3.1.94 \(\int \frac {(f+g x) (a+b \text {ArcSin}(c x))}{(d+e x)^4} \, dx\) [94]

Optimal. Leaf size=257 \[ \frac {b c (e f-d g) \sqrt {1-c^2 x^2}}{6 e \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {b c \left (c^2 d f-e g\right ) \sqrt {1-c^2 x^2}}{2 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {(e f-d g) (a+b \text {ArcSin}(c x))}{3 e^2 (d+e x)^3}-\frac {g (a+b \text {ArcSin}(c x))}{2 e^2 (d+e x)^2}+\frac {b c^3 \left (e^2 (e f-4 d g)+c^2 d^2 (2 e f+d g)\right ) \text {ArcTan}\left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{6 e^2 \left (c^2 d^2-e^2\right )^{5/2}} \]

[Out]

-1/3*(-d*g+e*f)*(a+b*arcsin(c*x))/e^2/(e*x+d)^3-1/2*g*(a+b*arcsin(c*x))/e^2/(e*x+d)^2+1/6*b*c^3*(e^2*(-4*d*g+e
*f)+c^2*d^2*(d*g+2*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)^(5/2)+1/
6*b*c*(-d*g+e*f)*(-c^2*x^2+1)^(1/2)/e/(c^2*d^2-e^2)/(e*x+d)^2+1/2*b*c*(c^2*d*f-e*g)*(-c^2*x^2+1)^(1/2)/(c^2*d^
2-e^2)^2/(e*x+d)

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Rubi [A]
time = 0.30, antiderivative size = 257, normalized size of antiderivative = 1.00, number of steps used = 6, 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))}{3 e^2 (d+e x)^3}-\frac {g (a+b \text {ArcSin}(c x))}{2 e^2 (d+e x)^2}+\frac {b c^3 \text {ArcTan}\left (\frac {c^2 d x+e}{\sqrt {1-c^2 x^2} \sqrt {c^2 d^2-e^2}}\right ) \left (c^2 d^2 (d g+2 e f)+e^2 (e f-4 d g)\right )}{6 e^2 \left (c^2 d^2-e^2\right )^{5/2}}+\frac {b c \sqrt {1-c^2 x^2} \left (c^2 d f-e g\right )}{2 \left (c^2 d^2-e^2\right )^2 (d+e x)}+\frac {b c \sqrt {1-c^2 x^2} (e f-d g)}{6 e \left (c^2 d^2-e^2\right ) (d+e x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((f + g*x)*(a + b*ArcSin[c*x]))/(d + e*x)^4,x]

[Out]

(b*c*(e*f - d*g)*Sqrt[1 - c^2*x^2])/(6*e*(c^2*d^2 - e^2)*(d + e*x)^2) + (b*c*(c^2*d*f - e*g)*Sqrt[1 - c^2*x^2]
)/(2*(c^2*d^2 - e^2)^2*(d + e*x)) - ((e*f - d*g)*(a + b*ArcSin[c*x]))/(3*e^2*(d + e*x)^3) - (g*(a + b*ArcSin[c
*x]))/(2*e^2*(d + e*x)^2) + (b*c^3*(e^2*(e*f - 4*d*g) + c^2*d^2*(2*e*f + d*g))*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*
d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(6*e^2*(c^2*d^2 - e^2)^(5/2))

