3.2.4 \(\int \frac {(f+g x+h x^2) (a+b \text {ArcSin}(c x))}{(d+e x)^5} \, dx\) [104]

Optimal. Leaf size=470 \[ \frac {b c \left (e^2 f-d e g+d^2 h\right ) \sqrt {1-c^2 x^2}}{12 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^3}-\frac {b c \left (4 e^2 (e g-2 d h)-c^2 d \left (5 e^2 f-d e g-3 d^2 h\right )\right ) \sqrt {1-c^2 x^2}}{24 e^2 \left (c^2 d^2-e^2\right )^2 (d+e x)^2}+\frac {b c \left (12 e^4 h+c^4 d^2 \left (11 e^2 f+d e g-d^2 h\right )+4 c^2 e^2 \left (e^2 f-4 d e g+d^2 h\right )\right ) \sqrt {1-c^2 x^2}}{24 e^2 \left (c^2 d^2-e^2\right )^3 (d+e x)}-\frac {\left (e^2 f-d e g+d^2 h\right ) (a+b \text {ArcSin}(c x))}{4 e^3 (d+e x)^4}-\frac {(e g-2 d h) (a+b \text {ArcSin}(c x))}{3 e^3 (d+e x)^3}-\frac {h (a+b \text {ArcSin}(c x))}{2 e^3 (d+e x)^2}-\frac {b c^3 \left (4 e^4 (e g-5 d h)-c^2 d e^2 \left (9 e^2 f-13 d e g-7 d^2 h\right )-2 c^4 d^3 \left (3 e^2 f+d e g+d^2 h\right )\right ) \text {ArcTan}\left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{24 e^3 \left (c^2 d^2-e^2\right )^{7/2}} \]

[Out]

-1/4*(d^2*h-d*e*g+e^2*f)*(a+b*arcsin(c*x))/e^3/(e*x+d)^4-1/3*(-2*d*h+e*g)*(a+b*arcsin(c*x))/e^3/(e*x+d)^3-1/2*
h*(a+b*arcsin(c*x))/e^3/(e*x+d)^2-1/24*b*c^3*(4*e^4*(-5*d*h+e*g)-c^2*d*e^2*(-7*d^2*h-13*d*e*g+9*e^2*f)-2*c^4*d
^3*(d^2*h+d*e*g+3*e^2*f))*arctan((c^2*d*x+e)/(c^2*d^2-e^2)^(1/2)/(-c^2*x^2+1)^(1/2))/e^3/(c^2*d^2-e^2)^(7/2)+1
/12*b*c*(d^2*h-d*e*g+e^2*f)*(-c^2*x^2+1)^(1/2)/e^2/(c^2*d^2-e^2)/(e*x+d)^3-1/24*b*c*(4*e^2*(-2*d*h+e*g)-c^2*d*
(-3*d^2*h-d*e*g+5*e^2*f))*(-c^2*x^2+1)^(1/2)/e^2/(c^2*d^2-e^2)^2/(e*x+d)^2+1/24*b*c*(12*e^4*h+c^4*d^2*(-d^2*h+
d*e*g+11*e^2*f)+4*c^2*e^2*(d^2*h-4*d*e*g+e^2*f))*(-c^2*x^2+1)^(1/2)/e^2/(c^2*d^2-e^2)^3/(e*x+d)

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Rubi [A]
time = 0.65, antiderivative size = 470, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 8, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {712, 4837, 12, 1665, 849, 821, 739, 210} \begin {gather*} -\frac {(a+b \text {ArcSin}(c x)) \left (d^2 h-d e g+e^2 f\right )}{4 e^3 (d+e x)^4}-\frac {(e g-2 d h) (a+b \text {ArcSin}(c x))}{3 e^3 (d+e x)^3}-\frac {h (a+b \text {ArcSin}(c x))}{2 e^3 (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 (-2 c^4 d^3 \left (d^2 h+d e g+3 e^2 f\right )-c^2 d e^2 \left (-7 d^2 h-13 d e g+9 e^2 f\right )+4 e^4 (e g-5 d h)\right )}{24 e^3 \left (c^2 d^2-e^2\right )^{7/2}}-\frac {b c \sqrt {1-c^2 x^2} \left (4 e^2 (e g-2 d h)-c^2 d \left (-3 d^2 h-d e g+5 e^2 f\right )\right )}{24 e^2 \left (c^2 d^2-e^2\right )^2 (d+e x)^2}+\frac {b c \sqrt {1-c^2 x^2} \left (d^2 h-d e g+e^2 f\right )}{12 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^3}+\frac {b c \sqrt {1-c^2 x^2} \left (c^4 d^2 \left (d^2 (-h)+d e g+11 e^2 f\right )+4 c^2 e^2 \left (d^2 h-4 d e g+e^2 f\right )+12 e^4 h\right )}{24 e^2 \left (c^2 d^2-e^2\right )^3 (d+e x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((f + g*x + h*x^2)*(a + b*ArcSin[c*x]))/(d + e*x)^5,x]

