3.6.44 \(\int \frac {x^5}{a c+b c x^2+d \sqrt {a+b x^2}} \, dx\) [544]

Optimal. Leaf size=134 \[ -\frac {\left (2 a c^2-d^2\right ) x^2}{2 b^2 c^3}+\frac {d \left (2 a c^2-d^2\right ) \sqrt {a+b x^2}}{b^3 c^4}-\frac {d \left (a+b x^2\right )^{3/2}}{3 b^3 c^2}+\frac {\left (a+b x^2\right )^2}{4 b^3 c}+\frac {\left (a c^2-d^2\right )^2 \log \left (d+c \sqrt {a+b x^2}\right )}{b^3 c^5} \]

[Out]

-1/2*(2*a*c^2-d^2)*x^2/b^2/c^3-1/3*d*(b*x^2+a)^(3/2)/b^3/c^2+1/4*(b*x^2+a)^2/b^3/c+(a*c^2-d^2)^2*ln(d+c*(b*x^2
+a)^(1/2))/b^3/c^5+d*(2*a*c^2-d^2)*(b*x^2+a)^(1/2)/b^3/c^4

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Rubi [A]
time = 0.25, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2186, 711} \begin {gather*} -\frac {d \left (a+b x^2\right )^{3/2}}{3 b^3 c^2}+\frac {\left (a c^2-d^2\right )^2 \log \left (c \sqrt {a+b x^2}+d\right )}{b^3 c^5}+\frac {d \sqrt {a+b x^2} \left (2 a c^2-d^2\right )}{b^3 c^4}+\frac {\left (a+b x^2\right )^2}{4 b^3 c}-\frac {x^2 \left (2 a c^2-d^2\right )}{2 b^2 c^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^5/(a*c + b*c*x^2 + d*Sqrt[a + b*x^2]),x]

[Out]

-1/2*((2*a*c^2 - d^2)*x^2)/(b^2*c^3) + (d*(2*a*c^2 - d^2)*Sqrt[a + b*x^2])/(b^3*c^4) - (d*(a + b*x^2)^(3/2))/(
3*b^3*c^2) + (a + b*x^2)^2/(4*b^3*c) + ((a*c^2 - d^2)^2*Log[d + c*Sqrt[a + b*x^2]])/(b^3*c^5)

Rule 711

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

Rule 2186

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

Rubi steps

\begin {align*} \int \frac {x^5}{a c+b c x^2+d \sqrt {a+b x^2}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^2}{a c+b c x+d \sqrt {a+b x}} \, dx,x,x^2\right )\\ &=\frac {\text {Subst}\left (\int \frac {\left (a-x^2\right )^2}{d+c x} \, dx,x,\sqrt {a+b x^2}\right )}{b^3}\\ &=\frac {\text {Subst}\left (\int \left (\frac {2 a c^2 d-d^3}{c^4}-\frac {\left (2 a c^2-d^2\right ) x}{c^3}-\frac {d x^2}{c^2}+\frac {x^3}{c}+\frac {\left (a c^2-d^2\right )^2}{c^4 (d+c x)}\right ) \, dx,x,\sqrt {a+b x^2}\right )}{b^3}\\ &=-\frac {\left (2 a c^2-d^2\right ) x^2}{2 b^2 c^3}+\frac {d \left (2 a c^2-d^2\right ) \sqrt {a+b x^2}}{b^3 c^4}-\frac {d \left (a+b x^2\right )^{3/2}}{3 b^3 c^2}+\frac {\left (a+b x^2\right )^2}{4 b^3 c}+\frac {\left (a c^2-d^2\right )^2 \log \left (d+c \sqrt {a+b x^2}\right )}{b^3 c^5}\\ \end {align*}

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Mathematica [A]
time = 0.16, size = 125, normalized size = 0.93 \begin {gather*} \frac {-4 c d \sqrt {a+b x^2} \left (-5 a c^2+3 d^2+b c^2 x^2\right )+3 c^2 \left (-3 a^2 c^2+2 a \left (d^2-b c^2 x^2\right )+b x^2 \left (2 d^2+b c^2 x^2\right )\right )+12 \left (-a c^2+d^2\right )^2 \log \left (d+c \sqrt {a+b x^2}\right )}{12 b^3 c^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^5/(a*c + b*c*x^2 + d*Sqrt[a + b*x^2]),x]

