3.21.19 \(\int \frac {(f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{3/2}}{(d+e x)^{13/2}} \, dx\)

Optimal. Leaf size=387 \[ -\frac {c^3 (-8 b e g+13 c d g+3 c e f) \tanh ^{-1}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {d+e x} \sqrt {2 c d-b e}}\right )}{64 e^2 (2 c d-b e)^{5/2}}-\frac {c^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-8 b e g+13 c d g+3 c e f)}{64 e^2 (d+e x)^{3/2} (2 c d-b e)^2}+\frac {c \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-8 b e g+13 c d g+3 c e f)}{32 e^2 (d+e x)^{5/2} (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{4 e^2 (d+e x)^{13/2} (2 c d-b e)}-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-8 b e g+13 c d g+3 c e f)}{24 e^2 (d+e x)^{9/2} (2 c d-b e)} \]

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Rubi [A]  time = 0.65, antiderivative size = 387, normalized size of antiderivative = 1.00, number of steps used = 6, 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} \begin {gather*} -\frac {c^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-8 b e g+13 c d g+3 c e f)}{64 e^2 (d+e x)^{3/2} (2 c d-b e)^2}-\frac {c^3 (-8 b e g+13 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 )}{64 e^2 (2 c d-b e)^{5/2}}+\frac {c \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-8 b e g+13 c d g+3 c e f)}{32 e^2 (d+e x)^{5/2} (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 )^{5/2}}{4 e^2 (d+e x)^{13/2} (2 c d-b e)}-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-8 b e g+13 c d g+3 c e f)}{24 e^2 (d+e x)^{9/2} (2 c d-b e)} \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)^(13/2),x]

[Out]

(c*(3*c*e*f + 13*c*d*g - 8*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(32*e^2*(2*c*d - b*e)*(d + e*x)^(
5/2)) - (c^2*(3*c*e*f + 13*c*d*g - 8*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)^2
*(d + e*x)^(3/2)) - ((3*c*e*f + 13*c*d*g - 8*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(24*e^2*(2*c*
d - b*e)*(d + e*x)^(9/2)) - ((e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(4*e^2*(2*c*d - b*e)*(d
+ e*x)^(13/2)) - (c^3*(3*c*e*f + 13*c*d*g - 8*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])])/(64*e^2*(2*c*d - b*e)^(5/2))

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]

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 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 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)^{13/2}} \, dx &=-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{4 e^2 (2 c d-b e) (d+e x)^{13/2}}+\frac {(3 c e f+13 c d g-8 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}{8 e (2 c d-b e)}\\ &=-\frac {(3 c e f+13 c d g-8 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{24 e^2 (2 c d-b e) (d+e x)^{9/2}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{4 e^2 (2 c d-b e) (d+e x)^{13/2}}-\frac {(c (3 c e f+13 c d g-8 b e g)) \int \frac {\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{7/2}} \, dx}{16 e (2 c d-b e)}\\ &=\frac {c (3 c e f+13 c d g-8 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{32 e^2 (2 c d-b e) (d+e x)^{5/2}}-\frac {(3 c e f+13 c d g-8 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{24 e^2 (2 c d-b e) (d+e x)^{9/2}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{4 e^2 (2 c d-b e) (d+e x)^{13/2}}+\frac {\left (c^2 (3 c e f+13 c d g-8 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}{64 e (2 c d-b e)}\\ &=\frac {c (3 c e f+13 c d g-8 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{32 e^2 (2 c d-b e) (d+e x)^{5/2}}-\frac {c^2 (3 c e f+13 c d g-8 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)^2 (d+e x)^{3/2}}-\frac {(3 c e f+13 c d g-8 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{24 e^2 (2 c d-b e) (d+e x)^{9/2}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{4 e^2 (2 c d-b e) (d+e x)^{13/2}}+\frac {\left (c^3 (3 c e f+13 c d g-8 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}{128 e (2 c d-b e)^2}\\ &=\frac {c (3 c e f+13 c d g-8 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{32 e^2 (2 c d-b e) (d+e x)^{5/2}}-\frac {c^2 (3 c e f+13 c d g-8 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)^2 (d+e x)^{3/2}}-\frac {(3 c e f+13 c d g-8 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{24 e^2 (2 c d-b e) (d+e x)^{9/2}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{4 e^2 (2 c d-b e) (d+e x)^{13/2}}+\frac {\left (c^3 (3 c e f+13 c d g-8 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 )}{64 (2 c d-b e)^2}\\ &=\frac {c (3 c e f+13 c d g-8 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{32 e^2 (2 c d-b e) (d+e x)^{5/2}}-\frac {c^2 (3 c e f+13 c d g-8 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)^2 (d+e x)^{3/2}}-\frac {(3 c e f+13 c d g-8 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{24 e^2 (2 c d-b e) (d+e x)^{9/2}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{4 e^2 (2 c d-b e) (d+e x)^{13/2}}-\frac {c^3 (3 c e f+13 c d g-8 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 )}{64 e^2 (2 c d-b e)^{5/2}}\\ \end {align*}

