3.7.43 \(\int \frac {x (a+b \text {ArcSin}(c x))}{(d+e x^2)^3} \, dx\) [643]

Optimal. Leaf size=133 \[ \frac {b c x \sqrt {1-c^2 x^2}}{8 d \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {a+b \text {ArcSin}(c x)}{4 e \left (d+e x^2\right )^2}+\frac {b c \left (2 c^2 d+e\right ) \text {ArcTan}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{8 d^{3/2} e \left (c^2 d+e\right )^{3/2}} \]

[Out]

1/4*(-a-b*arcsin(c*x))/e/(e*x^2+d)^2+1/8*b*c*(2*c^2*d+e)*arctan(x*(c^2*d+e)^(1/2)/d^(1/2)/(-c^2*x^2+1)^(1/2))/
d^(3/2)/e/(c^2*d+e)^(3/2)+1/8*b*c*x*(-c^2*x^2+1)^(1/2)/d/(c^2*d+e)/(e*x^2+d)

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Rubi [A]
time = 0.07, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {4813, 390, 385, 211} \begin {gather*} -\frac {a+b \text {ArcSin}(c x)}{4 e \left (d+e x^2\right )^2}+\frac {b c \left (2 c^2 d+e\right ) \text {ArcTan}\left (\frac {x \sqrt {c^2 d+e}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{8 d^{3/2} e \left (c^2 d+e\right )^{3/2}}+\frac {b c x \sqrt {1-c^2 x^2}}{8 d \left (c^2 d+e\right ) \left (d+e x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x*(a + b*ArcSin[c*x]))/(d + e*x^2)^3,x]

[Out]

(b*c*x*Sqrt[1 - c^2*x^2])/(8*d*(c^2*d + e)*(d + e*x^2)) - (a + b*ArcSin[c*x])/(4*e*(d + e*x^2)^2) + (b*c*(2*c^
2*d + e)*ArcTan[(Sqrt[c^2*d + e]*x)/(Sqrt[d]*Sqrt[1 - c^2*x^2])])/(8*d^(3/2)*e*(c^2*d + e)^(3/2))

Rule 211

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

Rule 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 390

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*x*(a + b*x^n)^(p + 1)*
((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - a*d))), x] + Dist[(b*c + n*(p + 1)*(b*c - a*d))/(a*n*(p + 1)*(b*c - a
*d)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n, q}, x] && NeQ[b*c - a*d, 0] && Eq
Q[n*(p + q + 2) + 1, 0] && (LtQ[p, -1] ||  !LtQ[q, -1]) && NeQ[p, -1]

Rule 4813

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)
*((a + b*ArcSin[c*x])/(2*e*(p + 1))), x] - Dist[b*(c/(2*e*(p + 1))), Int[(d + e*x^2)^(p + 1)/Sqrt[1 - c^2*x^2]
, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[c^2*d + e, 0] && NeQ[p, -1]

Rubi steps

\begin {align*} \int \frac {x \left (a+b \sin ^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx &=-\frac {a+b \sin ^{-1}(c x)}{4 e \left (d+e x^2\right )^2}+\frac {(b c) \int \frac {1}{\sqrt {1-c^2 x^2} \left (d+e x^2\right )^2} \, dx}{4 e}\\ &=\frac {b c x \sqrt {1-c^2 x^2}}{8 d \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {a+b \sin ^{-1}(c x)}{4 e \left (d+e x^2\right )^2}+\frac {\left (b c \left (2 c^2 d+e\right )\right ) \int \frac {1}{\sqrt {1-c^2 x^2} \left (d+e x^2\right )} \, dx}{8 d e \left (c^2 d+e\right )}\\ &=\frac {b c x \sqrt {1-c^2 x^2}}{8 d \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {a+b \sin ^{-1}(c x)}{4 e \left (d+e x^2\right )^2}+\frac {\left (b c \left (2 c^2 d+e\right )\right ) \text {Subst}\left (\int \frac {1}{d-\left (-c^2 d-e\right ) x^2} \, dx,x,\frac {x}{\sqrt {1-c^2 x^2}}\right )}{8 d e \left (c^2 d+e\right )}\\ &=\frac {b c x \sqrt {1-c^2 x^2}}{8 d \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {a+b \sin ^{-1}(c x)}{4 e \left (d+e x^2\right )^2}+\frac {b c \left (2 c^2 d+e\right ) \tan ^{-1}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{8 d^{3/2} e \left (c^2 d+e\right )^{3/2}}\\ \end {align*}