[Out]

(-4*c*d*Sqrt[a + b*x^2]*(-5*a*c^2 + 3*d^2 + b*c^2*x^2) + 3*c^2*(-3*a^2*c^2 + 2*a*(d^2 - b*c^2*x^2) + b*x^2*(2*
d^2 + b*c^2*x^2)) + 12*(-(a*c^2) + d^2)^2*Log[d + c*Sqrt[a + b*x^2]])/(12*b^3*c^5)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(4946\) vs. \(2(122)=244\).
time = 0.06, size = 4947, normalized size = 36.92

method	result	size
default	\(\text {Expression too large to display}\)	\(4947\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(a*c+b*c*x^2+d*(b*x^2+a)^(1/2)),x,method=_RETURNVERBOSE)

[Out]

-1/2/b^2/((-a*b)^(1/2)*c^2+(-(a*c^2-d^2)*b*c^2)^(1/2))/((-a*b)^(1/2)*c^2-(-(a*c^2-d^2)*b*c^2)^(1/2))/c^4*d^7/(
d^2/c^2)^(1/2)*ln((2*d^2/c^2+2/c^2*(-(a*c^2-d^2)*b*c^2)^(1/2)*(x-(-(a*c^2-d^2)*b*c^2)^(1/2)/b/c^2)+2*(d^2/c^2)
^(1/2)*(b*(x-(-(a*c^2-d^2)*b*c^2)^(1/2)/b/c^2)^2+2/c^2*(-(a*c^2-d^2)*b*c^2)^(1/2)*(x-(-(a*c^2-d^2)*b*c^2)^(1/2
)/b/c^2)+d^2/c^2)^(1/2))/(x-(-(a*c^2-d^2)*b*c^2)^(1/2)/b/c^2))-1/2*d/b^2*c^2*a^2/((-a*b)^(1/2)*c^2+(-(a*c^2-d^
2)*b*c^2)^(1/2))/((-a*b)^(1/2)*c^2-(-(a*c^2-d^2)*b*c^2)^(1/2))*(b*(x-1/b*(-a*b)^(1/2))^2+2*(-a*b)^(1/2)*(x-1/b
*(-a*b)^(1/2)))^(1/2)-1/2*d/b^2*c^2*a^2/((-a*b)^(1/2)*c^2+(-(a*c^2-d^2)*b*c^2)^(1/2))/((-a*b)^(1/2)*c^2-(-(a*c
^2-d^2)*b*c^2)^(1/2))*(b*(x+1/b*(-a*b)^(1/2))^2-2*(-a*b)^(1/2)*(x+1/b*(-a*b)^(1/2)))^(1/2)+1/2*d/b^2/((-a*b)^(
1/2)*c^2+(-(a*c^2-d^2)*b*c^2)^(1/2))/((-a*b)^(1/2)*c^2-(-(a*c^2-d^2)*b*c^2)^(1/2))*c^2*(b*(x+(-(a*c^2-d^2)*b*c
^2)^(1/2)/b/c^2)^2-2/c^2*(-(a*c^2-d^2)*b*c^2)^(1/2)*(x+(-(a*c^2-d^2)*b*c^2)^(1/2)/b/c^2)+d^2/c^2)^(1/2)*a^2-1/
2*d/b^(5/2)/((-a*b)^(1/2)*c^2+(-(a*c^2-d^2)*b*c^2)^(1/2))/((-a*b)^(1/2)*c^2-(-(a*c^2-d^2)*b*c^2)^(1/2))*(-(a*c
^2-d^2)*b*c^2)^(1/2)*ln((-1/c^2*(-(a*c^2-d^2)*b*c^2)^(1/2)+(x+(-(a*c^2-d^2)*b*c^2)^(1/2)/b/c^2)*b)/b^(1/2)+(b*
(x+(-(a*c^2-d^2)*b*c^2)^(1/2)/b/c^2)^2-2/c^2*(-(a*c^2-d^2)*b*c^2)^(1/2)*(x+(-(a*c^2-d^2)*b*c^2)^(1/2)/b/c^2)+d
