3.1.96 \(\int \frac {(f+g x) (a+b \text {ArcSin}(c x))}{(d+e x)^6} \, dx\) [96]

Optimal. Leaf size=457 \[ \frac {b c (e f-d g) \sqrt {1-c^2 x^2}}{20 e \left (c^2 d^2-e^2\right ) (d+e x)^4}-\frac {b c \left (5 e^2 g-c^2 d (7 e f-2 d g)\right ) \sqrt {1-c^2 x^2}}{60 e \left (c^2 d^2-e^2\right )^2 (d+e x)^3}+\frac {b c^3 \left (e^2 (9 e f-34 d g)+c^2 d^2 (26 e f-d g)\right ) \sqrt {1-c^2 x^2}}{120 e \left (c^2 d^2-e^2\right )^3 (d+e x)^2}-\frac {b c^3 \left (4 e^4 g-c^2 d e^2 (11 e f-18 d g)-c^4 d^3 (10 e f+d g)\right ) \sqrt {1-c^2 x^2}}{24 e \left (c^2 d^2-e^2\right )^4 (d+e x)}-\frac {(e f-d g) (a+b \text {ArcSin}(c x))}{5 e^2 (d+e x)^5}-\frac {g (a+b \text {ArcSin}(c x))}{4 e^2 (d+e x)^4}+\frac {b c^5 \left (c^2 d^2 e^2 (24 e f-19 d g)+3 e^4 (e f-6 d g)+2 c^4 d^4 (4 e f+d g)\right ) \text {ArcTan}\left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{40 e^2 \left (c^2 d^2-e^2\right )^{9/2}} \]

[Out]

-1/5*(-d*g+e*f)*(a+b*arcsin(c*x))/e^2/(e*x+d)^5-1/4*g*(a+b*arcsin(c*x))/e^2/(e*x+d)^4+1/40*b*c^5*(c^2*d^2*e^2*
(-19*d*g+24*e*f)+3*e^4*(-6*d*g+e*f)+2*c^4*d^4*(d*g+4*e*f))*arctan((c^2*d*x+e)/(c^2*d^2-e^2)^(1/2)/(-c^2*x^2+1)
^(1/2))/e^2/(c^2*d^2-e^2)^(9/2)+1/20*b*c*(-d*g+e*f)*(-c^2*x^2+1)^(1/2)/e/(c^2*d^2-e^2)/(e*x+d)^4-1/60*b*c*(5*e
^2*g-c^2*d*(-2*d*g+7*e*f))*(-c^2*x^2+1)^(1/2)/e/(c^2*d^2-e^2)^2/(e*x+d)^3+1/120*b*c^3*(e^2*(-34*d*g+9*e*f)+c^2
*d^2*(-d*g+26*e*f))*(-c^2*x^2+1)^(1/2)/e/(c^2*d^2-e^2)^3/(e*x+d)^2-1/24*b*c^3*(4*e^4*g-c^2*d*e^2*(-18*d*g+11*e
*f)-c^4*d^3*(d*g+10*e*f))*(-c^2*x^2+1)^(1/2)/e/(c^2*d^2-e^2)^4/(e*x+d)

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Rubi [A]
time = 0.68, antiderivative size = 457, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {45, 4837, 12, 849, 821, 739, 210} \begin {gather*} -\frac {(e f-d g) (a+b \text {ArcSin}(c x))}{5 e^2 (d+e x)^5}-\frac {g (a+b \text {ArcSin}(c x))}{4 e^2 (d+e x)^4}+\frac {b c^5 \text {ArcTan}\left (\frac {c^2 d x+e}{\sqrt {1-c^2 x^2} \sqrt {c^2 d^2-e^2}}\right ) \left (2 c^4 d^4 (d g+4 e f)+c^2 d^2 e^2 (24 e f-19 d g)+3 e^4 (e f-6 d g)\right )}{40 e^2 \left (c^2 d^2-e^2\right )^{9/2}}-\frac {b c \sqrt {1-c^2 x^2} \left (5 e^2 g-c^2 d (7 e f-2 d g)\right )}{60 e \left (c^2 d^2-e^2\right )^2 (d+e x)^3}+\frac {b c \sqrt {1-c^2 x^2} (e f-d g)}{20 e \left (c^2 d^2-e^2\right ) (d+e x)^4}+\frac {b c^3 \sqrt {1-c^2 x^2} \left (c^2 d^2 (26 e f-d g)+e^2 (9 e f-34 d g)\right )}{120 e \left (c^2 d^2-e^2\right )^3 (d+e x)^2}-\frac {b c^3 \sqrt {1-c^2 x^2} \left (c^4 \left (-d^3\right ) (d g+10 e f)-c^2 d e^2 (11 e f-18 d g)+4 e^4 g\right )}{24 e \left (c^2 d^2-e^2\right )^4 (d+e x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((f + g*x)*(a + b*ArcSin[c*x]))/(d + e*x)^6,x]

