3.20.60 \(\int \frac {(f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{3/2}}{(d+e x)^5} \, dx\)

Optimal. Leaf size=214 \[ \frac {c^{3/2} g \tan ^{-1}\left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{e^2}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e^2 (d+e x)^5 (2 c d-b e)}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^3}+\frac {2 c g \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 (d+e x)} \]

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Rubi [A]  time = 0.44, antiderivative size = 214, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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)} \end {gather*}

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 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])

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 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 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]

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*}

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Mathematica [C]  time = 0.28, size = 150, normalized size = 0.70 \begin {gather*} \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)} \end {gather*}

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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IntegrateAlgebraic [B]  time = 153.13, size = 13691, normalized size = 63.98 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((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]

Result too large to show

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fricas [B]  time = 9.81, size = 879, normalized size = 4.11 \begin {gather*} \left [\frac {15 \, {\left ({\left (2 \, c^{2} d e^{3} - b c e^{4}\right )} g x^{3} + 3 \, {\left (2 \, c^{2} d^{2} e^{2} - b c d e^{3}\right )} g x^{2} + 3 \, {\left (2 \, c^{2} d^{3} e - b c d^{2} e^{2}\right )} g x + {\left (2 \, c^{2} d^{4} - b c d^{3} e\right )} g\right )} \sqrt {-c} \log \left (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} {\left (2 \, c e x + b e\right )} \sqrt {-c}\right ) - 4 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left ({\left (3 \, c^{2} e^{3} f - {\left (43 \, c^{2} d e^{2} - 20 \, b c e^{3}\right )} g\right )} x^{2} + 3 \, {\left (c^{2} d^{2} e - 2 \, b c d e^{2} + b^{2} e^{3}\right )} f - {\left (23 \, c^{2} d^{3} - 6 \, b c d^{2} e - 2 \, b^{2} d e^{2}\right )} g - {\left (6 \, {\left (c^{2} d e^{2} - b c e^{3}\right )} f + {\left (54 \, c^{2} d^{2} e - 14 \, b c d e^{2} - 5 \, b^{2} e^{3}\right )} g\right )} x\right )}}{30 \, {\left (2 \, c d^{4} e^{2} - b d^{3} e^{3} + {\left (2 \, c d e^{5} - b e^{6}\right )} x^{3} + 3 \, {\left (2 \, c d^{2} e^{4} - b d e^{5}\right )} x^{2} + 3 \, {\left (2 \, c d^{3} e^{3} - b d^{2} e^{4}\right )} x\right )}}, -\frac {15 \, {\left ({\left (2 \, c^{2} d e^{3} - b c e^{4}\right )} g x^{3} + 3 \, {\left (2 \, c^{2} d^{2} e^{2} - b c d e^{3}\right )} g x^{2} + 3 \, {\left (2 \, c^{2} d^{3} e - b c d^{2} e^{2}\right )} g x + {\left (2 \, c^{2} d^{4} - b c d^{3} e\right )} g\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (2 \, c e x + b e\right )} \sqrt {c}}{2 \, {\left (c^{2} e^{2} x^{2} + b c e^{2} x - c^{2} d^{2} + b c d e\right )}}\right ) + 2 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left ({\left (3 \, c^{2} e^{3} f - {\left (43 \, c^{2} d e^{2} - 20 \, b c e^{3}\right )} g\right )} x^{2} + 3 \, {\left (c^{2} d^{2} e - 2 \, b c d e^{2} + b^{2} e^{3}\right )} f - {\left (23 \, c^{2} d^{3} - 6 \, b c d^{2} e - 2 \, b^{2} d e^{2}\right )} g - {\left (6 \, {\left (c^{2} d e^{2} - b c e^{3}\right )} f + {\left (54 \, c^{2} d^{2} e - 14 \, b c d e^{2} - 5 \, b^{2} e^{3}\right )} g\right )} x\right )}}{15 \, {\left (2 \, c d^{4} e^{2} - b d^{3} e^{3} + {\left (2 \, c d e^{5} - b e^{6}\right )} x^{3} + 3 \, {\left (2 \, c d^{2} e^{4} - b d e^{5}\right )} x^{2} + 3 \, {\left (2 \, c d^{3} e^{3} - b d^{2} e^{4}\right )} x\right )}}\right ] \end {gather*}