^2/c^2)^(1/2))*a^2-1/2/b^(5/2)/((-a*b)^(1/2)*c^2+(-(a*c^2-d^2)*b*c^2)^(1/2))/((-a*b)^(1/2)*c^2-(-(a*c^2-d^2)*b
*c^2)^(1/2))/c^4*(-(a*c^2-d^2)*b*c^2)^(1/2)*ln((-1/c^2*(-(a*c^2-d^2)*b*c^2)^(1/2)+(x+(-(a*c^2-d^2)*b*c^2)^(1/2
)/b/c^2)*b)/b^(1/2)+(b*(x+(-(a*c^2-d^2)*b*c^2)^(1/2)/b/c^2)^2-2/c^2*(-(a*c^2-d^2)*b*c^2)^(1/2)*(x+(-(a*c^2-d^2
)*b*c^2)^(1/2)/b/c^2)+d^2/c^2)^(1/2))*d^5-1/2/b^2/((-a*b)^(1/2)*c^2+(-(a*c^2-d^2)*b*c^2)^(1/2))/((-a*b)^(1/2)*
c^2-(-(a*c^2-d^2)*b*c^2)^(1/2))*d^3/(d^2/c^2)^(1/2)*ln((2*d^2/c^2-2/c^2*(-(a*c^2-d^2)*b*c^2)^(1/2)*(x+(-(a*c^2
-d^2)*b*c^2)^(1/2)/b/c^2)+2*(d^2/c^2)^(1/2)*(b*(x+(-(a*c^2-d^2)*b*c^2)^(1/2)/b/c^2)^2-2/c^2*(-(a*c^2-d^2)*b*c^
2)^(1/2)*(x+(-(a*c^2-d^2)*b*c^2)^(1/2)/b/c^2)+d^2/c^2)^(1/2))/(x+(-(a*c^2-d^2)*b*c^2)^(1/2)/b/c^2))*a^2-1/2/b^
2/((-a*b)^(1/2)*c^2+(-(a*c^2-d^2)*b*c^2)^(1/2))/((-a*b)^(1/2)*c^2-(-(a*c^2-d^2)*b*c^2)^(1/2))/c^4*d^7/(d^2/c^2
)^(1/2)*ln((2*d^2/c^2-2/c^2*(-(a*c^2-d^2)*b*c^2)^(1/2)*(x+(-(a*c^2-d^2)*b*c^2)^(1/2)/b/c^2)+2*(d^2/c^2)^(1/2)*
(b*(x+(-(a*c^2-d^2)*b*c^2)^(1/2)/b/c^2)^2-2/c^2*(-(a*c^2-d^2)*b*c^2)^(1/2)*(x+(-(a*c^2-d^2)*b*c^2)^(1/2)/b/c^2
)+d^2/c^2)^(1/2))/(x+(-(a*c^2-d^2)*b*c^2)^(1/2)/b/c^2))-1/2*d/b^(5/2)*c^2*a^2/((-a*b)^(1/2)*c^2+(-(a*c^2-d^2)*
b*c^2)^(1/2))/((-a*b)^(1/2)*c^2-(-(a*c^2-d^2)*b*c^2)^(1/2))*(-a*b)^(1/2)*ln(((x-1/b*(-a*b)^(1/2))*b+(-a*b)^(1/
2))/b^(1/2)+(b*(x-1/b*(-a*b)^(1/2))^2+2*(-a*b)^(1/2)*(x-1/b*(-a*b)^(1/2)))^(1/2))+1/2*d/b^(5/2)*c^2*a^2/((-a*b
)^(1/2)*c^2+(-(a*c^2-d^2)*b*c^2)^(1/2))/((-a*b)^(1/2)*c^2-(-(a*c^2-d^2)*b*c^2)^(1/2))*(-a*b)^(1/2)*ln(((x+1/b*
(-a*b)^(1/2))*b-(-a*b)^(1/2))/b^(1/2)+(b*(x+1/b*(-a*b)^(1/2))^2-2*(-a*b)^(1/2)*(x+1/b*(-a*b)^(1/2)))^(1/2))+1/
b^(5/2)/((-a*b)^(1/2)*c^2+(-(a*c^2-d^2)*b*c^2)^(1/2))/((-a*b)^(1/2)*c^2-(-(a*c^2-d^2)*b*c^2)^(1/2))/c^2*(-(a*c
^2-d^2)*b*c^2)^(1/2)*ln((-1/c^2*(-(a*c^2-d^2)*b*c^2)^(1/2)+(x+(-(a*c^2-d^2)*b*c^2)^(1/2)/b/c^2)*b)/b^(1/2)+(b*