[Out]

(b*c*(e*f - d*g)*Sqrt[1 - c^2*x^2])/(20*e*(c^2*d^2 - e^2)*(d + e*x)^4) - (b*c*(5*e^2*g - c^2*d*(7*e*f - 2*d*g)
)*Sqrt[1 - c^2*x^2])/(60*e*(c^2*d^2 - e^2)^2*(d + e*x)^3) + (b*c^3*(e^2*(9*e*f - 34*d*g) + c^2*d^2*(26*e*f - d
*g))*Sqrt[1 - c^2*x^2])/(120*e*(c^2*d^2 - e^2)^3*(d + e*x)^2) - (b*c^3*(4*e^4*g - c^2*d*e^2*(11*e*f - 18*d*g)
- c^4*d^3*(10*e*f + d*g))*Sqrt[1 - c^2*x^2])/(24*e*(c^2*d^2 - e^2)^4*(d + e*x)) - ((e*f - d*g)*(a + b*ArcSin[c
*x]))/(5*e^2*(d + e*x)^5) - (g*(a + b*ArcSin[c*x]))/(4*e^2*(d + e*x)^4) + (b*c^5*(c^2*d^2*e^2*(24*e*f - 19*d*g
) + 3*e^4*(e*f - 6*d*g) + 2*c^4*d^4*(4*e*f + d*g))*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2]
)])/(40*e^2*(c^2*d^2 - e^2)^(9/2))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 739

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 821

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g
))*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 + a*e^2))), x] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e
^2), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0
] && EqQ[Simplify[m + 2*p + 3], 0]

Rule 849

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*f - d*g)*
(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/((m + 1)*(c*d^2 + a*e^2))), x] + Dist[1/((m + 1)*(c*d^2 + a*e^2)), Int[
(d + e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; Fr
eeQ[{a, c, d, e, f, g, p}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || Integer
sQ[2*m, 2*p])

Rule 4837

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*(Px_)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> With[{u = IntHide[Px*(d
+ e*x)^m, x]}, Dist[a + b*ArcSin[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]
] /; FreeQ[{a, b, c, d, e, m}, x] && PolynomialQ[Px, x]