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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giac [B]  time = 1.75, size = 849, normalized size = 3.97 \begin {gather*} \frac {2}{15} \, {\left (\sqrt {-c e^{2} + \frac {2 \, c d e^{2}}{x e + d} - \frac {b e^{3}}{x e + d}} {\left (\frac {30 \, {\left (4 \, c^{2} d^{2} e^{9} - 4 \, b c d e^{10} + b^{2} e^{11}\right )} C_{0}}{8 \, c^{3} d^{3} e^{12} - 12 \, b c^{2} d^{2} e^{13} + 6 \, b^{2} c d e^{14} - b^{3} e^{15}} - \frac {{\left (\frac {128 \, c^{4} d^{4} g e^{8} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 48 \, c^{4} d^{3} f e^{9} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 232 \, b c^{3} d^{3} g e^{9} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 72 \, b c^{3} d^{2} f e^{10} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 156 \, b^{2} c^{2} d^{2} g e^{10} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 36 \, b^{2} c^{2} d f e^{11} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 46 \, b^{3} c d g e^{11} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 6 \, b^{3} c f e^{12} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 5 \, b^{4} g e^{12} \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{8 \, c^{3} d^{3} e^{12} - 12 \, b c^{2} d^{2} e^{13} + 6 \, b^{2} c d e^{14} - b^{3} e^{15}} - \frac {3 \, {\left (16 \, c^{4} d^{5} g e^{9} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 16 \, c^{4} d^{4} f e^{10} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 32 \, b c^{3} d^{4} g e^{10} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 32 \, b c^{3} d^{3} f e^{11} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 24 \, b^{2} c^{2} d^{3} g e^{11} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 24 \, b^{2} c^{2} d^{2} f e^{12} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 8 \, b^{3} c d^{2} g e^{12} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 8 \, b^{3} c d f e^{13} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + b^{4} d g e^{13} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - b^{4} f e^{14} \mathrm {sgn}\left (\frac {1}{x e + d}\right )\right )} e^{\left (-1\right )}}{{\left (8 \, c^{3} d^{3} e^{12} - 12 \, b c^{2} d^{2} e^{13} + 6 \, b^{2} c d e^{14} - b^{3} e^{15}\right )} {\left (x e + d\right )}}\right )} e^{\left (-1\right )}}{x e + d} + \frac {172 \, c^{4} d^{3} g e^{7} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 12 \, c^{4} d^{2} f e^{8} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 252 \, b c^{3} d^{2} g e^{8} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 12 \, b c^{3} d f e^{9} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 123 \, b^{2} c^{2} d g e^{9} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 3 \, b^{2} c^{2} f e^{10} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 20 \, b^{3} c g e^{10} \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{8 \, c^{3} d^{3} e^{12} - 12 \, b c^{2} d^{2} e^{13} + 6 \, b^{2} c d e^{14} - b^{3} e^{15}}\right )} - \frac {{\left (43 \, \sqrt {-c e^{2}} c^{2} d g - 3 \, \sqrt {-c e^{2}} c^{2} f e - 20 \, \sqrt {-c e^{2}} b c g e\right )} \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{2 \, c d e^{5} - b e^{6}}\right )} e^{2} \end {gather*}

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]