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)^4} \, dx &=-\frac {(e f-d g) \left (a+b \sin ^{-1}(c x)\right )}{3 e^2 (d+e x)^3}-\frac {g \left (a+b \sin ^{-1}(c x)\right )}{2 e^2 (d+e x)^2}-(b c) \int \frac {-2 e f-d g-3 e g x}{6 e^2 (d+e x)^3 \sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {(e f-d g) \left (a+b \sin ^{-1}(c x)\right )}{3 e^2 (d+e x)^3}-\frac {g \left (a+b \sin ^{-1}(c x)\right )}{2 e^2 (d+e x)^2}-\frac {(b c) \int \frac {-2 e f-d g-3 e g x}{(d+e x)^3 \sqrt {1-c^2 x^2}} \, dx}{6 e^2}\\ &=\frac {b c (e f-d g) \sqrt {1-c^2 x^2}}{6 e \left (c^2 d^2-e^2\right ) (d+e x)^2}-\frac {(e f-d g) \left (a+b \sin ^{-1}(c x)\right )}{3 e^2 (d+e x)^3}-\frac {g \left (a+b \sin ^{-1}(c x)\right )}{2 e^2 (d+e x)^2}-\frac {(b c) \int \frac {2 \left (3 e^2 g-c^2 d (2 e f+d g)\right )+2 c^2 e (e f-d g) x}{(d+e x)^2 \sqrt {1-c^2 x^2}} \, dx}{12 e^2 \left (c^2 d^2-e^2\right )}\\ &=\frac {b c (e f-d g) \sqrt {1-c^2 x^2}}{6 e \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {b c \left (c^2 d f-e g\right ) \sqrt {1-c^2 x^2}}{2 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {(e f-d g) \left (a+b \sin ^{-1}(c x)\right )}{3 e^2 (d+e x)^3}-\frac {g \left (a+b \sin ^{-1}(c x)\right )}{2 e^2 (d+e x)^2}+\frac {\left (b c^3 \left (e^2 (e f-4 d g)+c^2 d^2 (2 e f+d g)\right )\right ) \int \frac {1}{(d+e x) \sqrt {1-c^2 x^2}} \, dx}{6 e^2 \left (c^2 d^2-e^2\right )^2}\\ &=\frac {b c (e f-d g) \sqrt {1-c^2 x^2}}{6 e \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {b c \left (c^2 d f-e g\right ) \sqrt {1-c^2 x^2}}{2 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {(e f-d g) \left (a+b \sin ^{-1}(c x)\right )}{3 e^2 (d+e x)^3}-\frac {g \left (a+b \sin ^{-1}(c x)\right )}{2 e^2 (d+e x)^2}-\frac {\left (b c^3 \left (e^2 (e f-4 d g)+c^2 d^2 (2 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 )}{6 e^2 \left (c^2 d^2-e^2\right )^2}\\ &=\frac {b c (e f-d g) \sqrt {1-c^2 x^2}}{6 e \left (c^2 d^2-e^2\right ) (d+e x)^2}+\frac {b c \left (c^2 d f-e g\right ) \sqrt {1-c^2 x^2}}{2 \left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {(e f-d g) \left (a+b \sin ^{-1}(c x)\right )}{3 e^2 (d+e x)^3}-\frac {g \left (a+b \sin ^{-1}(c x)\right )}{2 e^2 (d+e x)^2}+\frac {b c^3 \left (e^2 (e f-4 d g)+c^2 d^2 (2 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 )}{6 e^2 \left (c^2 d^2-e^2\right )^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 0.48, size = 321, normalized size = 1.25 \begin {gather*} \frac {\frac {a (-2 e f+2 d g)}{(d+e x)^3}-\frac {3 a g}{(d+e x)^2}+\frac {b c e \sqrt {1-c^2 x^2} \left (c^2 d \left (4 d e f-d^2 g+3 e^2 f x\right )-e^2 (2 d g+e (f+3 g x))\right )}{\left (-c^2 d^2+e^2\right )^2 (d+e x)^2}-\frac {b (2 e f+d g+3 e g x) \text {ArcSin}(c x)}{(d+e x)^3}+\frac {b c^3 \left (e^2 (e f-4 d g)+c^2 d^2 (2 e f+d g)\right ) \log (d+e x)}{(-c d+e)^2 (c d+e)^2 \sqrt {-c^2 d^2+e^2}}-\frac {b c^3 \left (e^2 (e f-4 d g)+c^2 d^2 (2 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)^2 (c d+e)^2 \sqrt {-c^2 d^2+e^2}}}{6 e^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((f + g*x)*(a + b*ArcSin[c*x]))/(d + e*x)^4,x]