Rubi steps

\begin {align*} \int \frac {(f+g x) \left (a+b \sin ^{-1}(c x)\right )}{(d+e x)^6} \, dx &=-\frac {(e f-d g) \left (a+b \sin ^{-1}(c x)\right )}{5 e^2 (d+e x)^5}-\frac {g \left (a+b \sin ^{-1}(c x)\right )}{4 e^2 (d+e x)^4}-(b c) \int \frac {-4 e f-d g-5 e g x}{20 e^2 (d+e x)^5 \sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {(e f-d g) \left (a+b \sin ^{-1}(c x)\right )}{5 e^2 (d+e x)^5}-\frac {g \left (a+b \sin ^{-1}(c x)\right )}{4 e^2 (d+e x)^4}-\frac {(b c) \int \frac {-4 e f-d g-5 e g x}{(d+e x)^5 \sqrt {1-c^2 x^2}} \, dx}{20 e^2}\\ &=\frac {b c (e f-d g) \sqrt {1-c^2 x^2}}{20 e \left (c^2 d^2-e^2\right ) (d+e x)^4}-\frac {(e f-d g) \left (a+b \sin ^{-1}(c x)\right )}{5 e^2 (d+e x)^5}-\frac {g \left (a+b \sin ^{-1}(c x)\right )}{4 e^2 (d+e x)^4}-\frac {(b c) \int \frac {4 \left (5 e^2 g-c^2 d (4 e f+d g)\right )+12 c^2 e (e f-d g) x}{(d+e x)^4 \sqrt {1-c^2 x^2}} \, dx}{80 e^2 \left (c^2 d^2-e^2\right )}\\ &=\frac {b c (e f-d g) \sqrt {1-c^2 x^2}}{20 e \left (c^2 d^2-e^2\right ) (d+e x)^4}-\frac {b c \left (5 e^2 g-c^2 d (7 e f-2 d g)\right ) \sqrt {1-c^2 x^2}}{60 e \left (c^2 d^2-e^2\right )^2 (d+e x)^3}-\frac {(e f-d g) \left (a+b \sin ^{-1}(c x)\right )}{5 e^2 (d+e x)^5}-\frac {g \left (a+b \sin ^{-1}(c x)\right )}{4 e^2 (d+e x)^4}-\frac {(b c) \int \frac {-12 c^2 \left (e^2 (3 e f-8 d g)+c^2 d^2 (4 e f+d g)\right )-8 c^2 e \left (5 e^2 g-c^2 d (7 e f-2 d g)\right ) x}{(d+e x)^3 \sqrt {1-c^2 x^2}} \, dx}{240 e^2 \left (c^2 d^2-e^2\right )^2}\\ &=\frac {b c (e f-d g) \sqrt {1-c^2 x^2}}{20 e \left (c^2 d^2-e^2\right ) (d+e x)^4}-\frac {b c \left (5 e^2 g-c^2 d (7 e f-2 d g)\right ) \sqrt {1-c^2 x^2}}{60 e \left (c^2 d^2-e^2\right )^2 (d+e x)^3}+\frac {b c^3 \left (e^2 (9 e f-34 d g)+c^2 d^2 (26 e f-d g)\right ) \sqrt {1-c^2 x^2}}{120 e \left (c^2 d^2-e^2\right )^3 (d+e x)^2}-\frac {(e f-d g) \left (a+b \sin ^{-1}(c x)\right )}{5 e^2 (d+e x)^5}-\frac {g \left (a+b \sin ^{-1}(c x)\right )}{4 e^2 (d+e x)^4}-\frac {(b c) \int \frac {8 c^2 \left (10 e^4 g-c^2 d e^2 (23 e f-28 d g)-3 c^4 d^3 (4 e f+d g)\right )+4 c^4 e \left (e^2 (9 e f-34 d g)+c^2 d^2 (26 e f-d g)\right ) x}{(d+e x)^2 \sqrt {1-c^2 x^2}} \, dx}{480 e^2 \left (c^2 d^2-e^2\right )^3}\\ &=\frac {b c (e f-d g) \sqrt {1-c^2 x^2}}{20 e \left (c^2 d^2-e^2\right ) (d+e x)^4}-\frac {b c \left (5 e^2 g-c^2 d (7 e f-2 d g)\right ) \sqrt {1-c^2 x^2}}{60 e \left (c^2 d^2-e^2\right )^2 (d+e x)^3}+\frac {b c^3 \left (e^2 (9 e f-34 d g)+c^2 d^2 (26 e f-d g)\right ) \sqrt {1-c^2 x^2}}{120 e \left (c^2 d^2-e^2\right )^3 (d+e x)^2}-\frac {b c^3 \left (4 e^4 g-c^2 d e^2 (11 e f-18 d g)-c^4 d^3 (10 e f+d g)\right ) \sqrt {1-c^2 x^2}}{24 e \left (c^2 d^2-e^2\right )^4 (d+e x)}-\frac {(e f-d g) \left (a+b \sin ^{-1}(c x)\right )}{5 e^2 (d+e x)^5}-\frac {g \left (a+b \sin ^{-1}(c x)\right )}{4 e^2 (d+e x)^4}+\frac {\left (b c^5 \left (c^2 d^2 e^2 (24 e f-19 d g)+3 e^4 (e f-6 d g)+2 c^4 d^4 (4 e f+d g)\right )\right ) \int \frac {1}{(d+e x) \sqrt {1-c^2 x^2}} \, dx}{40 e^2 \left (c^2 d^2-e^2\right )^4}\\ &=\frac {b c (e f-d g) \sqrt {1-c^2 x^2}}{20 e \left (c^2 d^2-e^2\right ) (d+e x)^4}-\frac {b c \left (5 e^2 g-c^2 d (7 e f-2 d g)\right ) \sqrt {1-c^2 x^2}}{60 e \left (c^2 d^2-e^2\right )^2 (d+e x)^3}+\frac {b c^3 \left (e^2 (9 e f-34 d g)+c^2 d^2 (26 e f-d g)\right ) \sqrt {1-c^2 x^2}}{120 e \left (c^2 d^2-e^2\right )^3 (d+e x)^2}-\frac {b c^3 \left (4 e^4 g-c^2 d e^2 (11 e f-18 d g)-c^4 d^3 (10 e f+d g)\right ) \sqrt {1-c^2 x^2}}{24 e \left (c^2 d^2-e^2\right )^4 (d+e x)}-\frac {(e f-d g) \left (a+b \sin ^{-1}(c x)\right )}{5 e^2 (d+e x)^5}-\frac {g \left (a+b \sin ^{-1}(c x)\right )}{4 e^2 (d+e x)^4}-\frac {\left (b c^5 \left (c^2 d^2 e^2 (24 e f-19 d g)+3 e^4 (e f-6 d g)+2 c^4 d^4 (4 e f+d g)\right )\right ) \text {Subst}\left (\int \frac {1}{-c^2 d^2+e^2-x^2} \, dx,x,\frac {e+c^2 d x}{\sqrt {1-c^2 x^2}}\right )}{40 e^2 \left (c^2 d^2-e^2\right )^4}\\ &=\frac {b c (e f-d g) \sqrt {1-c^2 x^2}}{20 e \left (c^2 d^2-e^2\right ) (d+e x)^4}-\frac {b c \left (5 e^2 g-c^2 d (7 e f-2 d g)\right ) \sqrt {1-c^2 x^2}}{60 e \left (c^2 d^2-e^2\right )^2 (d+e x)^3}+\frac {b c^3 \left (e^2 (9 e f-34 d g)+c^2 d^2 (26 e f-d g)\right ) \sqrt {1-c^2 x^2}}{120 e \left (c^2 d^2-e^2\right )^3 (d+e x)^2}-\frac {b c^3 \left (4 e^4 g-c^2 d e^2 (11 e f-18 d g)-c^4 d^3 (10 e f+d g)\right ) \sqrt {1-c^2 x^2}}{24 e \left (c^2 d^2-e^2\right )^4 (d+e x)}-\frac {(e f-d g) \left (a+b \sin ^{-1}(c x)\right )}{5 e^2 (d+e x)^5}-\frac {g \left (a+b \sin ^{-1}(c x)\right )}{4 e^2 (d+e x)^4}+\frac {b c^5 \left (c^2 d^2 e^2 (24 e f-19 d g)+3 e^4 (e f-6 d g)+2 c^4 d^4 (4 e f+d g)\right ) \tan ^{-1}\left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{40 e^2 \left (c^2 d^2-e^2\right )^{9/2}}\\ \end {align*}

