

Antiderivative was successfully verified.

[In]

Int[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2))/(d + e*x)^(17/2),x]

[Out]

-(c^2*(3*c*e*f + 17*c*d*g - 10*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(64*e^2*(2*c*d - b*e)*(d + e*
x)^(5/2)) + (c^3*(3*c*e*f + 17*c*d*g - 10*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(128*e^2*(2*c*d -
b*e)^2*(d + e*x)^(3/2)) + (c*(3*c*e*f + 17*c*d*g - 10*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(48*
e^2*(2*c*d - b*e)*(d + e*x)^(9/2)) - ((3*c*e*f + 17*c*d*g - 10*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5
/2))/(40*e^2*(2*c*d - b*e)*(d + e*x)^(13/2)) - ((e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(5*e^
2*(2*c*d - b*e)*(d + e*x)^(17/2)) + (c^4*(3*c*e*f + 17*c*d*g - 10*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])])/(128*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 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 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) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{17/2}} \, dx &=-\frac{(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{5 e^2 (2 c d-b e) (d+e x)^{17/2}}+\frac{(3 c e f+17 c d g-10 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)^{15/2}} \, dx}{10 e (2 c d-b e)}\\ &=-\frac{(3 c e f+17 c d g-10 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{40 e^2 (2 c d-b e) (d+e x)^{13/2}}-\frac{(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{5 e^2 (2 c d-b e) (d+e x)^{17/2}}-\frac{(c (3 c e f+17 c d g-10 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)^{11/2}} \, dx}{16 e (2 c d-b e)}\\ &=\frac{c (3 c e f+17 c d g-10 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{48 e^2 (2 c d-b e) (d+e x)^{9/2}}-\frac{(3 c e f+17 c d g-10 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{40 e^2 (2 c d-b e) (d+e x)^{13/2}}-\frac{(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{5 e^2 (2 c d-b e) (d+e x)^{17/2}}+\frac{\left (c^2 (3 c e f+17 c d g-10 b e g)\right ) \int \frac{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{7/2}} \, dx}{32 e (2 c d-b e)}\\ &=-\frac{c^2 (3 c e f+17 c d g-10 b e g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{64 e^2 (2 c d-b e) (d+e x)^{5/2}}+\frac{c (3 c e f+17 c d g-10 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{48 e^2 (2 c d-b e) (d+e x)^{9/2}}-\frac{(3 c e f+17 c d g-10 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{40 e^2 (2 c d-b e) (d+e x)^{13/2}}-\frac{(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{5 e^2 (2 c d-b e) (d+e x)^{17/2}}-\frac{\left (c^3 (3 c e f+17 c d g-10 b e g)\right ) \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}{128 e (2 c d-b e)}\\ &=-\frac{c^2 (3 c e f+17 c d g-10 b e g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{64 e^2 (2 c d-b e) (d+e x)^{5/2}}+\frac{c^3 (3 c e f+17 c d g-10 b e g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{128 e^2 (2 c d-b e)^2 (d+e x)^{3/2}}+\frac{c (3 c e f+17 c d g-10 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{48 e^2 (2 c d-b e) (d+e x)^{9/2}}-\frac{(3 c e f+17 c d g-10 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{40 e^2 (2 c d-b e) (d+e x)^{13/2}}-\frac{(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{5 e^2 (2 c d-b e) (d+e x)^{17/2}}-\frac{\left (c^4 (3 c e f+17 c d g-10 b e g)\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{256 e (2 c d-b e)^2}\\ &=-\frac{c^2 (3 c e f+17 c d g-10 b e g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{64 e^2 (2 c d-b e) (d+e x)^{5/2}}+\frac{c^3 (3 c e f+17 c d g-10 b e g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{128 e^2 (2 c d-b e)^2 (d+e x)^{3/2}}+\frac{c (3 c e f+17 c d g-10 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{48 e^2 (2 c d-b e) (d+e x)^{9/2}}-\frac{(3 c e f+17 c d g-10 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{40 e^2 (2 c d-b e) (d+e x)^{13/2}}-\frac{(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{5 e^2 (2 c d-b e) (d+e x)^{17/2}}-\frac{\left (c^4 (3 c e f+17 c d g-10 b e g)\right ) \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 )}{128 (2 c d-b e)^2}\\ &=-\frac{c^2 (3 c e f+17 c d g-10 b e g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{64 e^2 (2 c d-b e) (d+e x)^{5/2}}+\frac{c^3 (3 c e f+17 c d g-10 b e g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{128 e^2 (2 c d-b e)^2 (d+e x)^{3/2}}+\frac{c (3 c e f+17 c d g-10 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{48 e^2 (2 c d-b e) (d+e x)^{9/2}}-\frac{(3 c e f+17 c d g-10 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{40 e^2 (2 c d-b e) (d+e x)^{13/2}}-\frac{(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{5 e^2 (2 c d-b e) (d+e x)^{17/2}}+\frac{c^4 (3 c e f+17 c d g-10 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 )}{128 e^2 (2 c d-b e)^{5/2}}\\ \end{align*}