[Out]

((a*(-2*e*f + 2*d*g))/(d + e*x)^3 - (3*a*g)/(d + e*x)^2 + (b*c*e*Sqrt[1 - c^2*x^2]*(c^2*d*(4*d*e*f - d^2*g + 3
*e^2*f*x) - e^2*(2*d*g + e*(f + 3*g*x))))/((-(c^2*d^2) + e^2)^2*(d + e*x)^2) - (b*(2*e*f + d*g + 3*e*g*x)*ArcS
in[c*x])/(d + e*x)^3 + (b*c^3*(e^2*(e*f - 4*d*g) + c^2*d^2*(2*e*f + d*g))*Log[d + e*x])/((-(c*d) + e)^2*(c*d +
e)^2*Sqrt[-(c^2*d^2) + e^2]) - (b*c^3*(e^2*(e*f - 4*d*g) + c^2*d^2*(2*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)^2*(c*d + e)^2*Sqrt[-(c^2*d^2) + e^2]))/(6*e^2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1257\) vs. \(2(237)=474\).
time = 0.13, size = 1258, normalized size = 4.89 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x+f)*(a+b*arcsin(c*x))/(e*x+d)^4,x,method=_RETURNVERBOSE)

[Out]

1/c*(a*c^3*(1/3*c*(d*g-e*f)/e^2/(c*e*x+c*d)^3-1/2*g/e^2/(c*e*x+c*d)^2)+1/3*b*c^4*arcsin(c*x)/e^2/(c*e*x+c*d)^3
*d*g-1/3*b*c^4*arcsin(c*x)/e/(c*e*x+c*d)^3*f-1/2*b*c^3*arcsin(c*x)*g/e^2/(c*e*x+c*d)^2-1/6*b*c^4/e^3/(c^2*d^2-
e^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)*d*g+1/6*b*c^4/e^2/(c^2*d^2-e^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-1/2*b*c^5/e^2*d^2/(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)*g+1/2*b*c^5/e*d/(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)*f+1/2*b*c^6/e^3*d^3/(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))*g-1/2*b*c^6/e^2*d^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-2/3*b*c^4/e^3/(c^2*d^2-e^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))*d*g+1/6*b*c^4/e^2/(c^2*d^2-e^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+1/2*b*c^3/e^2*g/(c^2*d^2-e^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))