[Out]

(b*c*(e^2*f - d*e*g + d^2*h)*Sqrt[1 - c^2*x^2])/(12*e^2*(c^2*d^2 - e^2)*(d + e*x)^3) - (b*c*(4*e^2*(e*g - 2*d*
h) - c^2*d*(5*e^2*f - d*e*g - 3*d^2*h))*Sqrt[1 - c^2*x^2])/(24*e^2*(c^2*d^2 - e^2)^2*(d + e*x)^2) + (b*c*(12*e
^4*h + c^4*d^2*(11*e^2*f + d*e*g - d^2*h) + 4*c^2*e^2*(e^2*f - 4*d*e*g + d^2*h))*Sqrt[1 - c^2*x^2])/(24*e^2*(c
^2*d^2 - e^2)^3*(d + e*x)) - ((e^2*f - d*e*g + d^2*h)*(a + b*ArcSin[c*x]))/(4*e^3*(d + e*x)^4) - ((e*g - 2*d*h
)*(a + b*ArcSin[c*x]))/(3*e^3*(d + e*x)^3) - (h*(a + b*ArcSin[c*x]))/(2*e^3*(d + e*x)^2) - (b*c^3*(4*e^4*(e*g
- 5*d*h) - c^2*d*e^2*(9*e^2*f - 13*d*e*g - 7*d^2*h) - 2*c^4*d^3*(3*e^2*f + d*e*g + d^2*h))*ArcTan[(e + c^2*d*x
)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/(24*e^3*(c^2*d^2 - e^2)^(7/2))

Rule 12

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

Rule 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 712

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

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 1665

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, d
+ e*x, x], R = PolynomialRemainder[Pq, d + e*x, x]}, Simp[(e*R*(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*ExpandToSum[(m
+ 1)*(c*d^2 + a*e^2)*Q + c*d*R*(m + 1) - c*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, c, d, e, p}, x] && Po
lyQ[Pq, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1]