Mathematica [C]  time = 0.190557, size = 129, normalized size = 0.28 \[ \frac{((d+e x) (c (d-e x)-b e))^{7/2} \left (-\frac{c^4 (d+e x)^5 (-10 b e g+17 c d g+3 c e f) \, _2F_1\left (\frac{7}{2},5;\frac{9}{2};\frac{-c d+b e+c e x}{b e-2 c d}\right )}{(2 c d-b e)^5}+7 d g-7 e f\right )}{35 e^2 (d+e x)^{17/2} (2 c d-b e)} \]

Antiderivative was successfully verified.

[In]

Integrate[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2))/(d + e*x)^(17/2),x]

[Out]

(((d + e*x)*(-(b*e) + c*(d - e*x)))^(7/2)*(-7*e*f + 7*d*g - (c^4*(3*c*e*f + 17*c*d*g - 10*b*e*g)*(d + e*x)^5*H
ypergeometric2F1[7/2, 5, 9/2, (-(c*d) + b*e + c*e*x)/(-2*c*d + b*e)])/(2*c*d - b*e)^5))/(35*e^2*(2*c*d - b*e)*
(d + e*x)^(17/2))

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Maple [B]  time = 0.039, size = 2087, normalized size = 4.5 \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)^(5/2)/(e*x+d)^(17/2),x)

[Out]

