3.2256 \(\int \frac{(f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2}}{(d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=316 \[ \frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e^2 (d+e x)^{5/2}}+\frac{2 (2 c d-b e) (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^{3/2}}+\frac{2 (2 c d-b e)^2 (e f-d g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 \sqrt{d+e x}}-\frac{2 (2 c d-b e)^{5/2} (e f-d g) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{e^2}-\frac{2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{7 c e^2 (d+e x)^{7/2}} \]

[Out]

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

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Rubi [A]  time = 0.588462, antiderivative size = 316, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {794, 664, 660, 208} \[ \frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e^2 (d+e x)^{5/2}}+\frac{2 (2 c d-b e) (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^{3/2}}+\frac{2 (2 c d-b e)^2 (e f-d g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 \sqrt{d+e x}}-\frac{2 (2 c d-b e)^{5/2} (e f-d g) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{e^2}-\frac{2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{7 c e^2 (d+e x)^{7/2}} \]

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)^(7/2),x]

[Out]

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

Rule 794

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[(g*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] + Dist[(m*(g*(c*d - b*e) + c*e*f) + e*(p + 1)
*(2*c*f - b*g))/(c*e*(m + 2*p + 2)), Int[(d + e*x)^m*(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] && NeQ[m + 2*p + 2, 0] && (NeQ[m, 2] || Eq
Q[d, 0])

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

Mathematica [A]  time = 0.629811, size = 197, normalized size = 0.62 \[ \frac{2 ((d+e x) (c (d-e x)-b e))^{5/2} \left (\frac{7 c (e f-d g) \left (\sqrt{c (d-e x)-b e} \left (23 b^2 e^2+b c e (11 e x-81 d)+c^2 \left (73 d^2-16 d e x+3 e^2 x^2\right )\right )-15 (2 c d-b e)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{-b e+c d-c e x}}{\sqrt{2 c d-b e}}\right )\right )}{15 (c (d-e x)-b e)^{5/2}}+g (b e-c d+c e x)\right )}{7 c e^2 (d+e x)^{5/2}} \]

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)^(7/2),x]

[Out]

(2*((d + e*x)*(-(b*e) + c*(d - e*x)))^(5/2)*(g*(-(c*d) + b*e + c*e*x) + (7*c*(e*f - d*g)*(Sqrt[-(b*e) + c*(d -
e*x)]*(23*b^2*e^2 + b*c*e*(-81*d + 11*e*x) + c^2*(73*d^2 - 16*d*e*x + 3*e^2*x^2)) - 15*(2*c*d - b*e)^(5/2)*Ar
cTanh[Sqrt[c*d - b*e - c*e*x]/Sqrt[2*c*d - b*e]]))/(15*(-(b*e) + c*(d - e*x))^(5/2))))/(7*c*e^2*(d + e*x)^(5/2
))

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Maple [B]  time = 0.023, size = 956, normalized size = 3. \begin{align*}{\frac{2}{105\,c{e}^{2}}\sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}} \left ( 15\,{x}^{3}{c}^{3}{e}^{3}g\sqrt{-cex-be+cd}\sqrt{be-2\,cd}+45\,{x}^{2}b{c}^{2}{e}^{3}g\sqrt{-cex-be+cd}\sqrt{be-2\,cd}-66\,{x}^{2}{c}^{3}d{e}^{2}g\sqrt{-cex-be+cd}\sqrt{be-2\,cd}+21\,{x}^{2}{c}^{3}{e}^{3}f\sqrt{-cex-be+cd}\sqrt{be-2\,cd}+105\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ){b}^{3}cd{e}^{3}g-105\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ){b}^{3}c{e}^{4}f-630\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ){b}^{2}{c}^{2}{d}^{2}{e}^{2}g+630\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ){b}^{2}{c}^{2}d{e}^{3}f+1260\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) b{c}^{3}{d}^{3}eg-1260\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) b{c}^{3}{d}^{2}{e}^{2}f-840\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ){c}^{4}{d}^{4}g+840\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ){c}^{4}{d}^{3}ef+45\,x{b}^{2}c{e}^{3}g\sqrt{-cex-be+cd}\sqrt{be-2\,cd}-167\,xb{c}^{2}d{e}^{2}g\sqrt{-cex-be+cd}\sqrt{be-2\,cd}+77\,xb{c}^{2}{e}^{3}f\sqrt{-cex-be+cd}\sqrt{be-2\,cd}+157\,x{c}^{3}{d}^{2}eg\sqrt{-cex-be+cd}\sqrt{be-2\,cd}-112\,x{c}^{3}d{e}^{2}f\sqrt{-cex-be+cd}\sqrt{be-2\,cd}+15\,\sqrt{be-2\,cd}\sqrt{-cex-be+cd}{b}^{3}{e}^{3}g-206\,\sqrt{be-2\,cd}\sqrt{-cex-be+cd}{b}^{2}cd{e}^{2}g+161\,\sqrt{be-2\,cd}\sqrt{-cex-be+cd}{b}^{2}c{e}^{3}f+612\,\sqrt{be-2\,cd}\sqrt{-cex-be+cd}b{c}^{2}{d}^{2}eg-567\,\sqrt{be-2\,cd}\sqrt{-cex-be+cd}b{c}^{2}d{e}^{2}f-526\,\sqrt{be-2\,cd}\sqrt{-cex-be+cd}{c}^{3}{d}^{3}g+511\,\sqrt{be-2\,cd}\sqrt{-cex-be+cd}{c}^{3}{d}^{2}ef \right ){\frac{1}{\sqrt{ex+d}}}{\frac{1}{\sqrt{-cex-be+cd}}}{\frac{1}{\sqrt{be-2\,cd}}}} \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)^(7/2),x)

