3.6.53 \(\int \frac {x^5}{a c+b c x^3+d \sqrt {a+b x^3}} \, dx\) [553]

Optimal. Leaf size=73 \[ \frac {x^3}{3 b c}-\frac {2 d \sqrt {a+b x^3}}{3 b^2 c^2}-\frac {2 \left (a c^2-d^2\right ) \log \left (d+c \sqrt {a+b x^3}\right )}{3 b^2 c^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.14, antiderivative size = 73, 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 {2 d \sqrt {a+b x^3}}{3 b^2 c^2}-\frac {2 \left (a c^2-d^2\right ) \log \left (c \sqrt {a+b x^3}+d\right )}{3 b^2 c^3}+\frac {x^3}{3 b c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^5/(a*c + b*c*x^3 + d*Sqrt[a + b*x^3]),x]

[Out]

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

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

________________________________________________________________________________________

Mathematica [A]
time = 0.05, size = 66, normalized size = 0.90 \begin {gather*} \frac {c \left (a c+b c x^3-2 d \sqrt {a+b x^3}\right )+\left (-2 a c^2+2 d^2\right ) \log \left (d+c \sqrt {a+b x^3}\right )}{3 b^2 c^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^5/(a*c + b*c*x^3 + d*Sqrt[a + b*x^3]),x]

[Out]

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

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.07, size = 555, normalized size = 7.60

method	result	size
default	\(d \left (-\frac {2 a \sqrt {b \,x^{3}+a}}{3 d^{2} b^{2}}+\frac {\left (c^{2} a -d^{2}\right ) \left (\frac {2 \sqrt {b \,x^{3}+a}}{3 c^{2} b}+\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (b \,c^{2} \textit {\_Z}^{3}+c^{2} a -d^{2}\right )}{\sum }\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {2}\, \sqrt {\frac {i b \left (2 x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}-i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {b \left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}{-3 \left (-a \,b^{2}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {-\frac {i b \left (2 x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}{2 \left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \left (i \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha b -i \left (-a \,b^{2}\right )^{\frac {2}{3}} \sqrt {3}+2 \underline {\hspace {1.25 ex}}\alpha ^{2} b^{2}-\left (-a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha b -\left (-a \,b^{2}\right )^{\frac {2}{3}}\right ) \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, -\frac {c^{2} \left (2 i \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} b -i \left (-a \,b^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha +i \sqrt {3}\, a b -3 \left (-a \,b^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha -3 a b \right )}{2 b \,d^{2}}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{2 \sqrt {b \,x^{3}+a}}\right )}{3 b^{3} c^{2}}\right )}{d^{2} b}\right )-\frac {a \ln \left (b \,c^{2} x^{3}+c^{2} a -d^{2}\right )}{3 c \,b^{2}}+\frac {x^{3}}{3 b c}+\frac {d^{2} \ln \left (b \,c^{2} x^{3}+c^{2} a -d^{2}\right )}{3 b^{2} c^{3}}\)	\(555\)
elliptic	\(\frac {\sqrt {b \,x^{3}+a}\, \left (d +c \sqrt {b \,x^{3}+a}\right ) \left (\frac {x^{3}}{3 b c}-\frac {a \ln \left (b \,c^{2} x^{3}+c^{2} a -d^{2}\right )}{3 c \,b^{2}}+\frac {d^{2} \ln \left (b \,c^{2} x^{3}+c^{2} a -d^{2}\right )}{3 b^{2} c^{3}}-\frac {2 d \sqrt {b \,x^{3}+a}}{3 b^{2} c^{2}}+\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (b \,c^{2} \textit {\_Z}^{3}+c^{2} a -d^{2}\right )}{\sum }\frac {\left (c^{2} a -d^{2}\right ) \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {2}\, \sqrt {\frac {i b \left (2 x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}-i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {b \left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}{-3 \left (-a \,b^{2}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {-\frac {i b \left (2 x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}{2 \left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \left (i \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha b -i \left (-a \,b^{2}\right )^{\frac {2}{3}} \sqrt {3}+2 \underline {\hspace {1.25 ex}}\alpha ^{2} b^{2}-\left (-a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha b -\left (-a \,b^{2}\right )^{\frac {2}{3}}\right ) \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, -\frac {c^{2} \left (2 i \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} b -i \left (-a \,b^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha +i \sqrt {3}\, a b -3 \left (-a \,b^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha -3 a b \right )}{2 b \,d^{2}}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{2 \sqrt {b \,x^{3}+a}}\right )}{3 d \,b^{4} c^{2}}\right )}{a c +b c \,x^{3}+d \sqrt {b \,x^{3}+a}}\)	\(576\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(a*c+b*c*x^3+d*(b*x^3+a)^(1/2)),x,method=_RETURNVERBOSE)