1/1920*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*(-384*b^4*e^5*f*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)+150*arc
tan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x^5*b*c^4*e^6*g-255*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1
/2))*x^5*c^5*d*e^5*g-1275*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x^4*c^5*d^2*e^4*g-225*arctan((-c*e*
x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x^4*c^5*d*e^5*f-2550*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x^3*
c^5*d^3*e^3*g-450*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x^3*c^5*d^2*e^4*f+45*x^4*c^4*e^5*f*(b*e-2*c
*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)-2550*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x^2*c^5*d^4*e^2*g-450*a
rctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x^2*c^5*d^3*e^3*f-1275*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*
d)^(1/2))*x*c^5*d^5*e*g-225*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x*c^5*d^4*e^2*f+150*arctan((-c*e*
x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c^4*d^5*e*g-480*x*b^4*e^5*g*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)-96*
b^4*d*e^4*g*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)-951*c^4*d^4*e*f*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)-
255*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^5*d^6*g-269*c^4*d^5*g*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d
)^(1/2)-45*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x^5*c^5*e^6*f-45*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*
e-2*c*d)^(1/2))*c^5*d^5*e*f-30*x^3*b*c^3*e^5*f*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)-3760*x^3*c^4*d^2*e^3*g
*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)+240*x^3*c^4*d*e^4*f*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)-1360*x^
2*b^3*c*e^5*g*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)-744*x^2*b^2*c^2*e^5*f*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d
)^(1/2)+1046*x^2*c^4*d^3*e^2*g*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)-2526*x^2*c^4*d^2*e^3*f*(b*e-2*c*d)^(1/
2)*(-c*e*x-b*e+c*d)^(1/2)-150*x^4*b*c^3*e^5*g*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)+255*x^4*c^4*d*e^4*g*(b*
e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)-1180*x^3*b^2*c^2*e^5*g*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)-1008*x*b
^3*c*e^5*f*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)-1352*x*c^4*d^4*e*g*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2
)+2472*x*c^4*d^3*e^2*f*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)+416*b^3*c*d^2*e^3*g*(b*e-2*c*d)^(1/2)*(-c*e*x-
b*e+c*d)^(1/2)+2064*b^3*c*d*e^4*f*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)-628*b^2*c^2*d^3*e^2*g*(b*e-2*c*d)^(
1/2)*(-c*e*x-b*e+c*d)^(1/2)-3912*b^2*c^2*d^2*e^3*f*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)+472*b*c^3*d^4*e*g*
(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)+3138*b*c^3*d^3*e^2*f*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)+750*arc
tan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x^4*b*c^4*d*e^5*g+1500*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)
^(1/2))*x^3*b*c^4*d^2*e^4*g+1500*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x^2*b*c^4*d^3*e^3*g+750*arct
an((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x*b*c^4*d^4*e^2*g+5364*x^2*b^2*c^2*d*e^4*g*(b*e-2*c*d)^(1/2)*(-c*
e*x-b*e+c*d)^(1/2)-5946*x^2*b*c^3*d^2*e^3*g*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)+2886*x^2*b*c^3*d*e^4*f*(b
*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)+2128*x*b^3*c*d*e^4*g*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)-3300*x*b^
2*c^2*d^2*e^3*g*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)+4560*x*b^2*c^2*d*e^4*f*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+
c*d)^(1/2)+2514*x*b*c^3*d^3*e^2*g*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)-6234*x*b*c^3*d^2*e^3*f*(b*e-2*c*d)^
(1/2)*(-c*e*x-b*e+c*d)^(1/2)+4150*x^3*b*c^3*d*e^4*g*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2))/(e*x+d)^(11/2)/(
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{{\left (-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e\right )}^{\frac{5}{2}}{\left (g x + f\right )}}{{\left (e x + d\right )}^{\frac{17}{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)^(5/2)/(e*x+d)^(17/2),x, algorithm="maxima")

[Out]

integrate((-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(5/2)*(g*x + f)/(e*x + d)^(17/2), x)

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Fricas [B]  time = 2.10372, size = 5625, normalized size = 12.12 \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)^(17/2),x, algorithm="fricas")

[Out]

