3.545 \(\int \frac{x^3}{a c+b c x^2+d \sqrt{a+b x^2}} \, dx\)

Optimal. Leaf size=69 \[ -\frac{\left (a c^2-d^2\right ) \log \left (c \sqrt{a+b x^2}+d\right )}{b^2 c^3}-\frac{d \sqrt{a+b x^2}}{b^2 c^2}+\frac{x^2}{2 b c} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.214601, antiderivative size = 69, 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} \[ -\frac{\left (a c^2-d^2\right ) \log \left (c \sqrt{a+b x^2}+d\right )}{b^2 c^3}-\frac{d \sqrt{a+b x^2}}{b^2 c^2}+\frac{x^2}{2 b c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0892205, size = 65, normalized size = 0.94 \[ \frac{-\frac{\left (a c^2-d^2\right ) \log \left (c \sqrt{a+b x^2}+d\right )}{c^3}-\frac{d \sqrt{a+b x^2}}{c^2}+\frac{b x^2}{2 c}}{b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [B]  time = 0.019, size = 3410, normalized size = 49.4 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(a*c+b*c*x^2+d*(b*x^2+a)^(1/2)),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.29795, size = 84, normalized size = 1.22 \begin{align*} \frac{\frac{{\left (b x^{2} + a\right )} c - 2 \, \sqrt{b x^{2} + a} d}{c^{2}} - \frac{2 \,{\left (a c^{2} - d^{2}\right )} \log \left (\sqrt{b x^{2} + a} c + d\right )}{c^{3}}}{2 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(a*c+b*c*x^2+d*(b*x^2+a)^(1/2)),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [B]  time = 1.44253, size = 340, normalized size = 4.93 \begin{align*} \frac{2 \, b c^{2} x^{2} - 4 \, \sqrt{b x^{2} + a} c d - 2 \,{\left (a c^{2} - d^{2}\right )} \log \left (b c^{2} x^{2} + a c^{2} - d^{2}\right ) -{\left (a c^{2} - d^{2}\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 ) +{\left (a c^{2} - d^{2}\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 \, b^{2} c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(a*c+b*c*x^2+d*(b*x^2+a)^(1/2)),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 4.08136, size = 88, normalized size = 1.28 \begin{align*} \begin{cases} \frac{\frac{a + b x^{2}}{2 b c} - \frac{d \sqrt{a + b x^{2}}}{b c^{2}} - \frac{\left (a c^{2} - d^{2}\right ) \left (\begin{cases} \frac{\sqrt{a + b x^{2}}}{d} & \text{for}\: c = 0 \\\frac{\log{\left (c \sqrt{a + b x^{2}} + d \right )}}{c} & \text{otherwise} \end{cases}\right )}{b c^{2}}}{b} & \text{for}\: b \neq 0 \\\frac{x^{4}}{2 \left (2 \sqrt{a} d + 2 a c\right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(a*c+b*c*x**2+d*(b*x**2+a)**(1/2)),x)

[Out]

Piecewise((((a + b*x**2)/(2*b*c) - d*sqrt(a + b*x**2)/(b*c**2) - (a*c**2 - d**2)*Piecewise((sqrt(a + b*x**2)/d
, Eq(c, 0)), (log(c*sqrt(a + b*x**2) + d)/c, True))/(b*c**2))/b, Ne(b, 0)), (x**4/(2*(2*sqrt(a)*d + 2*a*c)), T
rue))

________________________________________________________________________________________

Giac [A]  time = 1.12227, size = 97, normalized size = 1.41 \begin{align*} -\frac{\frac{2 \,{\left (a c^{2} - d^{2}\right )} \log \left ({\left | \sqrt{b x^{2} + a} c + d \right |}\right )}{b c^{3}} - \frac{{\left (b x^{2} + a\right )} b c - 2 \, \sqrt{b x^{2} + a} b d}{b^{2} c^{2}}}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(a*c+b*c*x^2+d*(b*x^2+a)^(1/2)),x, algorithm="giac")

[Out]

-1/2*(2*(a*c^2 - d^2)*log(abs(sqrt(b*x^2 + a)*c + d))/(b*c^3) - ((b*x^2 + a)*b*c - 2*sqrt(b*x^2 + a)*b*d)/(b^2
*c^2))/b