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 {\left (f+g x+h x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{(d+e x)^5} \, dx &=-\frac {\left (e^2 f-d e g+d^2 h\right ) \left (a+b \sin ^{-1}(c x)\right )}{4 e^3 (d+e x)^4}-\frac {(e g-2 d h) \left (a+b \sin ^{-1}(c x)\right )}{3 e^3 (d+e x)^3}-\frac {h \left (a+b \sin ^{-1}(c x)\right )}{2 e^3 (d+e x)^2}-(b c) \int \frac {-3 e^2 f-d e g-d^2 h-4 e (e g+d h) x-6 e^2 h x^2}{12 e^3 (d+e x)^4 \sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {\left (e^2 f-d e g+d^2 h\right ) \left (a+b \sin ^{-1}(c x)\right )}{4 e^3 (d+e x)^4}-\frac {(e g-2 d h) \left (a+b \sin ^{-1}(c x)\right )}{3 e^3 (d+e x)^3}-\frac {h \left (a+b \sin ^{-1}(c x)\right )}{2 e^3 (d+e x)^2}-\frac {(b c) \int \frac {-3 e^2 f-d e g-d^2 h-4 e (e g+d h) x-6 e^2 h x^2}{(d+e x)^4 \sqrt {1-c^2 x^2}} \, dx}{12 e^3}\\ &=\frac {b c \left (e^2 f-d e g+d^2 h\right ) \sqrt {1-c^2 x^2}}{12 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^3}-\frac {\left (e^2 f-d e g+d^2 h\right ) \left (a+b \sin ^{-1}(c x)\right )}{4 e^3 (d+e x)^4}-\frac {(e g-2 d h) \left (a+b \sin ^{-1}(c x)\right )}{3 e^3 (d+e x)^3}-\frac {h \left (a+b \sin ^{-1}(c x)\right )}{2 e^3 (d+e x)^2}-\frac {(b c) \int \frac {3 \left (2 e^2 (2 e g-d h)-c^2 d \left (3 e^2 f+d e g+d^2 h\right )\right )+6 e \left (3 e^2 h+c^2 \left (e^2 f-d e g-2 d^2 h\right )\right ) x}{(d+e x)^3 \sqrt {1-c^2 x^2}} \, dx}{36 e^3 \left (c^2 d^2-e^2\right )}\\ &=\frac {b c \left (e^2 f-d e g+d^2 h\right ) \sqrt {1-c^2 x^2}}{12 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^3}-\frac {b c \left (4 e^2 (e g-2 d h)-c^2 d \left (5 e^2 f-d e g-3 d^2 h\right )\right ) \sqrt {1-c^2 x^2}}{24 e^2 \left (c^2 d^2-e^2\right )^2 (d+e x)^2}-\frac {\left (e^2 f-d e g+d^2 h\right ) \left (a+b \sin ^{-1}(c x)\right )}{4 e^3 (d+e x)^4}-\frac {(e g-2 d h) \left (a+b \sin ^{-1}(c x)\right )}{3 e^3 (d+e x)^3}-\frac {h \left (a+b \sin ^{-1}(c x)\right )}{2 e^3 (d+e x)^2}-\frac {(b c) \int \frac {-6 \left (6 e^4 h+2 c^2 e^2 \left (e^2 f-3 d e g-d^2 h\right )+c^4 d^2 \left (3 e^2 f+d e g+d^2 h\right )\right )-3 c^2 e \left (4 e^2 (e g-2 d h)-c^2 d \left (5 e^2 f-d e g-3 d^2 h\right )\right ) x}{(d+e x)^2 \sqrt {1-c^2 x^2}} \, dx}{72 e^3 \left (c^2 d^2-e^2\right )^2}\\ &=\frac {b c \left (e^2 f-d e g+d^2 h\right ) \sqrt {1-c^2 x^2}}{12 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^3}-\frac {b c \left (4 e^2 (e g-2 d h)-c^2 d \left (5 e^2 f-d e g-3 d^2 h\right )\right ) \sqrt {1-c^2 x^2}}{24 e^2 \left (c^2 d^2-e^2\right )^2 (d+e x)^2}+\frac {b c \left (12 e^4 h+c^4 d^2 \left (11 e^2 f+d e g-d^2 h\right )+4 c^2 e^2 \left (e^2 f-4 d e g+d^2 h\right )\right ) \sqrt {1-c^2 x^2}}{24 e^2 \left (c^2 d^2-e^2\right )^3 (d+e x)}-\frac {\left (e^2 f-d e g+d^2 h\right ) \left (a+b \sin ^{-1}(c x)\right )}{4 e^3 (d+e x)^4}-\frac {(e g-2 d h) \left (a+b \sin ^{-1}(c x)\right )}{3 e^3 (d+e x)^3}-\frac {h \left (a+b \sin ^{-1}(c x)\right )}{2 e^3 (d+e x)^2}-\frac {\left (b c^3 \left (4 e^4 (e g-5 d h)-c^2 d e^2 \left (9 e^2 f-13 d e g-7 d^2 h\right )-2 c^4 d^3 \left (3 e^2 f+d e g+d^2 h\right )\right )\right ) \int \frac {1}{(d+e x) \sqrt {1-c^2 x^2}} \, dx}{24 e^3 \left (c^2 d^2-e^2\right )^3}\\ &=\frac {b c \left (e^2 f-d e g+d^2 h\right ) \sqrt {1-c^2 x^2}}{12 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^3}-\frac {b c \left (4 e^2 (e g-2 d h)-c^2 d \left (5 e^2 f-d e g-3 d^2 h\right )\right ) \sqrt {1-c^2 x^2}}{24 e^2 \left (c^2 d^2-e^2\right )^2 (d+e x)^2}+\frac {b c \left (12 e^4 h+c^4 d^2 \left (11 e^2 f+d e g-d^2 h\right )+4 c^2 e^2 \left (e^2 f-4 d e g+d^2 h\right )\right ) \sqrt {1-c^2 x^2}}{24 e^2 \left (c^2 d^2-e^2\right )^3 (d+e x)}-\frac {\left (e^2 f-d e g+d^2 h\right ) \left (a+b \sin ^{-1}(c x)\right )}{4 e^3 (d+e x)^4}-\frac {(e g-2 d h) \left (a+b \sin ^{-1}(c x)\right )}{3 e^3 (d+e x)^3}-\frac {h \left (a+b \sin ^{-1}(c x)\right )}{2 e^3 (d+e x)^2}+\frac {\left (b c^3 \left (4 e^4 (e g-5 d h)-c^2 d e^2 \left (9 e^2 f-13 d e g-7 d^2 h\right )-2 c^4 d^3 \left (3 e^2 f+d e g+d^2 h\right )\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 )}{24 e^3 \left (c^2 d^2-e^2\right )^3}\\ &=\frac {b c \left (e^2 f-d e g+d^2 h\right ) \sqrt {1-c^2 x^2}}{12 e^2 \left (c^2 d^2-e^2\right ) (d+e x)^3}-\frac {b c \left (4 e^2 (e g-2 d h)-c^2 d \left (5 e^2 f-d e g-3 d^2 h\right )\right ) \sqrt {1-c^2 x^2}}{24 e^2 \left (c^2 d^2-e^2\right )^2 (d+e x)^2}+\frac {b c \left (12 e^4 h+c^4 d^2 \left (11 e^2 f+d e g-d^2 h\right )+4 c^2 e^2 \left (e^2 f-4 d e g+d^2 h\right )\right ) \sqrt {1-c^2 x^2}}{24 e^2 \left (c^2 d^2-e^2\right )^3 (d+e x)}-\frac {\left (e^2 f-d e g+d^2 h\right ) \left (a+b \sin ^{-1}(c x)\right )}{4 e^3 (d+e x)^4}-\frac {(e g-2 d h) \left (a+b \sin ^{-1}(c x)\right )}{3 e^3 (d+e x)^3}-\frac {h \left (a+b \sin ^{-1}(c x)\right )}{2 e^3 (d+e x)^2}-\frac {b c^3 \left (4 e^4 (e g-5 d h)-c^2 d e^2 \left (9 e^2 f-13 d e g-7 d^2 h\right )-2 c^4 d^3 \left (3 e^2 f+d e g+d^2 h\right )\right ) \tan ^{-1}\left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{24 e^3 \left (c^2 d^2-e^2\right )^{7/2}}\\ \end {align*}