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Mathematica [A]
time = 0.88, size = 494, normalized size = 1.08 \begin {gather*} \frac {\frac {3 a (-8 e f+8 d g)}{(d+e x)^5}-\frac {30 a g}{(d+e x)^4}+\frac {b c e \sqrt {1-c^2 x^2} \left (-6 \left (-c^2 d^2+e^2\right )^3 (e f-d g)-2 \left (-c^2 d^2+e^2\right )^2 \left (5 e^2 g+c^2 d (-7 e f+2 d g)\right ) (d+e x)-c^2 \left (c^2 d^2-e^2\right ) \left (c^2 d^2 (-26 e f+d g)+e^2 (-9 e f+34 d g)\right ) (d+e x)^2+5 c^2 \left (-4 e^4 g+c^2 d e^2 (11 e f-18 d g)+c^4 d^3 (10 e f+d g)\right ) (d+e x)^3\right )}{\left (-c^2 d^2+e^2\right )^4 (d+e x)^4}-\frac {6 b (4 e f+d g+5 e g x) \text {ArcSin}(c x)}{(d+e x)^5}+\frac {3 b c^5 \left (c^2 d^2 e^2 (24 e f-19 d g)+3 e^4 (e f-6 d g)+2 c^4 d^4 (4 e f+d g)\right ) \log (d+e x)}{(-c d+e)^4 (c d+e)^4 \sqrt {-c^2 d^2+e^2}}-\frac {3 b c^5 \left (c^2 d^2 e^2 (24 e f-19 d g)+3 e^4 (e f-6 d g)+2 c^4 d^4 (4 e f+d g)\right ) \log \left (e+c^2 d x+\sqrt {-c^2 d^2+e^2} \sqrt {1-c^2 x^2}\right )}{(-c d+e)^4 (c d+e)^4 \sqrt {-c^2 d^2+e^2}}}{120 e^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((f + g*x)*(a + b*ArcSin[c*x]))/(d + e*x)^6,x]

