3.544 \(\int \frac{x^5}{a c+b c x^2+d \sqrt{a+b x^2}} \, dx\)

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

[Out]

-((2*a*c^2 - d^2)*x^2)/(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)

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Rubi [A]  time = 0.372887, antiderivative size = 134, normalized size of antiderivative = 1., 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 = {2155, 697} \[ -\frac{x^2 \left (2 a c^2-d^2\right )}{2 b^2 c^3}+\frac{d \sqrt{a+b x^2} \left (2 a c^2-d^2\right )}{b^3 c^4}+\frac{\left (a c^2-d^2\right )^2 \log \left (c \sqrt{a+b x^2}+d\right )}{b^3 c^5}-\frac{d \left (a+b x^2\right )^{3/2}}{3 b^3 c^2}+\frac{\left (a+b x^2\right )^2}{4 b^3 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((2*a*c^2 - d^2)*x^2)/(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 2155

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]

Rule 697

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]

Rubi steps

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

Mathematica [A]  time = 0.218416, size = 126, normalized size = 0.94 \[ \frac{c \left (a \left (20 c^2 d \sqrt{a+b x^2}-6 b c^3 x^2\right )+2 b c d x^2 \left (3 d-2 c \sqrt{a+b x^2}\right )-12 d^3 \sqrt{a+b x^2}+3 b^2 c^3 x^4\right )+12 \left (d^2-a c^2\right )^2 \log \left (c \sqrt{a+b x^2}+d\right )}{12 b^3 c^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c*(3*b^2*c^3*x^4 - 12*d^3*Sqrt[a + b*x^2] + 2*b*c*d*x^2*(3*d - 2*c*Sqrt[a + b*x^2]) + a*(-6*b*c^3*x^2 + 20*c^
2*d*Sqrt[a + b*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]  time = 0.077, size = 4947, normalized size = 36.9 \begin{align*} \text{output too large to display} \end{align*}

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)

[Out]

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

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Maxima [A]  time = 1.4181, size = 169, normalized size = 1.26 \begin{align*} \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{align*}

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 = 1.48454, size = 498, normalized size = 3.72 \begin{align*} \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{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{5}}{a c + b c x^{2} + d \sqrt{a + b x^{2}}}\, dx \end{align*}

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 = 1.13328, size = 209, normalized size = 1.56 \begin{align*} \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{align*}

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)