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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)^4,x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 929 vs. \(2 (240) = 480\).
time = 38.35, size = 1887, normalized size = 7.34 \begin {gather*} \left [-\frac {2 \, a c^{6} d^{7} g - 6 \, a c^{4} d^{5} g e^{2} + 6 \, a c^{2} d^{3} g e^{4} - 2 \, a d g e^{6} + {\left (b c^{5} d^{6} g + b c^{3} f x^{3} e^{6} - {\left (4 \, b c^{3} d g x^{3} - 3 \, b c^{3} d f x^{2}\right )} e^{5} + {\left (2 \, b c^{5} d^{2} f x^{3} - 12 \, b c^{3} d^{2} g x^{2} + 3 \, b c^{3} d^{2} f x\right )} e^{4} + {\left (b c^{5} d^{3} g x^{3} + 6 \, b c^{5} d^{3} f x^{2} - 12 \, b c^{3} d^{3} g x + b c^{3} d^{3} f\right )} e^{3} + {\left (3 \, b c^{5} d^{4} g x^{2} + 6 \, b c^{5} d^{4} f x - 4 \, b c^{3} d^{4} g\right )} e^{2} + {\left (3 \, b c^{5} d^{5} g x + 2 \, b c^{5} d^{5} f\right )} e\right )} \sqrt {-c^{2} d^{2} + e^{2}} \log \left (\frac {2 \, c^{4} d^{2} x^{2} + 2 \, c^{2} d x e - c^{2} d^{2} - 2 \, \sqrt {-c^{2} d^{2} + e^{2}} {\left (c^{2} d x + e\right )} \sqrt {-c^{2} x^{2} + 1} - {\left (c^{2} x^{2} - 2\right )} e^{2}}{x^{2} e^{2} + 2 \, d x e + d^{2}}\right ) + 2 \, {\left (b c^{6} d^{7} g - 3 \, b c^{4} d^{5} g e^{2} + 3 \, b c^{2} d^{3} g e^{4} - b d g e^{6} - {\left (3 \, b g x + 2 \, b f\right )} e^{7} + 3 \, {\left (3 \, b c^{2} d^{2} g x + 2 \, b c^{2} d^{2} f\right )} e^{5} - 3 \, {\left (3 \, b c^{4} d^{4} g x + 2 \, b c^{4} d^{4} f\right )} e^{3} + {\left (3 \, b c^{6} d^{6} g x + 2 \, b c^{6} d^{6} f\right )} e\right )} \arcsin \left (c x\right ) - 2 \, {\left (3 \, a g x + 2 \, a f\right )} e^{7} + 6 \, {\left (3 \, a c^{2} d^{2} g x + 2 \, a c^{2} d^{2} f\right )} e^{5} - 6 \, {\left (3 \, a c^{4} d^{4} g x + 2 \, a c^{4} d^{4} f\right )} e^{3} + 2 \, {\left (3 \, a c^{6} d^{6} g x + 2 \, a c^{6} d^{6} f\right )} e + 2 \, {\left (b c^{5} d^{6} g e - {\left (3 \, b c g x^{2} + b c f x\right )} e^{7} + {\left (3 \, b c^{3} d f x^{2} - 5 \, b c d g x - b c d f\right )} e^{6} + {\left (3 \, b c^{3} d^{2} g x^{2} + 8 \, b c^{3} d^{2} f x - 2 \, b c d^{2} g\right )} e^{5} - {\left (3 \, b c^{5} d^{3} f x^{2} - 4 \, b c^{3} d^{3} g x - 5 \, b c^{3} d^{3} f\right )} e^{4} - {\left (7 \, b c^{5} d^{4} f x - b c^{3} d^{4} g\right )} e^{3} + {\left (b c^{5} d^{5} g x - 4 \, b c^{5} d^{5} f\right )} e^{2}\right )} \sqrt {-c^{2} x^{2} + 1}}{12 \, {\left (3 \, c^{6} d^{8} x e^{3} + c^{6} d^{9} e^{2} - x^{3} e^{11} - 3 \, d x^{2} e^{10} + 3 \, {\left (c^{2} d^{2} x^{3} - d^{2} x\right )} e^{9} + {\left (9 \, c^{2} d^{3} x^{2} - d^{3}\right )} e^{8} - 3 \, {\left (c^{4} d^{4} x^{3} - 3 \, c^{2} d^{4} x\right )} e^{7} - 3 \, {\left (3 \, c^{4} d^{5} x^{2} - c^{2} d^{5}\right )} e^{6} + {\left (c^{6} d^{6} x^{3} - 9 \, c^{4} d^{6} x\right )} e^{5} + 3 \, {\left (c^{6} d^{7} x^{2} - c^{4} d^{7}\right )} e^{4}\right )}}, -\frac {a c^{6} d^{7} g - 3 \, a c^{4} d^{5} g e^{2} + 3 \, a c^{2} d^{3} g e^{4} - a d g e^{6} - {\left (b c^{5} d^{6} g + b c^{3} f x^{3} e^{6} - {\left (4 \, b c^{3} d g x^{3} - 3 \, b c^{3} d f x^{2}\right )} e^{5} + {\left (2 \, b c^{5} d^{2} f x^{3} - 12 \, b c^{3} d^{2} g x^{2} + 3 \, b c^{3} d^{2} f x\right )} e^{4} + {\left (b c^{5} d^{3} g x^{3} + 6 \, b c^{5} d^{3} f x^{2} - 12 \, b c^{3} d^{3} g x + b c^{3} d^{3} f\right )} e^{3} + {\left (3 \, b c^{5} d^{4} g x^{2} + 6 \, b c^{5} d^{4} f x - 4 \, b c^{3} d^{4} g\right )} e^{2} + {\left (3 \, b c^{5} d^{5} g x + 2 \, b c^{5} d^{5} f\right )} e\right )} \sqrt {c^{2} d^{2} - e^{2}} \arctan \left (-\frac {\sqrt {c^{2} d^{2} - e^{2}} {\left (c^{2} d x + e\right )} \sqrt {-c^{2} x^{2} + 1}}{c^{4} d^{2} x^{2} - c^{2} d^{2} - {\left (c^{2} x^{2} - 1\right )} e^{2}}\right ) + {\left (b c^{6} d^{7} g - 3 \, b c^{4} d^{5} g e^{2} + 3 \, b c^{2} d^{3} g e^{4} - b d g e^{6} - {\left (3 \, b g x + 2 \, b f\right )} e^{7} + 3 \, {\left (3 \, b c^{2} d^{2} g x + 2 \, b c^{2} d^{2} f\right )} e^{5} - 3 \, {\left (3 \, b c^{4} d^{4} g x + 2 \, b c^{4} d^{4} f\right )} e^{3} + {\left (3 \, b c^{6} d^{6} g x + 2 \, b c^{6} d^{6} f\right )} e\right )} \arcsin \left (c x\right ) - {\left (3 \, a g x + 2 \, a f\right )} e^{7} + 3 \, {\left (3 \, a c^{2} d^{2} g x + 2 \, a c^{2} d^{2} f\right )} e^{5} - 3 \, {\left (3 \, a c^{4} d^{4} g x + 2 \, a c^{4} d^{4} f\right )} e^{3} + {\left (3 \, a c^{6} d^{6} g x + 2 \, a c^{6} d^{6} f\right )} e + {\left (b c^{5} d^{6} g e - {\left (3 \, b c g x^{2} + b c f x\right )} e^{7} + {\left (3 \, b c^{3} d f x^{2} - 5 \, b c d g x - b c d f\right )} e^{6} + {\left (3 \, b c^{3} d^{2} g x^{2} + 8 \, b c^{3} d^{2} f x - 2 \, b c d^{2} g\right )} e^{5} - {\left (3 \, b c^{5} d^{3} f x^{2} - 4 \, b c^{3} d^{3} g x - 5 \, b c^{3} d^{3} f\right )} e^{4} - {\left (7 \, b c^{5} d^{4} f x - b c^{3} d^{4} g\right )} e^{3} + {\left (b c^{5} d^{5} g x - 4 \, b c^{5} d^{5} f\right )} e^{2}\right )} \sqrt {-c^{2} x^{2} + 1}}{6 \, {\left (3 \, c^{6} d^{8} x e^{3} + c^{6} d^{9} e^{2} - x^{3} e^{11} - 3 \, d x^{2} e^{10} + 3 \, {\left (c^{2} d^{2} x^{3} - d^{2} x\right )} e^{9} + {\left (9 \, c^{2} d^{3} x^{2} - d^{3}\right )} e^{8} - 3 \, {\left (c^{4} d^{4} x^{3} - 3 \, c^{2} d^{4} x\right )} e^{7} - 3 \, {\left (3 \, c^{4} d^{5} x^{2} - c^{2} d^{5}\right )} e^{6} + {\left (c^{6} d^{6} x^{3} - 9 \, c^{4} d^{6} x\right )} e^{5} + 3 \, {\left (c^{6} d^{7} x^{2} - c^{4} d^{7}\right )} e^{4}\right )}}\right ] \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)^4,x, algorithm="fricas")