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Mathematica [C]  time = 0.21, size = 128, normalized size = 0.33 \begin {gather*} \frac {((d+e x) (c (d-e x)-b e))^{5/2} \left (-\frac {c^3 (d+e x)^4 (-8 b e g+13 c d g+3 c e f) \, _2F_1\left (\frac {5}{2},4;\frac {7}{2};\frac {-c d+b e+c e x}{b e-2 c d}\right )}{(b e-2 c d)^4}+5 d g-5 e f\right )}{20 e^2 (d+e x)^{13/2} (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)^(13/2),x]

[Out]

(((d + e*x)*(-(b*e) + c*(d - e*x)))^(5/2)*(-5*e*f + 5*d*g - (c^3*(3*c*e*f + 13*c*d*g - 8*b*e*g)*(d + e*x)^4*Hy
pergeometric2F1[5/2, 4, 7/2, (-(c*d) + b*e + c*e*x)/(-2*c*d + b*e)])/(-2*c*d + b*e)^4))/(20*e^2*(2*c*d - b*e)*
(d + e*x)^(13/2))

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IntegrateAlgebraic [A]  time = 6.87, size = 500, normalized size = 1.29 \begin {gather*} \frac {\sqrt {-b e (d+e x)-c (d+e x)^2+2 c d (d+e x)} \left (64 b^3 e^3 g (d+e x)-48 b^3 d e^3 g+48 b^3 e^4 f+288 b^2 c d^2 e^2 g+72 b^2 c e^3 f (d+e x)-288 b^2 c d e^3 f-456 b^2 c d e^2 g (d+e x)+112 b^2 c e^2 g (d+e x)^2-576 b c^2 d^3 e g+576 b c^2 d^2 e^2 f+1056 b c^2 d^2 e g (d+e x)-288 b c^2 d e^2 f (d+e x)+6 b c^2 e^2 f (d+e x)^2-454 b c^2 d e g (d+e x)^2+24 b c^2 e g (d+e x)^3+384 c^3 d^4 g-384 c^3 d^3 e f-800 c^3 d^3 g (d+e x)+288 c^3 d^2 e f (d+e x)+460 c^3 d^2 g (d+e x)^2-12 c^3 d e f (d+e x)^2-9 c^3 e f (d+e x)^3-39 c^3 d g (d+e x)^3\right )}{192 e^2 (d+e x)^{9/2} (b e-2 c d)^2}+\frac {\left (-8 b c^3 e g+13 c^4 d g+3 c^4 e f\right ) \tan ^{-1}\left (\frac {\sqrt {b e-2 c d} \sqrt {(d+e x) (2 c d-b e)-c (d+e x)^2}}{\sqrt {d+e x} (b e+c (d+e x)-2 c d)}\right )}{64 e^2 (2 c d-b e)^2 \sqrt {b e-2 c d}} \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)^(13/2),x]

[Out]

