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

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

[Out]

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

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Rubi [A]  time = 0.21, antiderivative size = 69, 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 = {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 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]

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]

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*}

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Mathematica [A]  time = 0.09, size = 65, normalized size = 0.94 \[ \frac {-\frac {d \sqrt {a+b x^2}}{c^2}-\frac {\left (a c^2-d^2\right ) \log \left (c \sqrt {a+b x^2}+d\right )}{c^3}+\frac {b x^2}{2 c}}{b^2} \]

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

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fricas [B]  time = 0.44, size = 161, normalized size = 2.33 \[ \frac {2 \, b c^{2} x^{2} - 4 \, \sqrt {b x^{2} + a} c d - 2 \, {\left (a c^{2} - d^{2}\right )} \log \left (b c^{2} x^{2} + a c^{2} - d^{2}\right ) - {\left (a c^{2} - d^{2}\right )} \log \left (-\frac {b c^{2} x^{2} + a c^{2} + 2 \, \sqrt {b x^{2} + a} c d + d^{2}}{x^{2}}\right ) + {\left (a c^{2} - d^{2}\right )} \log \left (-\frac {b c^{2} x^{2} + a c^{2} - 2 \, \sqrt {b x^{2} + a} c d + d^{2}}{x^{2}}\right )}{4 \, b^{2} c^{3}} \]

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)

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giac [A]  time = 0.40, size = 72, normalized size = 1.04 \[ -\frac {\frac {2 \, {\left (a c^{2} - d^{2}\right )} \log \left ({\left | \sqrt {b x^{2} + a} c + d \right |}\right )}{b c^{3}} - \frac {{\left (b x^{2} + a\right )} b c - 2 \, \sqrt {b x^{2} + a} b d}{b^{2} c^{2}}}{2 \, b} \]

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

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maple [B]  time = 0.04, size = 3410, normalized size = 49.42 \[ \text {output too large to display} \]

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

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maxima [A]  time = 0.69, size = 62, normalized size = 0.90 \[ \frac {\frac {{\left (b x^{2} + a\right )} c - 2 \, \sqrt {b x^{2} + a} d}{c^{2}} - \frac {2 \, {\left (a c^{2} - d^{2}\right )} \log \left (\sqrt {b x^{2} + a} c + d\right )}{c^{3}}}{2 \, b^{2}} \]

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

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mupad [B]  time = 3.50, size = 123, normalized size = 1.78 \[ \frac {x^2}{2\,b\,c}-\frac {d\,\sqrt {b\,x^2+a}}{b^2\,c^2}+\frac {\mathrm {atanh}\left (\frac {c\,\left (a\,c^2-d^2\right )\,\sqrt {b\,x^2+a}}{d^3-a\,c^2\,d}\right )\,\left (a\,c^2-d^2\right )}{b^2\,c^3}-\frac {\ln \left (b\,c^2\,x^2+a\,c^2-d^2\right )\,\left (a\,c^2-d^2\right )}{2\,b^2\,c^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 6.51, size = 88, normalized size = 1.28 \[ \begin {cases} \frac {\frac {a + b x^{2}}{2 b c} - \frac {d \sqrt {a + b x^{2}}}{b c^{2}} - \frac {\left (a c^{2} - d^{2}\right ) \left (\begin {cases} \frac {\sqrt {a + b x^{2}}}{d} & \text {for}\: c = 0 \\\frac {\log {\left (c \sqrt {a + b x^{2}} + d \right )}}{c} & \text {otherwise} \end {cases}\right )}{b c^{2}}}{b} & \text {for}\: b \neq 0 \\\frac {x^{4}}{2 \left (2 \sqrt {a} d + 2 a c\right )} & \text {otherwise} \end {cases} \]

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))

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