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Mathematica [A]
time = 0.39, size = 141, normalized size = 1.06 \begin {gather*} \frac {1}{8} \left (\frac {-\frac {2 a}{e}+\frac {b c x \sqrt {1-c^2 x^2} \left (d+e x^2\right )}{d \left (c^2 d+e\right )}}{\left (d+e x^2\right )^2}-\frac {2 b \text {ArcSin}(c x)}{e \left (d+e x^2\right )^2}+\frac {b c \left (2 c^2 d+e\right ) \text {ArcTan}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{d^{3/2} e \left (c^2 d+e\right )^{3/2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x*(a + b*ArcSin[c*x]))/(d + e*x^2)^3,x]

[Out]

(((-2*a)/e + (b*c*x*Sqrt[1 - c^2*x^2]*(d + e*x^2))/(d*(c^2*d + e)))/(d + e*x^2)^2 - (2*b*ArcSin[c*x])/(e*(d +
e*x^2)^2) + (b*c*(2*c^2*d + e)*ArcTan[(Sqrt[c^2*d + e]*x)/(Sqrt[d]*Sqrt[1 - c^2*x^2])])/(d^(3/2)*e*(c^2*d + e)
^(3/2)))/8

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1020\) vs. \(2(118)=236\).
time = 0.12, size = 1021, normalized size = 7.68