(x+(-(a*c^2-d^2)*b*c^2)^(1/2)/b/c^2)^2-2/c^2*(-(a*c^2-d^2)*b*c^2)^(1/2)*(x+(-(a*c^2-d^2)*b*c^2)^(1/2)/b/c^2)+d
^2/c^2)^(1/2))*a*d^3+1/b^2/((-a*b)^(1/2)*c^2+(-(a*c^2-d^2)*b*c^2)^(1/2))/((-a*b)^(1/2)*c^2-(-(a*c^2-d^2)*b*c^2
)^(1/2))/c^2*d^5/(d^2/c^2)^(1/2)*ln((2*d^2/c^2-2/c^2*(-(a*c^2-d^2)*b*c^2)^(1/2)*(x+(-(a*c^2-d^2)*b*c^2)^(1/2)/
b/c^2)+2*(d^2/c^2)^(1/2)*(b*(x+(-(a*c^2-d^2)*b*c^2)^(1/2)/b/c^2)^2-2/c^2*(-(a*c^2-d^2)*b*c^2)^(1/2)*(x+(-(a*c^
2-d^2)*b*c^2)^(1/2)/b/c^2)+d^2/c^2)^(1/2))/(x+(-(a*c^2-d^2)*b*c^2)^(1/2)/b/c^2))*a-1/b^(5/2)/((-a*b)^(1/2)*c^2
+(-(a*c^2-d^2)*b*c^2)^(1/2))/((-a*b)^(1/2)*c^2-(-(a*c^2-d^2)*b*c^2)^(1/2))/c^2*(-(a*c^2-d^2)*b*c^2)^(1/2)*ln((
1/c^2*(-(a*c^2-d^2)*b*c^2)^(1/2)+(x-(-(a*c^2-d^2)*b*c^2)^(1/2)/b/c^2)*b)/b^(1/2)+(b*(x-(-(a*c^2-d^2)*b*c^2)^(1
/2)/b/c^2)^2+2/c^2*(-(a*c^2-d^2)*b*c^2)^(1/2)*(x-(-(a*c^2-d^2)*b*c^2)^(1/2)/b/c^2)+d^2/c^2)^(1/2))*a*d^3+1/b^2
/((-a*b)^(1/2)*c^2+(-(a*c^2-d^2)*b*c^2)^(1/2))/((-a*b)^(1/2)*c^2-(-(a*c^2-d^2)*b*c^2)^(1/2))/c^2*d^5/(d^2/c^2)
^(1/2)*ln((2*d^2/c^2+2/c^2*(-(a*c^2-d^2)*b*c^2)^(1/2)*(x-(-(a*c^2-d^2)*b*c^2)^(1/2)/b/c^2)+2*(d^2/c^2)^(1/2)*(
b*(x-(-(a*c^2-d^2)*b*c^2)^(1/2)/b/c^2)^2+2/c^2*(-(a*c^2-d^2)*b*c^2)^(1/2)*(x-(-(a*c^2-d^2)*b*c^2)^(1/2)/b/c^2)
+d^2/c^2)^(1/2))/(x-(-(a*c^2-d^2)*b*c^2)^(1/2)/b/c^2))*a+1/2/b^2/c^3*x^2*d^2+1/2/b^3/c^5*d^4*ln(b*c^2*x^2+a*c^
2-d^2)-1/b^2/((-a*b)^(1/2)*c^2+(-(a*c^2-d^2)*b*c^2)^(1/2))/((-a*b)^(1/2)*c^2-(-(a*c^2-d^2)*b*c^2)^(1/2))*(b*(x
-(-(a*c^2-d^2)*b*c^2)^(1/2)/b/c^2)^2+2/c^2*(-(a*c^2-d^2)*b*c^2)^(1/2)*(x-(-(a*c^2-d^2)*b*c^2)^(1/2)/b/c^2)+d^2
/c^2)^(1/2)*a*d^3+1/2/b^2/((-a*b)^(1/2)*c^2+(-(a*c^2-d^2)*b*c^2)^(1/2))/((-a*b)^(1/2)*c^2-(-(a*c^2-d^2)*b*c^2)
^(1/2))/c^2*(b*(x-(-(a*c^2-d^2)*b*c^2)^(1/2)/b/c^2)^2+2/c^2*(-(a*c^2-d^2)*b*c^2)^(1/2)*(x-(-(a*c^2-d^2)*b*c^2)
^(1/2)/b/c^2)+d^2/c^2)^(1/2)*d^5-1/b^2/((-a*b)^...