[-1/3840*(15*(3*c^5*d^6*e*f + (3*c^5*e^7*f + (17*c^5*d*e^6 - 10*b*c^4*e^7)*g)*x^6 + 6*(3*c^5*d*e^6*f + (17*c^5
*d^2*e^5 - 10*b*c^4*d*e^6)*g)*x^5 + 15*(3*c^5*d^2*e^5*f + (17*c^5*d^3*e^4 - 10*b*c^4*d^2*e^5)*g)*x^4 + 20*(3*c
^5*d^3*e^4*f + (17*c^5*d^4*e^3 - 10*b*c^4*d^3*e^4)*g)*x^3 + 15*(3*c^5*d^4*e^3*f + (17*c^5*d^5*e^2 - 10*b*c^4*d
^4*e^3)*g)*x^2 + (17*c^5*d^7 - 10*b*c^4*d^6*e)*g + 6*(3*c^5*d^5*e^2*f + (17*c^5*d^6*e - 10*b*c^4*d^5*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)*(15*(3*(2*c^5*d*e^5 - b*c^4*e^6)*f + (34*c^5*d^2*e^4 - 37*b*c^4*d*e^5 + 10*b^2*c^3*e^6)*g)*x^4 + 1
0*(3*(16*c^5*d^2*e^4 - 10*b*c^4*d*e^5 + b^2*c^3*e^6)*f - (752*c^5*d^3*e^3 - 1206*b*c^4*d^2*e^4 + 651*b^2*c^3*d
*e^5 - 118*b^3*c^2*e^6)*g)*x^3 - 2*(3*(842*c^5*d^3*e^3 - 1383*b*c^4*d^2*e^4 + 729*b^2*c^3*d*e^5 - 124*b^3*c^2*
e^6)*f - (1046*c^5*d^4*e^2 - 6469*b*c^4*d^3*e^3 + 8337*b^2*c^3*d^2*e^4 - 4042*b^3*c^2*d*e^5 + 680*b^4*c*e^6)*g
)*x^2 - 3*(634*c^5*d^5*e - 2409*b*c^4*d^4*e^2 + 3654*b^2*c^3*d^3*e^3 - 2680*b^3*c^2*d^2*e^4 + 944*b^4*c*d*e^5
- 128*b^5*e^6)*f - (538*c^5*d^6 - 1213*b*c^4*d^5*e + 1728*b^2*c^3*d^4*e^2 - 1460*b^3*c^2*d^3*e^3 + 608*b^4*c*d
^2*e^4 - 96*b^5*d*e^5)*g + 2*(3*(824*c^5*d^4*e^2 - 2490*b*c^4*d^3*e^3 + 2559*b^2*c^3*d^2*e^4 - 1096*b^3*c^2*d*
e^5 + 168*b^4*c*e^6)*f - (1352*c^5*d^5*e - 3190*b*c^4*d^4*e^2 + 4557*b^2*c^3*d^3*e^3 - 3778*b^3*c^2*d^2*e^4 +
1544*b^4*c*d*e^5 - 240*b^5*e^6)*g)*x)*sqrt(e*x + d))/(8*c^3*d^9*e^2 - 12*b*c^2*d^8*e^3 + 6*b^2*c*d^7*e^4 - b^3
*d^6*e^5 + (8*c^3*d^3*e^8 - 12*b*c^2*d^2*e^9 + 6*b^2*c*d*e^10 - b^3*e^11)*x^6 + 6*(8*c^3*d^4*e^7 - 12*b*c^2*d^
3*e^8 + 6*b^2*c*d^2*e^9 - b^3*d*e^10)*x^5 + 15*(8*c^3*d^5*e^6 - 12*b*c^2*d^4*e^7 + 6*b^2*c*d^3*e^8 - b^3*d^2*e
^9)*x^4 + 20*(8*c^3*d^6*e^5 - 12*b*c^2*d^5*e^6 + 6*b^2*c*d^4*e^7 - b^3*d^3*e^8)*x^3 + 15*(8*c^3*d^7*e^4 - 12*b
*c^2*d^6*e^5 + 6*b^2*c*d^5*e^6 - b^3*d^4*e^7)*x^2 + 6*(8*c^3*d^8*e^3 - 12*b*c^2*d^7*e^4 + 6*b^2*c*d^6*e^5 - b^
3*d^5*e^6)*x), 1/1920*(15*(3*c^5*d^6*e*f + (3*c^5*e^7*f + (17*c^5*d*e^6 - 10*b*c^4*e^7)*g)*x^6 + 6*(3*c^5*d*e^