[Out]

[-1/12*(2*a*c^6*d^7*g - 6*a*c^4*d^5*g*e^2 + 6*a*c^2*d^3*g*e^4 - 2*a*d*g*e^6 + (b*c^5*d^6*g + b*c^3*f*x^3*e^6 -
(4*b*c^3*d*g*x^3 - 3*b*c^3*d*f*x^2)*e^5 + (2*b*c^5*d^2*f*x^3 - 12*b*c^3*d^2*g*x^2 + 3*b*c^3*d^2*f*x)*e^4 + (b
*c^5*d^3*g*x^3 + 6*b*c^5*d^3*f*x^2 - 12*b*c^3*d^3*g*x + b*c^3*d^3*f)*e^3 + (3*b*c^5*d^4*g*x^2 + 6*b*c^5*d^4*f*
x - 4*b*c^3*d^4*g)*e^2 + (3*b*c^5*d^5*g*x + 2*b*c^5*d^5*f)*e)*sqrt(-c^2*d^2 + e^2)*log((2*c^4*d^2*x^2 + 2*c^2*
d*x*e - c^2*d^2 - 2*sqrt(-c^2*d^2 + e^2)*(c^2*d*x + e)*sqrt(-c^2*x^2 + 1) - (c^2*x^2 - 2)*e^2)/(x^2*e^2 + 2*d*
x*e + d^2)) + 2*(b*c^6*d^7*g - 3*b*c^4*d^5*g*e^2 + 3*b*c^2*d^3*g*e^4 - b*d*g*e^6 - (3*b*g*x + 2*b*f)*e^7 + 3*(
3*b*c^2*d^2*g*x + 2*b*c^2*d^2*f)*e^5 - 3*(3*b*c^4*d^4*g*x + 2*b*c^4*d^4*f)*e^3 + (3*b*c^6*d^6*g*x + 2*b*c^6*d^
6*f)*e)*arcsin(c*x) - 2*(3*a*g*x + 2*a*f)*e^7 + 6*(3*a*c^2*d^2*g*x + 2*a*c^2*d^2*f)*e^5 - 6*(3*a*c^4*d^4*g*x +
2*a*c^4*d^4*f)*e^3 + 2*(3*a*c^6*d^6*g*x + 2*a*c^6*d^6*f)*e + 2*(b*c^5*d^6*g*e - (3*b*c*g*x^2 + b*c*f*x)*e^7 +
(3*b*c^3*d*f*x^2 - 5*b*c*d*g*x - b*c*d*f)*e^6 + (3*b*c^3*d^2*g*x^2 + 8*b*c^3*d^2*f*x - 2*b*c*d^2*g)*e^5 - (3*
b*c^5*d^3*f*x^2 - 4*b*c^3*d^3*g*x - 5*b*c^3*d^3*f)*e^4 - (7*b*c^5*d^4*f*x - b*c^3*d^4*g)*e^3 + (b*c^5*d^5*g*x
- 4*b*c^5*d^5*f)*e^2)*sqrt(-c^2*x^2 + 1))/(3*c^6*d^8*x*e^3 + c^6*d^9*e^2 - x^3*e^11 - 3*d*x^2*e^10 + 3*(c^2*d^
2*x^3 - d^2*x)*e^9 + (9*c^2*d^3*x^2 - d^3)*e^8 - 3*(c^4*d^4*x^3 - 3*c^2*d^4*x)*e^7 - 3*(3*c^4*d^5*x^2 - c^2*d^
5)*e^6 + (c^6*d^6*x^3 - 9*c^4*d^6*x)*e^5 + 3*(c^6*d^7*x^2 - c^4*d^7)*e^4), -1/6*(a*c^6*d^7*g - 3*a*c^4*d^5*g*e
^2 + 3*a*c^2*d^3*g*e^4 - a*d*g*e^6 - (b*c^5*d^6*g + b*c^3*f*x^3*e^6 - (4*b*c^3*d*g*x^3 - 3*b*c^3*d*f*x^2)*e^5
+ (2*b*c^5*d^2*f*x^3 - 12*b*c^3*d^2*g*x^2 + 3*b*c^3*d^2*f*x)*e^4 + (b*c^5*d^3*g*x^3 + 6*b*c^5*d^3*f*x^2 - 12*b