method	result	size
derivativedivides	\(\frac {-\frac {a \,c^{6}}{4 e \left (c^{2} e \,x^{2}+c^{2} d \right )^{2}}-\frac {b \,c^{6} \arcsin \left (c x \right )}{4 \left (c^{2} e \,x^{2}+c^{2} d \right )^{2} e}+\frac {b \,c^{4} \sqrt {-\left (c x +\frac {\sqrt {-c^{2} e d}}{e}\right )^{2}+\frac {2 \sqrt {-c^{2} e d}\, \left (c x +\frac {\sqrt {-c^{2} e d}}{e}\right )}{e}+\frac {c^{2} d +e}{e}}}{16 e d \left (c^{2} d +e \right ) \left (c x +\frac {\sqrt {-c^{2} e d}}{e}\right )}-\frac {b \,c^{4} \sqrt {-c^{2} e d}\, \ln \left (\frac {\frac {2 c^{2} d +2 e}{e}+\frac {2 \sqrt {-c^{2} e d}\, \left (c x +\frac {\sqrt {-c^{2} e d}}{e}\right )}{e}+2 \sqrt {\frac {c^{2} d +e}{e}}\, \sqrt {-\left (c x +\frac {\sqrt {-c^{2} e d}}{e}\right )^{2}+\frac {2 \sqrt {-c^{2} e d}\, \left (c x +\frac {\sqrt {-c^{2} e d}}{e}\right )}{e}+\frac {c^{2} d +e}{e}}}{c x +\frac {\sqrt {-c^{2} e d}}{e}}\right )}{16 e^{2} d \left (c^{2} d +e \right ) \sqrt {\frac {c^{2} d +e}{e}}}-\frac {b \,c^{4} \ln \left (\frac {\frac {2 c^{2} d +2 e}{e}-\frac {2 \sqrt {-c^{2} e d}\, \left (c x -\frac {\sqrt {-c^{2} e d}}{e}\right )}{e}+2 \sqrt {\frac {c^{2} d +e}{e}}\, \sqrt {-\left (c x -\frac {\sqrt {-c^{2} e d}}{e}\right )^{2}-\frac {2 \sqrt {-c^{2} e d}\, \left (c x -\frac {\sqrt {-c^{2} e d}}{e}\right )}{e}+\frac {c^{2} d +e}{e}}}{c x -\frac {\sqrt {-c^{2} e d}}{e}}\right )}{16 e d \sqrt {-c^{2} e d}\, \sqrt {\frac {c^{2} d +e}{e}}}+\frac {b \,c^{4} \sqrt {-\left (c x -\frac {\sqrt {-c^{2} e d}}{e}\right )^{2}-\frac {2 \sqrt {-c^{2} e d}\, \left (c x -\frac {\sqrt {-c^{2} e d}}{e}\right )}{e}+\frac {c^{2} d +e}{e}}}{16 e d \left (c^{2} d +e \right ) \left (c x -\frac {\sqrt {-c^{2} e d}}{e}\right )}+\frac {b \,c^{4} \sqrt {-c^{2} e d}\, \ln \left (\frac {\frac {2 c^{2} d +2 e}{e}-\frac {2 \sqrt {-c^{2} e d}\, \left (c x -\frac {\sqrt {-c^{2} e d}}{e}\right )}{e}+2 \sqrt {\frac {c^{2} d +e}{e}}\, \sqrt {-\left (c x -\frac {\sqrt {-c^{2} e d}}{e}\right )^{2}-\frac {2 \sqrt {-c^{2} e d}\, \left (c x -\frac {\sqrt {-c^{2} e d}}{e}\right )}{e}+\frac {c^{2} d +e}{e}}}{c x -\frac {\sqrt {-c^{2} e d}}{e}}\right )}{16 e^{2} d \left (c^{2} d +e \right ) \sqrt {\frac {c^{2} d +e}{e}}}+\frac {b \,c^{4} \ln \left (\frac {\frac {2 c^{2} d +2 e}{e}+\frac {2 \sqrt {-c^{2} e d}\, \left (c x +\frac {\sqrt {-c^{2} e d}}{e}\right )}{e}+2 \sqrt {\frac {c^{2} d +e}{e}}\, \sqrt {-\left (c x +\frac {\sqrt {-c^{2} e d}}{e}\right )^{2}+\frac {2 \sqrt {-c^{2} e d}\, \left (c x +\frac {\sqrt {-c^{2} e d}}{e}\right )}{e}+\frac {c^{2} d +e}{e}}}{c x +\frac {\sqrt {-c^{2} e d}}{e}}\right )}{16 e d \sqrt {-c^{2} e d}\, \sqrt {\frac {c^{2} d +e}{e}}}}{c^{2}}\)	\(1021\)
default	\(\frac {-\frac {a \,c^{6}}{4 e \left (c^{2} e \,x^{2}+c^{2} d \right )^{2}}-\frac {b \,c^{6} \arcsin \left (c x \right )}{4 \left (c^{2} e \,x^{2}+c^{2} d \right )^{2} e}+\frac {b \,c^{4} \sqrt {-\left (c x +\frac {\sqrt {-c^{2} e d}}{e}\right )^{2}+\frac {2 \sqrt {-c^{2} e d}\, \left (c x +\frac {\sqrt {-c^{2} e d}}{e}\right )}{e}+\frac {c^{2} d +e}{e}}}{16 e d \left (c^{2} d +e \right ) \left (c x +\frac {\sqrt {-c^{2} e d}}{e}\right )}-\frac {b \,c^{4} \sqrt {-c^{2} e d}\, \ln \left (\frac {\frac {2 c^{2} d +2 e}{e}+\frac {2 \sqrt {-c^{2} e d}\, \left (c x +\frac {\sqrt {-c^{2} e d}}{e}\right )}{e}+2 \sqrt {\frac {c^{2} d +e}{e}}\, \sqrt {-\left (c x +\frac {\sqrt {-c^{2} e d}}{e}\right )^{2}+\frac {2 \sqrt {-c^{2} e d}\, \left (c x +\frac {\sqrt {-c^{2} e d}}{e}\right )}{e}+\frac {c^{2} d +e}{e}}}{c x +\frac {\sqrt {-c^{2} e d}}{e}}\right )}{16 e^{2} d \left (c^{2} d +e \right ) \sqrt {\frac {c^{2} d +e}{e}}}-\frac {b \,c^{4} \ln \left (\frac {\frac {2 c^{2} d +2 e}{e}-\frac {2 \sqrt {-c^{2} e d}\, \left (c x -\frac {\sqrt {-c^{2} e d}}{e}\right )}{e}+2 \sqrt {\frac {c^{2} d +e}{e}}\, \sqrt {-\left (c x -\frac {\sqrt {-c^{2} e d}}{e}\right )^{2}-\frac {2 \sqrt {-c^{2} e d}\, \left (c x -\frac {\sqrt {-c^{2} e d}}{e}\right )}{e}+\frac {c^{2} d +e}{e}}}{c x -\frac {\sqrt {-c^{2} e d}}{e}}\right )}{16 e d \sqrt {-c^{2} e d}\, \sqrt {\frac {c^{2} d +e}{e}}}+\frac {b \,c^{4} \sqrt {-\left (c x -\frac {\sqrt {-c^{2} e d}}{e}\right )^{2}-\frac {2 \sqrt {-c^{2} e d}\, \left (c x -\frac {\sqrt {-c^{2} e d}}{e}\right )}{e}+\frac {c^{2} d +e}{e}}}{16 e d \left (c^{2} d +e \right ) \left (c x -\frac {\sqrt {-c^{2} e d}}{e}\right )}+\frac {b \,c^{4} \sqrt {-c^{2} e d}\, \ln \left (\frac {\frac {2 c^{2} d +2 e}{e}-\frac {2 \sqrt {-c^{2} e d}\, \left (c x -\frac {\sqrt {-c^{2} e d}}{e}\right )}{e}+2 \sqrt {\frac {c^{2} d +e}{e}}\, \sqrt {-\left (c x -\frac {\sqrt {-c^{2} e d}}{e}\right )^{2}-\frac {2 \sqrt {-c^{2} e d}\, \left (c x -\frac {\sqrt {-c^{2} e d}}{e}\right )}{e}+\frac {c^{2} d +e}{e}}}{c x -\frac {\sqrt {-c^{2} e d}}{e}}\right )}{16 e^{2} d \left (c^{2} d +e \right ) \sqrt {\frac {c^{2} d +e}{e}}}+\frac {b \,c^{4} \ln \left (\frac {\frac {2 c^{2} d +2 e}{e}+\frac {2 \sqrt {-c^{2} e d}\, \left (c x +\frac {\sqrt {-c^{2} e d}}{e}\right )}{e}+2 \sqrt {\frac {c^{2} d +e}{e}}\, \sqrt {-\left (c x +\frac {\sqrt {-c^{2} e d}}{e}\right )^{2}+\frac {2 \sqrt {-c^{2} e d}\, \left (c x +\frac {\sqrt {-c^{2} e d}}{e}\right )}{e}+\frac {c^{2} d +e}{e}}}{c x +\frac {\sqrt {-c^{2} e d}}{e}}\right )}{16 e d \sqrt {-c^{2} e d}\, \sqrt {\frac {c^{2} d +e}{e}}}}{c^{2}}\)	\(1021\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a+b*arcsin(c*x))/(e*x^2+d)^3,x,method=_RETURNVERBOSE)