(Sqrt[2*c*d*(d + e*x) - b*e*(d + e*x) - c*(d + e*x)^2]*(-384*c^3*d^3*e*f + 576*b*c^2*d^2*e^2*f - 288*b^2*c*d*e
^3*f + 48*b^3*e^4*f + 384*c^3*d^4*g - 576*b*c^2*d^3*e*g + 288*b^2*c*d^2*e^2*g - 48*b^3*d*e^3*g + 288*c^3*d^2*e
*f*(d + e*x) - 288*b*c^2*d*e^2*f*(d + e*x) + 72*b^2*c*e^3*f*(d + e*x) - 800*c^3*d^3*g*(d + e*x) + 1056*b*c^2*d
^2*e*g*(d + e*x) - 456*b^2*c*d*e^2*g*(d + e*x) + 64*b^3*e^3*g*(d + e*x) - 12*c^3*d*e*f*(d + e*x)^2 + 6*b*c^2*e
^2*f*(d + e*x)^2 + 460*c^3*d^2*g*(d + e*x)^2 - 454*b*c^2*d*e*g*(d + e*x)^2 + 112*b^2*c*e^2*g*(d + e*x)^2 - 9*c
^3*e*f*(d + e*x)^3 - 39*c^3*d*g*(d + e*x)^3 + 24*b*c^2*e*g*(d + e*x)^3))/(192*e^2*(-2*c*d + b*e)^2*(d + e*x)^(
9/2)) + ((3*c^4*e*f + 13*c^4*d*g - 8*b*c^3*e*g)*ArcTan[(Sqrt[-2*c*d + b*e]*Sqrt[(2*c*d - b*e)*(d + e*x) - c*(d
+ e*x)^2])/(Sqrt[d + e*x]*(-2*c*d + b*e + c*(d + e*x)))])/(64*e^2*(2*c*d - b*e)^2*Sqrt[-2*c*d + b*e])

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fricas [B]  time = 0.50, size = 2104, normalized size = 5.44

result too large to display

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)^(13/2),x, algorithm="fricas")

[Out]