*c^3*d^3*g*x + b*c^3*d^3*f)*e^3 + (3*b*c^5*d^4*g*x^2 + 6*b*c^5*d^4*f*x - 4*b*c^3*d^4*g)*e^2 + (3*b*c^5*d^5*g*x
+ 2*b*c^5*d^5*f)*e)*sqrt(c^2*d^2 - e^2)*arctan(-sqrt(c^2*d^2 - e^2)*(c^2*d*x + e)*sqrt(-c^2*x^2 + 1)/(c^4*d^2
*x^2 - c^2*d^2 - (c^2*x^2 - 1)*e^2)) + (b*c^6*d^7*g - 3*b*c^4*d^5*g*e^2 + 3*b*c^2*d^3*g*e^4 - b*d*g*e^6 - (3*b
*g*x + 2*b*f)*e^7 + 3*(3*b*c^2*d^2*g*x + 2*b*c^2*d^2*f)*e^5 - 3*(3*b*c^4*d^4*g*x + 2*b*c^4*d^4*f)*e^3 + (3*b*c
^6*d^6*g*x + 2*b*c^6*d^6*f)*e)*arcsin(c*x) - (3*a*g*x + 2*a*f)*e^7 + 3*(3*a*c^2*d^2*g*x + 2*a*c^2*d^2*f)*e^5 -
3*(3*a*c^4*d^4*g*x + 2*a*c^4*d^4*f)*e^3 + (3*a*c^6*d^6*g*x + 2*a*c^6*d^6*f)*e + (b*c^5*d^6*g*e - (3*b*c*g*x^2
+ b*c*f*x)*e^7 + (3*b*c^3*d*f*x^2 - 5*b*c*d*g*x - b*c*d*f)*e^6 + (3*b*c^3*d^2*g*x^2 + 8*b*c^3*d^2*f*x - 2*b*c
*d^2*g)*e^5 - (3*b*c^5*d^3*f*x^2 - 4*b*c^3*d^3*g*x - 5*b*c^3*d^3*f)*e^4 - (7*b*c^5*d^4*f*x - b*c^3*d^4*g)*e^3
+ (b*c^5*d^5*g*x - 4*b*c^5*d^5*f)*e^2)*sqrt(-c^2*x^2 + 1))/(3*c^6*d^8*x*e^3 + c^6*d^9*e^2 - x^3*e^11 - 3*d*x^2
*e^10 + 3*(c^2*d^2*x^3 - d^2*x)*e^9 + (9*c^2*d^3*x^2 - d^3)*e^8 - 3*(c^4*d^4*x^3 - 3*c^2*d^4*x)*e^7 - 3*(3*c^4
*d^5*x^2 - c^2*d^5)*e^6 + (c^6*d^6*x^3 - 9*c^4*d^6*x)*e^5 + 3*(c^6*d^7*x^2 - c^4*d^7)*e^4)]

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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 )^{4}}\, 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)**4,x)

[Out]

Integral((a + b*asin(c*x))*(f + g*x)/(d + e*x)**4, 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)^4,x, algorithm="giac")

[Out]

integrate((g*x + f)*(b*arcsin(c*x) + a)/(e*x + d)^4, 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 )}^4} \,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)^4,x)

[Out]

int(((f + g*x)*(a + b*asin(c*x)))/(d + e*x)^4, x)

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