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Mathematica [A]
time = 1.80, size = 575, normalized size = 1.22 \begin {gather*} -\frac {\frac {6 a \left (e^2 f-d e g+d^2 h\right )}{(d+e x)^4}+\frac {8 a (e g-2 d h)}{(d+e x)^3}+\frac {12 a h}{(d+e x)^2}+\frac {b c e \sqrt {1-c^2 x^2} \left (c^4 d^2 \left (-2 d^4 h+11 e^4 f x^2+d e^3 x (27 f+g x)-d^3 e (2 g+5 h x)+d^2 e^2 (18 f+x (g-h x))\right )+2 e^4 \left (3 d^2 h+d e (g+8 h x)+e^2 (f+2 x (g+3 h x))\right )+c^2 e^2 \left (11 d^4 h+4 e^4 f x^2+d e^3 x (3 f-16 g x)+d^3 e (-15 g+19 h x)+d^2 e^2 (-5 f+x (-35 g+4 h x))\right )\right )}{\left (-c^2 d^2+e^2\right )^3 (d+e x)^3}+\frac {2 b \left (d^2 h+d e (g+4 h x)+e^2 \left (3 f+4 g x+6 h x^2\right )\right ) \text {ArcSin}(c x)}{(d+e x)^4}-\frac {b c^3 \left (-4 e^4 (e g-5 d h)+c^2 d e^2 \left (9 e^2 f-13 d e g-7 d^2 h\right )+2 c^4 d^3 \left (3 e^2 f+d e g+d^2 h\right )\right ) \log (d+e x)}{(c d-e)^3 (c d+e)^3 \sqrt {-c^2 d^2+e^2}}+\frac {b c^3 \left (-4 e^4 (e g-5 d h)+c^2 d e^2 \left (9 e^2 f-13 d e g-7 d^2 h\right )+2 c^4 d^3 \left (3 e^2 f+d e g+d^2 h\right )\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)^3 (c d+e)^3 \sqrt {-c^2 d^2+e^2}}}{24 e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((f + g*x + h*x^2)*(a + b*ArcSin[c*x]))/(d + e*x)^5,x]