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Maxima [A]
time = 0.27, size = 125, normalized size = 0.93 \begin {gather*} \frac {\frac {3 \, {\left (b x^{2} + a\right )}^{2} c^{3} - 4 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} c^{2} d - 6 \, {\left (2 \, a c^{3} - c d^{2}\right )} {\left (b x^{2} + a\right )} + 12 \, {\left (2 \, a c^{2} d - d^{3}\right )} \sqrt {b x^{2} + a}}{c^{4}} + \frac {12 \, {\left (a^{2} c^{4} - 2 \, a c^{2} d^{2} + d^{4}\right )} \log \left (\sqrt {b x^{2} + a} c + d\right )}{c^{5}}}{12 \, b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(a*c+b*c*x^2+d*(b*x^2+a)^(1/2)),x, algorithm="maxima")

[Out]

1/12*((3*(b*x^2 + a)^2*c^3 - 4*(b*x^2 + a)^(3/2)*c^2*d - 6*(2*a*c^3 - c*d^2)*(b*x^2 + a) + 12*(2*a*c^2*d - d^3
)*sqrt(b*x^2 + a))/c^4 + 12*(a^2*c^4 - 2*a*c^2*d^2 + d^4)*log(sqrt(b*x^2 + a)*c + d)/c^5)/b^3

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Fricas [A]
time = 0.38, size = 233, normalized size = 1.74 \begin {gather*} \frac {3 \, b^{2} c^{4} x^{4} - 6 \, {\left (a b c^{4} - b c^{2} d^{2}\right )} x^{2} + 6 \, {\left (a^{2} c^{4} - 2 \, a c^{2} d^{2} + d^{4}\right )} \log \left (b c^{2} x^{2} + a c^{2} - d^{2}\right ) + 3 \, {\left (a^{2} c^{4} - 2 \, a c^{2} d^{2} + d^{4}\right )} \log \left (-\frac {b c^{2} x^{2} + a c^{2} + 2 \, \sqrt {b x^{2} + a} c d + d^{2}}{x^{2}}\right ) - 3 \, {\left (a^{2} c^{4} - 2 \, a c^{2} d^{2} + d^{4}\right )} \log \left (-\frac {b c^{2} x^{2} + a c^{2} - 2 \, \sqrt {b x^{2} + a} c d + d^{2}}{x^{2}}\right ) - 4 \, {\left (b c^{3} d x^{2} - 5 \, a c^{3} d + 3 \, c d^{3}\right )} \sqrt {b x^{2} + a}}{12 \, b^{3} c^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(a*c+b*c*x^2+d*(b*x^2+a)^(1/2)),x, algorithm="fricas")

[Out]