6*f + (17*c^5*d^2*e^5 - 10*b*c^4*d*e^6)*g)*x^5 + 15*(3*c^5*d^2*e^5*f + (17*c^5*d^3*e^4 - 10*b*c^4*d^2*e^5)*g)*
x^4 + 20*(3*c^5*d^3*e^4*f + (17*c^5*d^4*e^3 - 10*b*c^4*d^3*e^4)*g)*x^3 + 15*(3*c^5*d^4*e^3*f + (17*c^5*d^5*e^2
- 10*b*c^4*d^4*e^3)*g)*x^2 + (17*c^5*d^7 - 10*b*c^4*d^6*e)*g + 6*(3*c^5*d^5*e^2*f + (17*c^5*d^6*e - 10*b*c^4*
d^5*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)*(15*(3*(2*c^5*d*e^5
- b*c^4*e^6)*f + (34*c^5*d^2*e^4 - 37*b*c^4*d*e^5 + 10*b^2*c^3*e^6)*g)*x^4 + 10*(3*(16*c^5*d^2*e^4 - 10*b*c^4
*d*e^5 + b^2*c^3*e^6)*f - (752*c^5*d^3*e^3 - 1206*b*c^4*d^2*e^4 + 651*b^2*c^3*d*e^5 - 118*b^3*c^2*e^6)*g)*x^3
- 2*(3*(842*c^5*d^3*e^3 - 1383*b*c^4*d^2*e^4 + 729*b^2*c^3*d*e^5 - 124*b^3*c^2*e^6)*f - (1046*c^5*d^4*e^2 - 64
69*b*c^4*d^3*e^3 + 8337*b^2*c^3*d^2*e^4 - 4042*b^3*c^2*d*e^5 + 680*b^4*c*e^6)*g)*x^2 - 3*(634*c^5*d^5*e - 2409
*b*c^4*d^4*e^2 + 3654*b^2*c^3*d^3*e^3 - 2680*b^3*c^2*d^2*e^4 + 944*b^4*c*d*e^5 - 128*b^5*e^6)*f - (538*c^5*d^6
- 1213*b*c^4*d^5*e + 1728*b^2*c^3*d^4*e^2 - 1460*b^3*c^2*d^3*e^3 + 608*b^4*c*d^2*e^4 - 96*b^5*d*e^5)*g + 2*(3
*(824*c^5*d^4*e^2 - 2490*b*c^4*d^3*e^3 + 2559*b^2*c^3*d^2*e^4 - 1096*b^3*c^2*d*e^5 + 168*b^4*c*e^6)*f - (1352*
c^5*d^5*e - 3190*b*c^4*d^4*e^2 + 4557*b^2*c^3*d^3*e^3 - 3778*b^3*c^2*d^2*e^4 + 1544*b^4*c*d*e^5 - 240*b^5*e^6)
*g)*x)*sqrt(e*x + d))/(8*c^3*d^9*e^2 - 12*b*c^2*d^8*e^3 + 6*b^2*c*d^7*e^4 - b^3*d^6*e^5 + (8*c^3*d^3*e^8 - 12*
b*c^2*d^2*e^9 + 6*b^2*c*d*e^10 - b^3*e^11)*x^6 + 6*(8*c^3*d^4*e^7 - 12*b*c^2*d^3*e^8 + 6*b^2*c*d^2*e^9 - b^3*d
*e^10)*x^5 + 15*(8*c^3*d^5*e^6 - 12*b*c^2*d^4*e^7 + 6*b^2*c*d^3*e^8 - b^3*d^2*e^9)*x^4 + 20*(8*c^3*d^6*e^5 - 1
2*b*c^2*d^5*e^6 + 6*b^2*c*d^4*e^7 - b^3*d^3*e^8)*x^3 + 15*(8*c^3*d^7*e^4 - 12*b*c^2*d^6*e^5 + 6*b^2*c*d^5*e^6
- b^3*d^4*e^7)*x^2 + 6*(8*c^3*d^8*e^3 - 12*b*c^2*d^7*e^4 + 6*b^2*c*d^6*e^5 - b^3*d^5*e^6)*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)**(5/2)/(e*x+d)**(17/2),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)^(17/2),x, algorithm="giac")

[Out]

Timed out