[-1/384*(3*(3*c^4*d^5*e*f + (3*c^4*e^6*f + (13*c^4*d*e^5 - 8*b*c^3*e^6)*g)*x^5 + 5*(3*c^4*d*e^5*f + (13*c^4*d^
2*e^4 - 8*b*c^3*d*e^5)*g)*x^4 + 10*(3*c^4*d^2*e^4*f + (13*c^4*d^3*e^3 - 8*b*c^3*d^2*e^4)*g)*x^3 + 10*(3*c^4*d^
3*e^3*f + (13*c^4*d^4*e^2 - 8*b*c^3*d^3*e^3)*g)*x^2 + (13*c^4*d^6 - 8*b*c^3*d^5*e)*g + 5*(3*c^4*d^4*e^2*f + (1
3*c^4*d^5*e - 8*b*c^3*d^4*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)*(3*(3*(2*c^4*d*e^4 - b*c^3*e^5)*f + (26*c^4*d^2*e^3 - 29*b*c^3*
d*e^4 + 8*b^2*c^2*e^5)*g)*x^3 + (3*(26*c^4*d^2*e^3 - 17*b*c^3*d*e^4 + 2*b^2*c^2*e^5)*f - (686*c^4*d^3*e^2 - 11
07*b*c^3*d^2*e^3 + 606*b^2*c^2*d*e^4 - 112*b^3*c*e^5)*g)*x^2 + 3*(78*c^4*d^4*e - 235*b*c^3*d^3*e^2 + 242*b^2*c
^2*d^2*e^3 - 104*b^3*c*d*e^4 + 16*b^4*e^5)*f - (10*c^4*d^5 + 95*b*c^3*d^4*e - 162*b^2*c^2*d^3*e^2 + 88*b^3*c*d
^2*e^3 - 16*b^4*d*e^4)*g - (3*(158*c^4*d^3*e^2 - 263*b*c^3*d^2*e^3 + 140*b^2*c^2*d*e^4 - 24*b^3*c*e^5)*f + (6*
c^4*d^4*e + 437*b*c^3*d^3*e^2 - 684*b^2*c^2*d^2*e^3 + 360*b^3*c*d*e^4 - 64*b^4*e^5)*g)*x)*sqrt(e*x + d))/(8*c^
3*d^8*e^2 - 12*b*c^2*d^7*e^3 + 6*b^2*c*d^6*e^4 - b^3*d^5*e^5 + (8*c^3*d^3*e^7 - 12*b*c^2*d^2*e^8 + 6*b^2*c*d*e
^9 - b^3*e^10)*x^5 + 5*(8*c^3*d^4*e^6 - 12*b*c^2*d^3*e^7 + 6*b^2*c*d^2*e^8 - b^3*d*e^9)*x^4 + 10*(8*c^3*d^5*e^
5 - 12*b*c^2*d^4*e^6 + 6*b^2*c*d^3*e^7 - b^3*d^2*e^8)*x^3 + 10*(8*c^3*d^6*e^4 - 12*b*c^2*d^5*e^5 + 6*b^2*c*d^4
*e^6 - b^3*d^3*e^7)*x^2 + 5*(8*c^3*d^7*e^3 - 12*b*c^2*d^6*e^4 + 6*b^2*c*d^5*e^5 - b^3*d^4*e^6)*x), -1/192*(3*(
3*c^4*d^5*e*f + (3*c^4*e^6*f + (13*c^4*d*e^5 - 8*b*c^3*e^6)*g)*x^5 + 5*(3*c^4*d*e^5*f + (13*c^4*d^2*e^4 - 8*b*
c^3*d*e^5)*g)*x^4 + 10*(3*c^4*d^2*e^4*f + (13*c^4*d^3*e^3 - 8*b*c^3*d^2*e^4)*g)*x^3 + 10*(3*c^4*d^3*e^3*f + (1
3*c^4*d^4*e^2 - 8*b*c^3*d^3*e^3)*g)*x^2 + (13*c^4*d^6 - 8*b*c^3*d^5*e)*g + 5*(3*c^4*d^4*e^2*f + (13*c^4*d^5*e
- 8*b*c^3*d^4*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)*(3*(3*(2*
c^4*d*e^4 - b*c^3*e^5)*f + (26*c^4*d^2*e^3 - 29*b*c^3*d*e^4 + 8*b^2*c^2*e^5)*g)*x^3 + (3*(26*c^4*d^2*e^3 - 17*
b*c^3*d*e^4 + 2*b^2*c^2*e^5)*f - (686*c^4*d^3*e^2 - 1107*b*c^3*d^2*e^3 + 606*b^2*c^2*d*e^4 - 112*b^3*c*e^5)*g)
*x^2 + 3*(78*c^4*d^4*e - 235*b*c^3*d^3*e^2 + 242*b^2*c^2*d^2*e^3 - 104*b^3*c*d*e^4 + 16*b^4*e^5)*f - (10*c^4*d
^5 + 95*b*c^3*d^4*e - 162*b^2*c^2*d^3*e^2 + 88*b^3*c*d^2*e^3 - 16*b^4*d*e^4)*g - (3*(158*c^4*d^3*e^2 - 263*b*c
^3*d^2*e^3 + 140*b^2*c^2*d*e^4 - 24*b^3*c*e^5)*f + (6*c^4*d^4*e + 437*b*c^3*d^3*e^2 - 684*b^2*c^2*d^2*e^3 + 36
0*b^3*c*d*e^4 - 64*b^4*e^5)*g)*x)*sqrt(e*x + d))/(8*c^3*d^8*e^2 - 12*b*c^2*d^7*e^3 + 6*b^2*c*d^6*e^4 - b^3*d^5
*e^5 + (8*c^3*d^3*e^7 - 12*b*c^2*d^2*e^8 + 6*b^2*c*d*e^9 - b^3*e^10)*x^5 + 5*(8*c^3*d^4*e^6 - 12*b*c^2*d^3*e^7
+ 6*b^2*c*d^2*e^8 - b^3*d*e^9)*x^4 + 10*(8*c^3*d^5*e^5 - 12*b*c^2*d^4*e^6 + 6*b^2*c*d^3*e^7 - b^3*d^2*e^8)*x^
3 + 10*(8*c^3*d^6*e^4 - 12*b*c^2*d^5*e^5 + 6*b^2*c*d^4*e^6 - b^3*d^3*e^7)*x^2 + 5*(8*c^3*d^7*e^3 - 12*b*c^2*d^
6*e^4 + 6*b^2*c*d^5*e^5 - b^3*d^4*e^6)*x)]