2/15*(sqrt(-c*e^2 + 2*c*d*e^2/(x*e + d) - b*e^3/(x*e + d))*(30*(4*c^2*d^2*e^9 - 4*b*c*d*e^10 + b^2*e^11)*C_0/(
8*c^3*d^3*e^12 - 12*b*c^2*d^2*e^13 + 6*b^2*c*d*e^14 - b^3*e^15) - ((128*c^4*d^4*g*e^8*sgn(1/(x*e + d)) - 48*c^
4*d^3*f*e^9*sgn(1/(x*e + d)) - 232*b*c^3*d^3*g*e^9*sgn(1/(x*e + d)) + 72*b*c^3*d^2*f*e^10*sgn(1/(x*e + d)) + 1
56*b^2*c^2*d^2*g*e^10*sgn(1/(x*e + d)) - 36*b^2*c^2*d*f*e^11*sgn(1/(x*e + d)) - 46*b^3*c*d*g*e^11*sgn(1/(x*e +
d)) + 6*b^3*c*f*e^12*sgn(1/(x*e + d)) + 5*b^4*g*e^12*sgn(1/(x*e + d)))/(8*c^3*d^3*e^12 - 12*b*c^2*d^2*e^13 +
6*b^2*c*d*e^14 - b^3*e^15) - 3*(16*c^4*d^5*g*e^9*sgn(1/(x*e + d)) - 16*c^4*d^4*f*e^10*sgn(1/(x*e + d)) - 32*b*
c^3*d^4*g*e^10*sgn(1/(x*e + d)) + 32*b*c^3*d^3*f*e^11*sgn(1/(x*e + d)) + 24*b^2*c^2*d^3*g*e^11*sgn(1/(x*e + d)
) - 24*b^2*c^2*d^2*f*e^12*sgn(1/(x*e + d)) - 8*b^3*c*d^2*g*e^12*sgn(1/(x*e + d)) + 8*b^3*c*d*f*e^13*sgn(1/(x*e
+ d)) + b^4*d*g*e^13*sgn(1/(x*e + d)) - b^4*f*e^14*sgn(1/(x*e + d)))*e^(-1)/((8*c^3*d^3*e^12 - 12*b*c^2*d^2*e
^13 + 6*b^2*c*d*e^14 - b^3*e^15)*(x*e + d)))*e^(-1)/(x*e + d) + (172*c^4*d^3*g*e^7*sgn(1/(x*e + d)) - 12*c^4*d
^2*f*e^8*sgn(1/(x*e + d)) - 252*b*c^3*d^2*g*e^8*sgn(1/(x*e + d)) + 12*b*c^3*d*f*e^9*sgn(1/(x*e + d)) + 123*b^2
*c^2*d*g*e^9*sgn(1/(x*e + d)) - 3*b^2*c^2*f*e^10*sgn(1/(x*e + d)) - 20*b^3*c*g*e^10*sgn(1/(x*e + d)))/(8*c^3*d
^3*e^12 - 12*b*c^2*d^2*e^13 + 6*b^2*c*d*e^14 - b^3*e^15)) - (43*sqrt(-c*e^2)*c^2*d*g - 3*sqrt(-c*e^2)*c^2*f*e
- 20*sqrt(-c*e^2)*b*c*g*e)*sgn(1/(x*e + d))/(2*c*d*e^5 - b*e^6))*e^2

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maple [B]  time = 0.07, size = 1023, normalized size = 4.78

result too large to display

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]

-2/5*(-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)-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))

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

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 >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(b*e-2*c*d>0)', see `assume?` f
or more details)Is b*e-2*c*d zero or nonzero?

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (f+g\,x\right )\,{\left (c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x\right )}^{3/2}}{{\left (d+e\,x\right )}^5} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((f + g*x)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(3/2))/(d + e*x)^5,x)

[Out]

int(((f + g*x)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(3/2))/(d + e*x)^5, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {3}{2}} \left (f + g x\right )}{\left (d + e x\right )^{5}}\, dx \end {gather*}

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]

Integral((-(d + e*x)*(b*e - c*d + c*e*x))**(3/2)*(f + g*x)/(d + e*x)**5, x)

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