[Out]

-1/24*((6*a*(e^2*f - d*e*g + d^2*h))/(d + e*x)^4 + (8*a*(e*g - 2*d*h))/(d + e*x)^3 + (12*a*h)/(d + e*x)^2 + (b
*c*e*Sqrt[1 - c^2*x^2]*(c^4*d^2*(-2*d^4*h + 11*e^4*f*x^2 + d*e^3*x*(27*f + g*x) - d^3*e*(2*g + 5*h*x) + d^2*e^
2*(18*f + x*(g - h*x))) + 2*e^4*(3*d^2*h + d*e*(g + 8*h*x) + e^2*(f + 2*x*(g + 3*h*x))) + c^2*e^2*(11*d^4*h +
4*e^4*f*x^2 + d*e^3*x*(3*f - 16*g*x) + d^3*e*(-15*g + 19*h*x) + d^2*e^2*(-5*f + x*(-35*g + 4*h*x)))))/((-(c^2*
d^2) + e^2)^3*(d + e*x)^3) + (2*b*(d^2*h + d*e*(g + 4*h*x) + e^2*(3*f + 4*g*x + 6*h*x^2))*ArcSin[c*x])/(d + e*
x)^4 - (b*c^3*(-4*e^4*(e*g - 5*d*h) + c^2*d*e^2*(9*e^2*f - 13*d*e*g - 7*d^2*h) + 2*c^4*d^3*(3*e^2*f + d*e*g +
d^2*h))*Log[d + e*x])/((c*d - e)^3*(c*d + e)^3*Sqrt[-(c^2*d^2) + e^2]) + (b*c^3*(-4*e^4*(e*g - 5*d*h) + c^2*d*
e^2*(9*e^2*f - 13*d*e*g - 7*d^2*h) + 2*c^4*d^3*(3*e^2*f + d*e*g + d^2*h))*Log[e + c^2*d*x + Sqrt[-(c^2*d^2) +
e^2]*Sqrt[1 - c^2*x^2]])/((c*d - e)^3*(c*d + e)^3*Sqrt[-(c^2*d^2) + e^2]))/e^3

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2964\) vs. \(2(444)=888\).
time = 0.11, size = 2965, normalized size = 6.31

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(2965\)
default	\(\text {Expression too large to display}\)	\(2965\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((h*x^2+g*x+f)*(a+b*arcsin(c*x))/(e*x+d)^5,x,method=_RETURNVERBOSE)

[Out]

1/c*(-1/4*b*c^5*arcsin(c*x)/e^3/(c*e*x+c*d)^4*d^2*h+1/4*b*c^5*arcsin(c*x)/e^2/(c*e*x+c*d)^4*d*g+1/12*b*c^5/e^3
/(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)*f-1/6*b*c^5/e/(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-1/3*b*c^4*arcsin(c*x)*g/e
^2/(c*e*x+c*d)^3-1/3*b*c^4/e^4/(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*h+5/8*b*c^7/e^3*d^4/(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)*h-7/6*b*c^5/e^3/(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)*d^2*h+1/12*b*c^5/e^5/(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)*d^2*h+5/24*b*c^6/e^4*d^3/(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)*h-5/8*b*c^8/e^4*d^5/(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))*h-5/6*b*c^4/e^4/(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*h+11/8*b
*c^6/e^4*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))*h-1/4*b*c^5*arc
sin(c*x)/e/(c*e*x+c*d)^4*f+1/2*b*c^3/e^3*h/(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)+2/3*b*c^4*arcsin(c*x)/e^3/(c*e*x+c*d)^3*d*h-1/2*b*c^3*arcsin(c*x)*h/e^3/(c*e*x+c*d)^2+a*c^
3*(1/3*c*(2*d*h-e*g)/e^3/(c*e*x+c*d)^3-1/2*h/e^3/(c*e*x+c*d)^2-1/4*c^2*(d^2*h-d*e*g+e^2*f)/e^3/(c*e*x+c*d)^4)+
1/6*b*c^4/e^3*g/(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)+1/6*b
*c^4/e^3*g/(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))+3/8*b*c^6/e^2*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))*f+2/3*b*c^5/e^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)*d*g-1/12*b*c^5/e^4/(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)*d*g-5/24*b*c^6/e^3*d^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)*g+5/24*b*c^6/e^2*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)*f-5/8*b*c^7/e^2*d^3/(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)*g+5/8*b*c^7/e*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)*f+5/8*b*c^8/e^3*d^4/(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))*g-5/8*b*c^8/e^2*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))*f-7/8*b*c^6/e^3*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))*g)