[Out]

((3*a*(-8*e*f + 8*d*g))/(d + e*x)^5 - (30*a*g)/(d + e*x)^4 + (b*c*e*Sqrt[1 - c^2*x^2]*(-6*(-(c^2*d^2) + e^2)^3
*(e*f - d*g) - 2*(-(c^2*d^2) + e^2)^2*(5*e^2*g + c^2*d*(-7*e*f + 2*d*g))*(d + e*x) - c^2*(c^2*d^2 - e^2)*(c^2*
d^2*(-26*e*f + d*g) + e^2*(-9*e*f + 34*d*g))*(d + e*x)^2 + 5*c^2*(-4*e^4*g + c^2*d*e^2*(11*e*f - 18*d*g) + c^4
*d^3*(10*e*f + d*g))*(d + e*x)^3))/((-(c^2*d^2) + e^2)^4*(d + e*x)^4) - (6*b*(4*e*f + d*g + 5*e*g*x)*ArcSin[c*
x])/(d + e*x)^5 + (3*b*c^5*(c^2*d^2*e^2*(24*e*f - 19*d*g) + 3*e^4*(e*f - 6*d*g) + 2*c^4*d^4*(4*e*f + d*g))*Log
[d + e*x])/((-(c*d) + e)^4*(c*d + e)^4*Sqrt[-(c^2*d^2) + e^2]) - (3*b*c^5*(c^2*d^2*e^2*(24*e*f - 19*d*g) + 3*e
^4*(e*f - 6*d*g) + 2*c^4*d^4*(4*e*f + d*g))*Log[e + c^2*d*x + Sqrt[-(c^2*d^2) + e^2]*Sqrt[1 - c^2*x^2]])/((-(c
*d) + e)^4*(c*d + e)^4*Sqrt[-(c^2*d^2) + e^2]))/(120*e^2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2419\) vs. \(2(429)=858\).
time = 0.11, size = 2420, normalized size = 5.30

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(2420\)
default	\(\text {Expression too large to display}\)	\(2420\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x+f)*(a+b*arcsin(c*x))/(e*x+d)^6,x,method=_RETURNVERBOSE)

[Out]