[Out]

d*(-2/3*a/d^2/b^2*(b*x^3+a)^(1/2)+(a*c^2-d^2)/d^2/b*(2/3/c^2/b*(b*x^3+a)^(1/2)+1/3*I/b^3/c^2*2^(1/2)*sum((-a*b
^2)^(1/3)*(1/2*I*b*(2*x+1/b*((-a*b^2)^(1/3)-I*3^(1/2)*(-a*b^2)^(1/3)))/(-a*b^2)^(1/3))^(1/2)*(b*(x-1/b*(-a*b^2
)^(1/3))/(-3*(-a*b^2)^(1/3)+I*3^(1/2)*(-a*b^2)^(1/3)))^(1/2)*(-1/2*I*b*(2*x+1/b*((-a*b^2)^(1/3)+I*3^(1/2)*(-a*
b^2)^(1/3)))/(-a*b^2)^(1/3))^(1/2)/(b*x^3+a)^(1/2)*(I*(-a*b^2)^(1/3)*3^(1/2)*_alpha*b-I*(-a*b^2)^(2/3)*3^(1/2)
+2*_alpha^2*b^2-(-a*b^2)^(1/3)*_alpha*b-(-a*b^2)^(2/3))*EllipticPi(1/3*3^(1/2)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*
I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2),-1/2/b*c^2*(2*I*(-a*b^2)^(1/3)*3^(1/2)*_alpha^2*b-
I*(-a*b^2)^(2/3)*3^(1/2)*_alpha+I*3^(1/2)*a*b-3*(-a*b^2)^(2/3)*_alpha-3*a*b)/d^2,(I*3^(1/2)/b*(-a*b^2)^(1/3)/(
-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2)),_alpha=RootOf(_Z^3*b*c^2+a*c^2-d^2))))-1/3*a/c/b
^2*ln(b*c^2*x^3+a*c^2-d^2)+1/3*x^3/b/c+1/3/b^2/c^3*d^2*ln(b*c^2*x^3+a*c^2-d^2)

________________________________________________________________________________________

Maxima [A]
time = 0.27, size = 62, normalized size = 0.85 \begin {gather*} \frac {\frac {{\left (b x^{3} + a\right )} c - 2 \, \sqrt {b x^{3} + a} d}{c^{2}} - \frac {2 \, {\left (a c^{2} - d^{2}\right )} \log \left (\sqrt {b x^{3} + a} c + d\right )}{c^{3}}}{3 \, b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(a*c+b*c*x^3+d*(b*x^3+a)^(1/2)),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.35, size = 118, normalized size = 1.62 \begin {gather*} \frac {b c^{2} x^{3} - 2 \, \sqrt {b x^{3} + a} c d - {\left (a c^{2} - d^{2}\right )} \log \left (b c^{2} x^{3} + a c^{2} - d^{2}\right ) - {\left (a c^{2} - d^{2}\right )} \log \left (\sqrt {b x^{3} + a} c + d\right ) + {\left (a c^{2} - d^{2}\right )} \log \left (\sqrt {b x^{3} + a} c - d\right )}{3 \, b^{2} c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(a*c+b*c*x^3+d*(b*x^3+a)^(1/2)),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]
time = 3.21, size = 95, normalized size = 1.30 \begin {gather*} \begin {cases} \frac {2 \left (\frac {a + b x^{3}}{6 b c} - \frac {d \sqrt {a + b x^{3}}}{3 b c^{2}} - \frac {\left (a c^{2} - d^{2}\right ) \left (\begin {cases} \frac {\sqrt {a + b x^{3}}}{d} & \text {for}\: c = 0 \\\frac {\log {\left (c \sqrt {a + b x^{3}} + d \right )}}{c} & \text {otherwise} \end {cases}\right )}{3 b c^{2}}\right )}{b} & \text {for}\: b \neq 0 \\\frac {x^{6}}{2 \cdot \left (3 \sqrt {a} d + 3 a c\right )} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(a*c+b*c*x**3+d*(b*x**3+a)**(1/2)),x)

[Out]

Piecewise((2*((a + b*x**3)/(6*b*c) - d*sqrt(a + b*x**3)/(3*b*c**2) - (a*c**2 - d**2)*Piecewise((sqrt(a + b*x**
3)/d, Eq(c, 0)), (log(c*sqrt(a + b*x**3) + d)/c, True))/(3*b*c**2))/b, Ne(b, 0)), (x**6/(2*(3*sqrt(a)*d + 3*a*
c)), True))

________________________________________________________________________________________

Giac [A]
time = 3.69, size = 72, normalized size = 0.99 \begin {gather*} -\frac {\frac {2 \, {\left (a c^{2} - d^{2}\right )} \log \left ({\left | \sqrt {b x^{3} + a} c + d \right |}\right )}{b c^{3}} - \frac {{\left (b x^{3} + a\right )} b c - 2 \, \sqrt {b x^{3} + a} b d}{b^{2} c^{2}}}{3 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(a*c+b*c*x^3+d*(b*x^3+a)^(1/2)),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

Mupad [B]
time = 3.56, size = 119, normalized size = 1.63 \begin {gather*} \frac {x^3}{3\,b\,c}-\frac {2\,d\,\sqrt {b\,x^3+a}}{3\,b^2\,c^2}+\frac {\ln \left (\frac {d-c\,\sqrt {b\,x^3+a}}{d+c\,\sqrt {b\,x^3+a}}\right )\,\left (a\,c^2-d^2\right )}{3\,b^2\,c^3}-\frac {\ln \left (b\,c^2\,x^3+a\,c^2-d^2\right )\,\left (a\,c^2-d^2\right )}{3\,b^2\,c^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(a*c + d*(a + b*x^3)^(1/2) + b*c*x^3),x)

[Out]

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

________________________________________________________________________________________