1/12*(3*b^2*c^4*x^4 - 6*(a*b*c^4 - b*c^2*d^2)*x^2 + 6*(a^2*c^4 - 2*a*c^2*d^2 + d^4)*log(b*c^2*x^2 + a*c^2 - d^
2) + 3*(a^2*c^4 - 2*a*c^2*d^2 + d^4)*log(-(b*c^2*x^2 + a*c^2 + 2*sqrt(b*x^2 + a)*c*d + d^2)/x^2) - 3*(a^2*c^4
- 2*a*c^2*d^2 + d^4)*log(-(b*c^2*x^2 + a*c^2 - 2*sqrt(b*x^2 + a)*c*d + d^2)/x^2) - 4*(b*c^3*d*x^2 - 5*a*c^3*d
+ 3*c*d^3)*sqrt(b*x^2 + a))/(b^3*c^5)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{5}}{a c + b c x^{2} + d \sqrt {a + b x^{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(a*c+b*c*x**2+d*(b*x**2+a)**(1/2)),x)

[Out]

Integral(x**5/(a*c + b*c*x**2 + d*sqrt(a + b*x**2)), x)

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Giac [A]
time = 4.00, size = 155, normalized size = 1.16 \begin {gather*} \frac {{\left (a^{2} c^{4} - 2 \, a c^{2} d^{2} + d^{4}\right )} \log \left ({\left | \sqrt {b x^{2} + a} c + d \right |}\right )}{b^{3} c^{5}} + \frac {3 \, {\left (b x^{2} + a\right )}^{2} b^{9} c^{3} - 12 \, {\left (b x^{2} + a\right )} a b^{9} c^{3} - 4 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{9} c^{2} d + 24 \, \sqrt {b x^{2} + a} a b^{9} c^{2} d + 6 \, {\left (b x^{2} + a\right )} b^{9} c d^{2} - 12 \, \sqrt {b x^{2} + a} b^{9} d^{3}}{12 \, b^{12} c^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(a*c+b*c*x^2+d*(b*x^2+a)^(1/2)),x, algorithm="giac")

[Out]

(a^2*c^4 - 2*a*c^2*d^2 + d^4)*log(abs(sqrt(b*x^2 + a)*c + d))/(b^3*c^5) + 1/12*(3*(b*x^2 + a)^2*b^9*c^3 - 12*(
b*x^2 + a)*a*b^9*c^3 - 4*(b*x^2 + a)^(3/2)*b^9*c^2*d + 24*sqrt(b*x^2 + a)*a*b^9*c^2*d + 6*(b*x^2 + a)*b^9*c*d^
2 - 12*sqrt(b*x^2 + a)*b^9*d^3)/(b^12*c^4)

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Mupad [B]
time = 3.64, size = 167, normalized size = 1.25 \begin {gather*} \frac {x^4}{4\,b\,c}-\sqrt {b\,x^2+a}\,\left (\frac {d^3}{b^3\,c^4}-\frac {2\,a\,d}{b^3\,c^2}\right )-\frac {d\,{\left (b\,x^2+a\right )}^{3/2}}{3\,b^3\,c^2}-\frac {x^2\,\left (a\,c^2-d^2\right )}{2\,b^2\,c^3}+\frac {\mathrm {atanh}\left (\frac {c\,\sqrt {b\,x^2+a}}{d}\right )\,{\left (a\,c^2-d^2\right )}^2}{b^3\,c^5}+\frac {\ln \left (b\,c^2\,x^2+a\,c^2-d^2\right )\,\left (a^2\,c^4-2\,a\,c^2\,d^2+d^4\right )}{2\,b^3\,c^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(a*c + d*(a + b*x^2)^(1/2) + b*c*x^2),x)

[Out]

x^4/(4*b*c) - (a + b*x^2)^(1/2)*(d^3/(b^3*c^4) - (2*a*d)/(b^3*c^2)) - (d*(a + b*x^2)^(3/2))/(3*b^3*c^2) - (x^2
*(a*c^2 - d^2))/(2*b^2*c^3) + (atanh((c*(a + b*x^2)^(1/2))/d)*(a*c^2 - d^2)^2)/(b^3*c^5) + (log(a*c^2 - d^2 +
b*c^2*x^2)*(d^4 + a^2*c^4 - 2*a*c^2*d^2))/(2*b^3*c^5)

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