1/c*(a*c^5*(-1/4*g/e^2/(c*e*x+c*d)^4+1/5*c*(d*g-e*f)/e^2/(c*e*x+c*d)^5)-1/4*b*c^5*arcsin(c*x)*g/e^2/(c*e*x+c*d
)^4+1/5*b*c^6*arcsin(c*x)/e^2/(c*e*x+c*d)^5*d*g-1/5*b*c^6*arcsin(c*x)/e/(c*e*x+c*d)^5*f+1/12*b*c^5/e^4*g/(c^2*
d^2-e^2)/(c*x+d*c/e)^3*(-(c*x+d*c/e)^2+2*d*c/e*(c*x+d*c/e)-(c^2*d^2-e^2)/e^2)^(1/2)+17/60*b*c^6/e^3*g*d/(c^2*d
^2-e^2)^2/(c*x+d*c/e)^2*(-(c*x+d*c/e)^2+2*d*c/e*(c*x+d*c/e)-(c^2*d^2-e^2)/e^2)^(1/2)+13/12*b*c^7/e^2*g*d^2/(c^
2*d^2-e^2)^3/(c*x+d*c/e)*(-(c*x+d*c/e)^2+2*d*c/e*(c*x+d*c/e)-(c^2*d^2-e^2)/e^2)^(1/2)-11/8*b*c^8/e^3*g*d^3/(c^
2*d^2-e^2)^3/(-(c^2*d^2-e^2)/e^2)^(1/2)*ln((-2*(c^2*d^2-e^2)/e^2+2*d*c/e*(c*x+d*c/e)+2*(-(c^2*d^2-e^2)/e^2)^(1
/2)*(-(c*x+d*c/e)^2+2*d*c/e*(c*x+d*c/e)-(c^2*d^2-e^2)/e^2)^(1/2))/(c*x+d*c/e))+9/20*b*c^6/e^3*g*d/(c^2*d^2-e^2
)^2/(-(c^2*d^2-e^2)/e^2)^(1/2)*ln((-2*(c^2*d^2-e^2)/e^2+2*d*c/e*(c*x+d*c/e)+2*(-(c^2*d^2-e^2)/e^2)^(1/2)*(-(c*
x+d*c/e)^2+2*d*c/e*(c*x+d*c/e)-(c^2*d^2-e^2)/e^2)^(1/2))/(c*x+d*c/e))-1/6*b*c^5/e^2*g/(c^2*d^2-e^2)^2/(c*x+d*c
/e)*(-(c*x+d*c/e)^2+2*d*c/e*(c*x+d*c/e)-(c^2*d^2-e^2)/e^2)^(1/2)-1/20*b*c^6/e^5/(c^2*d^2-e^2)/(c*x+d*c/e)^4*(-
(c*x+d*c/e)^2+2*d*c/e*(c*x+d*c/e)-(c^2*d^2-e^2)/e^2)^(1/2)*d*g+1/20*b*c^6/e^4/(c^2*d^2-e^2)/(c*x+d*c/e)^4*(-(c
*x+d*c/e)^2+2*d*c/e*(c*x+d*c/e)-(c^2*d^2-e^2)/e^2)^(1/2)*f-7/60*b*c^7/e^4*d^2/(c^2*d^2-e^2)^2/(c*x+d*c/e)^3*(-
(c*x+d*c/e)^2+2*d*c/e*(c*x+d*c/e)-(c^2*d^2-e^2)/e^2)^(1/2)*g+7/60*b*c^7/e^3*d/(c^2*d^2-e^2)^2/(c*x+d*c/e)^3*(-
(c*x+d*c/e)^2+2*d*c/e*(c*x+d*c/e)-(c^2*d^2-e^2)/e^2)^(1/2)*f-7/24*b*c^8/e^3*d^3/(c^2*d^2-e^2)^3/(c*x+d*c/e)^2*
(-(c*x+d*c/e)^2+2*d*c/e*(c*x+d*c/e)-(c^2*d^2-e^2)/e^2)^(1/2)*g+7/24*b*c^8/e^2*d^2/(c^2*d^2-e^2)^3/(c*x+d*c/e)^
2*(-(c*x+d*c/e)^2+2*d*c/e*(c*x+d*c/e)-(c^2*d^2-e^2)/e^2)^(1/2)*f-7/8*b*c^9/e^2*d^4/(c^2*d^2-e^2)^4/(c*x+d*c/e)
*(-(c*x+d*c/e)^2+2*d*c/e*(c*x+d*c/e)-(c^2*d^2-e^2)/e^2)^(1/2)*g+7/8*b*c^9/e*d^3/(c^2*d^2-e^2)^4/(c*x+d*c/e)*(-
(c*x+d*c/e)^2+2*d*c/e*(c*x+d*c/e)-(c^2*d^2-e^2)/e^2)^(1/2)*f+7/8*b*c^10/e^3*d^5/(c^2*d^2-e^2)^4/(-(c^2*d^2-e^2
)/e^2)^(1/2)*ln((-2*(c^2*d^2-e^2)/e^2+2*d*c/e*(c*x+d*c/e)+2*(-(c^2*d^2-e^2)/e^2)^(1/2)*(-(c*x+d*c/e)^2+2*d*c/e
*(c*x+d*c/e)-(c^2*d^2-e^2)/e^2)^(1/2))/(c*x+d*c/e))*g-7/8*b*c^10/e^2*d^4/(c^2*d^2-e^2)^4/(-(c^2*d^2-e^2)/e^2)^
(1/2)*ln((-2*(c^2*d^2-e^2)/e^2+2*d*c/e*(c*x+d*c/e)+2*(-(c^2*d^2-e^2)/e^2)^(1/2)*(-(c*x+d*c/e)^2+2*d*c/e*(c*x+d
*c/e)-(c^2*d^2-e^2)/e^2)^(1/2))/(c*x+d*c/e))*f+3/4*b*c^8/e^2*d^2/(c^2*d^2-e^2)^3/(-(c^2*d^2-e^2)/e^2)^(1/2)*ln
((-2*(c^2*d^2-e^2)/e^2+2*d*c/e*(c*x+d*c/e)+2*(-(c^2*d^2-e^2)/e^2)^(1/2)*(-(c*x+d*c/e)^2+2*d*c/e*(c*x+d*c/e)-(c
^2*d^2-e^2)/e^2)^(1/2))/(c*x+d*c/e))*f-11/24*b*c^7/e*d/(c^2*d^2-e^2)^3/(c*x+d*c/e)*(-(c*x+d*c/e)^2+2*d*c/e*(c*
x+d*c/e)-(c^2*d^2-e^2)/e^2)^(1/2)*f-3/40*b*c^6/e^2/(c^2*d^2-e^2)^2/(c*x+d*c/e)^2*(-(c*x+d*c/e)^2+2*d*c/e*(c*x+
d*c/e)-(c^2*d^2-e^2)/e^2)^(1/2)*f-3/40*b*c^6/e^2/(c^2*d^2-e^2)^2/(-(c^2*d^2-e^2)/e^2)^(1/2)*ln((-2*(c^2*d^2-e^
2)/e^2+2*d*c/e*(c*x+d*c/e)+2*(-(c^2*d^2-e^2)/e^2)^(1/2)*(-(c*x+d*c/e)^2+2*d*c/e*(c*x+d*c/e)-(c^2*d^2-e^2)/e^2)
^(1/2))/(c*x+d*c/e))*f)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)*(a+b*arcsin(c*x))/(e*x+d)^6,x, algorithm="maxima")