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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)^(13/2),x, algorithm="giac")

[Out]

Timed out

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maple [B]  time = 0.08, size = 1517, normalized size = 3.92

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)^(13/2),x)

[Out]

-1/192*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*(-39*c^4*d^5*g*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))+
9*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)*c^3*e^4*f*x^3+382*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)*b*c^2*d*
e^3*g*x^2-48*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)*b^3*e^4*f+232*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)*b
^2*c*d*e^3*g*x-220*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)*b*c^2*d^2*e^2*g*x+276*(-c*e*x-b*e+c*d)^(1/2)*(b*e-
2*c*d)^(1/2)*b*c^2*d*e^3*f*x-5*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)*c^3*d^4*g-9*c^4*e^5*f*x^4*arctan((-c*e
*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))-9*c^4*d^4*e*f*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))-16*(-c*e*x
-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)*b^3*d*e^3*g+117*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)*c^3*d^3*e*f+24*b*c^
3*e^5*g*x^4*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))-39*c^4*d*e^4*g*x^4*arctan((-c*e*x-b*e+c*d)^(1/2)/
(b*e-2*c*d)^(1/2))-156*c^4*d^2*e^3*g*x^3*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))-36*c^4*d*e^4*f*x^3*a
rctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))-234*c^4*d^3*e^2*g*x^2*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d
)^(1/2))-54*c^4*d^2*e^3*f*x^2*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))-156*c^4*d^4*e*g*x*arctan((-c*e*
x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))-36*c^4*d^3*e^2*f*x*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))+24*b*c
^3*d^4*e*g*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))-64*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)*b^3*e^
4*g*x+96*b*c^3*d^3*e^2*g*x*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))+39*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c
*d)^(1/2)*c^3*d*e^3*f*x^2-72*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)*b^2*c*e^4*f*x-3*(-c*e*x-b*e+c*d)^(1/2)*(
b*e-2*c*d)^(1/2)*c^3*d^3*e*g*x-237*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)*c^3*d^2*e^2*f*x+56*(-c*e*x-b*e+c*d
)^(1/2)*(b*e-2*c*d)^(1/2)*b^2*c*d^2*e^2*g+216*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)*b^2*c*d*e^3*f-50*(-c*e*
x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)*b*c^2*d^3*e*g-294*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)*b*c^2*d^2*e^2*f-
24*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)*b*c^2*e^4*g*x^3+39*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)*c^3*d*
e^3*g*x^3-112*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)*b^2*c*e^4*g*x^2-6*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1
/2)*b*c^2*e^4*f*x^2-343*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)*c^3*d^2*e^2*g*x^2+96*b*c^3*d*e^4*g*x^3*arctan
((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))+144*b*c^3*d^2*e^3*g*x^2*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(
1/2)))/(e*x+d)^(9/2)/(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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e\right )}^{\frac {3}{2}} {\left (g x + f\right )}}{{\left (e x + d\right )}^{\frac {13}{2}}}\,{d x} \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)^(13/2),x, algorithm="maxima")

[Out]

integrate((-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(3/2)*(g*x + f)/(e*x + d)^(13/2), x)

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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 )}^{13/2}} \,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)^(13/2),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)^(13/2), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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)**(13/2),x)

[Out]

Timed out

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