[Out]

1/c^2*(-1/4*a*c^6/e/(c^2*e*x^2+c^2*d)^2-1/4*b*c^6/(c^2*e*x^2+c^2*d)^2*arcsin(c*x)/e+1/16*b*c^4/e/d/(c^2*d+e)/(
c*x+(-c^2*e*d)^(1/2)/e)*(-(c*x+(-c^2*e*d)^(1/2)/e)^2+2*(-c^2*e*d)^(1/2)/e*(c*x+(-c^2*e*d)^(1/2)/e)+(c^2*d+e)/e
)^(1/2)-1/16*b*c^4/e^2/d*(-c^2*e*d)^(1/2)/(c^2*d+e)/((c^2*d+e)/e)^(1/2)*ln((2*(c^2*d+e)/e+2*(-c^2*e*d)^(1/2)/e
*(c*x+(-c^2*e*d)^(1/2)/e)+2*((c^2*d+e)/e)^(1/2)*(-(c*x+(-c^2*e*d)^(1/2)/e)^2+2*(-c^2*e*d)^(1/2)/e*(c*x+(-c^2*e
*d)^(1/2)/e)+(c^2*d+e)/e)^(1/2))/(c*x+(-c^2*e*d)^(1/2)/e))-1/16*b*c^4/e/d/(-c^2*e*d)^(1/2)/((c^2*d+e)/e)^(1/2)
*ln((2*(c^2*d+e)/e-2*(-c^2*e*d)^(1/2)/e*(c*x-(-c^2*e*d)^(1/2)/e)+2*((c^2*d+e)/e)^(1/2)*(-(c*x-(-c^2*e*d)^(1/2)
/e)^2-2*(-c^2*e*d)^(1/2)/e*(c*x-(-c^2*e*d)^(1/2)/e)+(c^2*d+e)/e)^(1/2))/(c*x-(-c^2*e*d)^(1/2)/e))+1/16*b*c^4/e
/d/(c^2*d+e)/(c*x-(-c^2*e*d)^(1/2)/e)*(-(c*x-(-c^2*e*d)^(1/2)/e)^2-2*(-c^2*e*d)^(1/2)/e*(c*x-(-c^2*e*d)^(1/2)/
e)+(c^2*d+e)/e)^(1/2)+1/16*b*c^4/e^2/d*(-c^2*e*d)^(1/2)/(c^2*d+e)/((c^2*d+e)/e)^(1/2)*ln((2*(c^2*d+e)/e-2*(-c^
2*e*d)^(1/2)/e*(c*x-(-c^2*e*d)^(1/2)/e)+2*((c^2*d+e)/e)^(1/2)*(-(c*x-(-c^2*e*d)^(1/2)/e)^2-2*(-c^2*e*d)^(1/2)/
e*(c*x-(-c^2*e*d)^(1/2)/e)+(c^2*d+e)/e)^(1/2))/(c*x-(-c^2*e*d)^(1/2)/e))+1/16*b*c^4/e/d/(-c^2*e*d)^(1/2)/((c^2
*d+e)/e)^(1/2)*ln((2*(c^2*d+e)/e+2*(-c^2*e*d)^(1/2)/e*(c*x+(-c^2*e*d)^(1/2)/e)+2*((c^2*d+e)/e)^(1/2)*(-(c*x+(-
c^2*e*d)^(1/2)/e)^2+2*(-c^2*e*d)^(1/2)/e*(c*x+(-c^2*e*d)^(1/2)/e)+(c^2*d+e)/e)^(1/2))/(c*x+(-c^2*e*d)^(1/2)/e)
))