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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((h*x^2+g*x+f)*(a+b*arcsin(c*x))/(e*x+d)^5,x, algorithm="maxima")

[Out]

-1/12*(4*x*e + d)*a*g/(x^4*e^6 + 4*d*x^3*e^5 + 6*d^2*x^2*e^4 + 4*d^3*x*e^3 + d^4*e^2) - 1/12*(6*x^2*e^2 + 4*d*
x*e + d^2)*a*h/(x^4*e^7 + 4*d*x^3*e^6 + 6*d^2*x^2*e^5 + 4*d^3*x*e^4 + d^4*e^3) - 1/4*a*f/(x^4*e^5 + 4*d*x^3*e^
4 + 6*d^2*x^2*e^3 + 4*d^3*x*e^2 + d^4*e) - 1/12*((6*b*h*x^2*e^2 + b*d^2*h + b*d*g*e + 3*b*f*e^2 + 4*(b*d*h*e +
b*g*e^2)*x)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + 12*(x^4*e^7 + 4*d*x^3*e^6 + 6*d^2*x^2*e^5 + 4*d^3*x*
e^4 + d^4*e^3)*integrate(1/12*(6*b*c*h*x^2*e^2 + b*c*d^2*h + b*c*d*g*e + 3*b*c*f*e^2 + 4*(b*c*d*h*e + b*c*g*e^
2)*x)*e^(1/2*log(c*x + 1) + 1/2*log(-c*x + 1))/(c^4*x^8*e^7 + 4*c^4*d*x^7*e^6 - 4*c^2*d^3*x^3*e^4 - c^2*d^4*x^
2*e^3 + (6*c^4*d^2*e^5 - c^2*e^7)*x^6 + 4*(c^4*d^3*e^4 - c^2*d*e^6)*x^5 + (c^4*d^4*e^3 - 6*c^2*d^2*e^5)*x^4 +
(c^2*x^6*e^7 + 4*c^2*d*x^5*e^6 + (6*c^2*d^2*e^5 - e^7)*x^4 - 4*d^3*x*e^4 - d^4*e^3 + 4*(c^2*d^3*e^4 - d*e^6)*x
^3 + (c^2*d^4*e^3 - 6*d^2*e^5)*x^2)*e^(log(c*x + 1) + log(-c*x + 1))), x))/(x^4*e^7 + 4*d*x^3*e^6 + 6*d^2*x^2*
e^5 + 4*d^3*x*e^4 + d^4*e^3)

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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((h*x^2+g*x+f)*(a+b*arcsin(c*x))/(e*x+d)^5,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 + h x^{2}\right )}{\left (d + e x\right )^{5}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((h*x**2+g*x+f)*(a+b*asin(c*x))/(e*x+d)**5,x)

[Out]

Integral((a + b*asin(c*x))*(f + g*x + h*x**2)/(d + e*x)**5, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((h*x^2+g*x+f)*(a+b*arcsin(c*x))/(e*x+d)^5,x, algorithm="giac")

[Out]

Exception raised: RuntimeError >> An error occurred running a Giac command:INPUT:sage2OUTPUT:sym2poly/r2sym(co
nst gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*asin(c*x))*(f + g*x + h*x^2))/(d + e*x)^5,x)

[Out]

int(((a + b*asin(c*x))*(f + g*x + h*x^2))/(d + e*x)^5, x)

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