[Out]

2/105*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*(15*x^3*c^3*e^3*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+45*x^2
*b*c^2*e^3*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-66*x^2*c^3*d*e^2*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1
/2)+21*x^2*c^3*e^3*f*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+105*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1
/2))*b^3*c*d*e^3*g-105*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b^3*c*e^4*f-630*arctan((-c*e*x-b*e+c*d
)^(1/2)/(b*e-2*c*d)^(1/2))*b^2*c^2*d^2*e^2*g+630*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b^2*c^2*d*e^
3*f+1260*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c^3*d^3*e*g-1260*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*
e-2*c*d)^(1/2))*b*c^3*d^2*e^2*f-840*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^4*d^4*g+840*arctan((-c*
e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^4*d^3*e*f+45*x*b^2*c*e^3*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-16
7*x*b*c^2*d*e^2*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+77*x*b*c^2*e^3*f*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)
^(1/2)+157*x*c^3*d^2*e*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-112*x*c^3*d*e^2*f*(-c*e*x-b*e+c*d)^(1/2)*(b*
e-2*c*d)^(1/2)+15*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)*b^3*e^3*g-206*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1
/2)*b^2*c*d*e^2*g+161*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)*b^2*c*e^3*f+612*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c
*d)^(1/2)*b*c^2*d^2*e*g-567*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)*b*c^2*d*e^2*f-526*(b*e-2*c*d)^(1/2)*(-c*e
*x-b*e+c*d)^(1/2)*c^3*d^3*g+511*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)*c^3*d^2*e*f)/(e*x+d)^(1/2)/(-c*e*x-b*
e+c*d)^(1/2)/c/e^2/(b*e-2*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{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)^(5/2)/(e*x+d)^(7/2),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)^(7/2), x)

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Fricas [A]  time = 1.74414, size = 2014, normalized size = 6.37 \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)^(7/2),x, algorithm="fricas")

[Out]

[-1/105*(105*sqrt(2*c*d - b*e)*((4*c^3*d^3*e - 4*b*c^2*d^2*e^2 + b^2*c*d*e^3)*f - (4*c^3*d^4 - 4*b*c^2*d^3*e +
b^2*c*d^2*e^2)*g + ((4*c^3*d^2*e^2 - 4*b*c^2*d*e^3 + b^2*c*e^4)*f - (4*c^3*d^3*e - 4*b*c^2*d^2*e^2 + b^2*c*d*
e^3)*g)*x)*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*(15*c^3*e^3*g*x^3 + 3*(7*c^3*e^3*f - (22
*c^3*d*e^2 - 15*b*c^2*e^3)*g)*x^2 + 7*(73*c^3*d^2*e - 81*b*c^2*d*e^2 + 23*b^2*c*e^3)*f - (526*c^3*d^3 - 612*b*
c^2*d^2*e + 206*b^2*c*d*e^2 - 15*b^3*e^3)*g - (7*(16*c^3*d*e^2 - 11*b*c^2*e^3)*f - (157*c^3*d^2*e - 167*b*c^2*
d*e^2 + 45*b^2*c*e^3)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(e*x + d))/(c*e^3*x + c*d*e^2), -2/
105*(105*sqrt(-2*c*d + b*e)*((4*c^3*d^3*e - 4*b*c^2*d^2*e^2 + b^2*c*d*e^3)*f - (4*c^3*d^4 - 4*b*c^2*d^3*e + b^
2*c*d^2*e^2)*g + ((4*c^3*d^2*e^2 - 4*b*c^2*d*e^3 + b^2*c*e^4)*f - (4*c^3*d^3*e - 4*b*c^2*d^2*e^2 + b^2*c*d*e^3
)*g)*x)*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)) - (15*c^3*e^3*g*x^3 + 3*(7*c^3*e^3*f - (22*c^3*d*e^2 - 15*b*c^2*e^3)*g)*x^2 + 7*(73*c^3*d^
2*e - 81*b*c^2*d*e^2 + 23*b^2*c*e^3)*f - (526*c^3*d^3 - 612*b*c^2*d^2*e + 206*b^2*c*d*e^2 - 15*b^3*e^3)*g - (7
*(16*c^3*d*e^2 - 11*b*c^2*e^3)*f - (157*c^3*d^2*e - 167*b*c^2*d*e^2 + 45*b^2*c*e^3)*g)*x)*sqrt(-c*e^2*x^2 - b*
e^2*x + c*d^2 - b*d*e)*sqrt(e*x + d))/(c*e^3*x + c*d*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)**(7/2),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)^(7/2),x, algorithm="giac")

[Out]

Timed out