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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(x*(a+b*arcsin(c*x))/(e*x^2+d)^3,x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 376 vs. \(2 (119) = 238\).
time = 8.01, size = 793, normalized size = 5.96 \begin {gather*} \left [-\frac {8 \, a c^{4} d^{4} + 16 \, a c^{2} d^{3} e + 8 \, a d^{2} e^{2} + {\left (2 \, b c^{3} d^{3} + b c x^{4} e^{3} + 2 \, {\left (b c^{3} d x^{4} + b c d x^{2}\right )} e^{2} + {\left (4 \, b c^{3} d^{2} x^{2} + b c d^{2}\right )} e\right )} \sqrt {-c^{2} d^{2} - d e} \log \left (\frac {8 \, c^{4} d^{2} x^{4} - 8 \, c^{2} d^{2} x^{2} + x^{4} e^{2} - 4 \, {\left (2 \, c^{2} d x^{3} + x^{3} e - d x\right )} \sqrt {-c^{2} d^{2} - d e} \sqrt {-c^{2} x^{2} + 1} + d^{2} + 2 \, {\left (4 \, c^{2} d x^{4} - 3 \, d x^{2}\right )} e}{x^{4} e^{2} + 2 \, d x^{2} e + d^{2}}\right ) + 8 \, {\left (b c^{4} d^{4} + 2 \, b c^{2} d^{3} e + b d^{2} e^{2}\right )} \arcsin \left (c x\right ) - 4 \, {\left (b c^{3} d^{3} x e + b c d x^{3} e^{3} + {\left (b c^{3} d^{2} x^{3} + b c d^{2} x\right )} e^{2}\right )} \sqrt {-c^{2} x^{2} + 1}}{32 \, {\left (c^{4} d^{6} e + d^{2} x^{4} e^{5} + 2 \, {\left (c^{2} d^{3} x^{4} + d^{3} x^{2}\right )} e^{4} + {\left (c^{4} d^{4} x^{4} + 4 \, c^{2} d^{4} x^{2} + d^{4}\right )} e^{3} + 2 \, {\left (c^{4} d^{5} x^{2} + c^{2} d^{5}\right )} e^{2}\right )}}, -\frac {4 \, a c^{4} d^{4} + 8 \, a c^{2} d^{3} e + 4 \, a d^{2} e^{2} + {\left (2 \, b c^{3} d^{3} + b c x^{4} e^{3} + 2 \, {\left (b c^{3} d x^{4} + b c d x^{2}\right )} e^{2} + {\left (4 \, b c^{3} d^{2} x^{2} + b c d^{2}\right )} e\right )} \sqrt {c^{2} d^{2} + d e} \arctan \left (\frac {{\left (2 \, c^{2} d x^{2} + x^{2} e - d\right )} \sqrt {c^{2} d^{2} + d e} \sqrt {-c^{2} x^{2} + 1}}{2 \, {\left (c^{4} d^{2} x^{3} - c^{2} d^{2} x + {\left (c^{2} d x^{3} - d x\right )} e\right )}}\right ) + 4 \, {\left (b c^{4} d^{4} + 2 \, b c^{2} d^{3} e + b d^{2} e^{2}\right )} \arcsin \left (c x\right ) - 2 \, {\left (b c^{3} d^{3} x e + b c d x^{3} e^{3} + {\left (b c^{3} d^{2} x^{3} + b c d^{2} x\right )} e^{2}\right )} \sqrt {-c^{2} x^{2} + 1}}{16 \, {\left (c^{4} d^{6} e + d^{2} x^{4} e^{5} + 2 \, {\left (c^{2} d^{3} x^{4} + d^{3} x^{2}\right )} e^{4} + {\left (c^{4} d^{4} x^{4} + 4 \, c^{2} d^{4} x^{2} + d^{4}\right )} e^{3} + 2 \, {\left (c^{4} d^{5} x^{2} + c^{2} d^{5}\right )} e^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*arcsin(c*x))/(e*x^2+d)^3,x, algorithm="fricas")