[Out]

-1/20*(5*x*e + d)*a*g/(x^5*e^7 + 5*d*x^4*e^6 + 10*d^2*x^3*e^5 + 10*d^3*x^2*e^4 + 5*d^4*x*e^3 + d^5*e^2) - 1/5*
a*f/(x^5*e^6 + 5*d*x^4*e^5 + 10*d^2*x^3*e^4 + 10*d^3*x^2*e^3 + 5*d^4*x*e^2 + d^5*e) - 1/20*((5*b*g*x*e + b*d*g
+ 4*b*f*e)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + 20*(x^5*e^7 + 5*d*x^4*e^6 + 10*d^2*x^3*e^5 + 10*d^3*x
^2*e^4 + 5*d^4*x*e^3 + d^5*e^2)*integrate(1/20*(5*b*c*g*x*e + b*c*d*g + 4*b*c*f*e)*e^(1/2*log(c*x + 1) + 1/2*l
og(-c*x + 1))/(c^4*x^9*e^7 + 5*c^4*d*x^8*e^6 - 5*c^2*d^4*x^3*e^3 - c^2*d^5*x^2*e^2 + (10*c^4*d^2*e^5 - c^2*e^7
)*x^7 + 5*(2*c^4*d^3*e^4 - c^2*d*e^6)*x^6 + 5*(c^4*d^4*e^3 - 2*c^2*d^2*e^5)*x^5 + (c^4*d^5*e^2 - 10*c^2*d^3*e^
4)*x^4 + (c^2*x^7*e^7 + 5*c^2*d*x^6*e^6 + (10*c^2*d^2*e^5 - e^7)*x^5 - 5*d^4*x*e^3 - d^5*e^2 + 5*(2*c^2*d^3*e^
4 - d*e^6)*x^4 + 5*(c^2*d^4*e^3 - 2*d^2*e^5)*x^3 + (c^2*d^5*e^2 - 10*d^3*e^4)*x^2)*e^(log(c*x + 1) + log(-c*x
+ 1))), x))/(x^5*e^7 + 5*d*x^4*e^6 + 10*d^2*x^3*e^5 + 10*d^3*x^2*e^4 + 5*d^4*x*e^3 + d^5*e^2)

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Fricas [F(-1)] Timed out
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)*(a+b*arcsin(c*x))/(e*x+d)^6,x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right ) \left (f + g x\right )}{\left (d + e x\right )^{6}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)*(a+b*asin(c*x))/(e*x+d)**6,x)

[Out]

Integral((a + b*asin(c*x))*(f + g*x)/(d + e*x)**6, x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)*(a+b*arcsin(c*x))/(e*x+d)^6,x, algorithm="giac")

[Out]

integrate((g*x + f)*(b*arcsin(c*x) + a)/(e*x + d)^6, 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 (a+b\,\mathrm {asin}\left (c\,x\right )\right )}{{\left (d+e\,x\right )}^6} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((f + g*x)*(a + b*asin(c*x)))/(d + e*x)^6,x)

[Out]

int(((f + g*x)*(a + b*asin(c*x)))/(d + e*x)^6, x)

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