[Out]

[-1/32*(8*a*c^4*d^4 + 16*a*c^2*d^3*e + 8*a*d^2*e^2 + (2*b*c^3*d^3 + b*c*x^4*e^3 + 2*(b*c^3*d*x^4 + b*c*d*x^2)*
e^2 + (4*b*c^3*d^2*x^2 + b*c*d^2)*e)*sqrt(-c^2*d^2 - d*e)*log((8*c^4*d^2*x^4 - 8*c^2*d^2*x^2 + x^4*e^2 - 4*(2*
c^2*d*x^3 + x^3*e - d*x)*sqrt(-c^2*d^2 - d*e)*sqrt(-c^2*x^2 + 1) + d^2 + 2*(4*c^2*d*x^4 - 3*d*x^2)*e)/(x^4*e^2
+ 2*d*x^2*e + d^2)) + 8*(b*c^4*d^4 + 2*b*c^2*d^3*e + b*d^2*e^2)*arcsin(c*x) - 4*(b*c^3*d^3*x*e + b*c*d*x^3*e^
3 + (b*c^3*d^2*x^3 + b*c*d^2*x)*e^2)*sqrt(-c^2*x^2 + 1))/(c^4*d^6*e + d^2*x^4*e^5 + 2*(c^2*d^3*x^4 + d^3*x^2)*
e^4 + (c^4*d^4*x^4 + 4*c^2*d^4*x^2 + d^4)*e^3 + 2*(c^4*d^5*x^2 + c^2*d^5)*e^2), -1/16*(4*a*c^4*d^4 + 8*a*c^2*d
^3*e + 4*a*d^2*e^2 + (2*b*c^3*d^3 + b*c*x^4*e^3 + 2*(b*c^3*d*x^4 + b*c*d*x^2)*e^2 + (4*b*c^3*d^2*x^2 + b*c*d^2
)*e)*sqrt(c^2*d^2 + d*e)*arctan(1/2*(2*c^2*d*x^2 + x^2*e - d)*sqrt(c^2*d^2 + d*e)*sqrt(-c^2*x^2 + 1)/(c^4*d^2*
x^3 - c^2*d^2*x + (c^2*d*x^3 - d*x)*e)) + 4*(b*c^4*d^4 + 2*b*c^2*d^3*e + b*d^2*e^2)*arcsin(c*x) - 2*(b*c^3*d^3
*x*e + b*c*d*x^3*e^3 + (b*c^3*d^2*x^3 + b*c*d^2*x)*e^2)*sqrt(-c^2*x^2 + 1))/(c^4*d^6*e + d^2*x^4*e^5 + 2*(c^2*
d^3*x^4 + d^3*x^2)*e^4 + (c^4*d^4*x^4 + 4*c^2*d^4*x^2 + d^4)*e^3 + 2*(c^4*d^5*x^2 + c^2*d^5)*e^2)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \left (a + b \operatorname {asin}{\left (c x \right )}\right )}{\left (d + e x^{2}\right )^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*asin(c*x))/(e*x**2+d)**3,x)

[Out]

Integral(x*(a + b*asin(c*x))/(d + e*x**2)**3, 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(x*(a+b*arcsin(c*x))/(e*x^2+d)^3,x, algorithm="giac")

[Out]

integrate((b*arcsin(c*x) + a)*x/(e*x^2 + d)^3, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}{{\left (e\,x^2+d\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*(a + b*asin(c*x)))/(d + e*x^2)^3,x)

[Out]

int((x*(a + b*asin(c*x)))/(d + e*x^2)^3, x)

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