Integrand size = 19, antiderivative size = 71 \[ \int \sqrt {c x} \sqrt {a+b x^3} \, dx=\frac {(c x)^{3/2} \sqrt {a+b x^3}}{3 c}+\frac {a \sqrt {c} \text {arctanh}\left (\frac {\sqrt {b} (c x)^{3/2}}{c^{3/2} \sqrt {a+b x^3}}\right )}{3 \sqrt {b}} \] Output:
1/3*(c*x)^(3/2)*(b*x^3+a)^(1/2)/c+1/3*a*c^(1/2)*arctanh(b^(1/2)*(c*x)^(3/2 )/c^(3/2)/(b*x^3+a)^(1/2))/b^(1/2)
Time = 0.16 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.92 \[ \int \sqrt {c x} \sqrt {a+b x^3} \, dx=\frac {\sqrt {c x} \left (x^{3/2} \sqrt {a+b x^3}+\frac {a \log \left (\sqrt {b} x^{3/2}+\sqrt {a+b x^3}\right )}{\sqrt {b}}\right )}{3 \sqrt {x}} \] Input:
Integrate[Sqrt[c*x]*Sqrt[a + b*x^3],x]
Output:
(Sqrt[c*x]*(x^(3/2)*Sqrt[a + b*x^3] + (a*Log[Sqrt[b]*x^(3/2) + Sqrt[a + b* x^3]])/Sqrt[b]))/(3*Sqrt[x])
Time = 0.37 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.01, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {811, 851, 807, 224, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \sqrt {c x} \sqrt {a+b x^3} \, dx\) |
\(\Big \downarrow \) 811 |
\(\displaystyle \frac {1}{2} a \int \frac {\sqrt {c x}}{\sqrt {b x^3+a}}dx+\frac {(c x)^{3/2} \sqrt {a+b x^3}}{3 c}\) |
\(\Big \downarrow \) 851 |
\(\displaystyle \frac {a \int \frac {c x}{\sqrt {b x^3+a}}d\sqrt {c x}}{c}+\frac {(c x)^{3/2} \sqrt {a+b x^3}}{3 c}\) |
\(\Big \downarrow \) 807 |
\(\displaystyle \frac {a \int \frac {1}{\sqrt {a+\frac {b x}{c^2}}}d(c x)^{3/2}}{3 c}+\frac {(c x)^{3/2} \sqrt {a+b x^3}}{3 c}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {a \int \frac {1}{1-\frac {b x}{c^2}}d\frac {(c x)^{3/2}}{\sqrt {a+\frac {b x}{c^2}}}}{3 c}+\frac {(c x)^{3/2} \sqrt {a+b x^3}}{3 c}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {a \sqrt {c} \text {arctanh}\left (\frac {\sqrt {b} (c x)^{3/2}}{c^{3/2} \sqrt {a+\frac {b x}{c^2}}}\right )}{3 \sqrt {b}}+\frac {(c x)^{3/2} \sqrt {a+b x^3}}{3 c}\) |
Input:
Int[Sqrt[c*x]*Sqrt[a + b*x^3],x]
Output:
((c*x)^(3/2)*Sqrt[a + b*x^3])/(3*c) + (a*Sqrt[c]*ArcTanh[(Sqrt[b]*(c*x)^(3 /2))/(c^(3/2)*Sqrt[a + (b*x)/c^2])])/(3*Sqrt[b])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c* x)^(m + 1)*((a + b*x^n)^p/(c*(m + n*p + 1))), x] + Simp[a*n*(p/(m + n*p + 1 )) Int[(c*x)^m*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x] && I GtQ[n, 0] && GtQ[p, 0] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b, c, n, m , p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^ n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]
Time = 0.74 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.11
method | result | size |
default | \(\frac {\sqrt {c x}\, \sqrt {b \,x^{3}+a}\, \left (\sqrt {c x \left (b \,x^{3}+a \right )}\, x \sqrt {b c}+\operatorname {arctanh}\left (\frac {\sqrt {c x \left (b \,x^{3}+a \right )}}{x^{2} \sqrt {b c}}\right ) a c \right )}{3 \sqrt {c x \left (b \,x^{3}+a \right )}\, \sqrt {b c}}\) | \(79\) |
risch | \(\frac {x^{2} \sqrt {b \,x^{3}+a}\, c}{3 \sqrt {c x}}+\frac {a \,\operatorname {arctanh}\left (\frac {\sqrt {c x \left (b \,x^{3}+a \right )}}{x^{2} \sqrt {b c}}\right ) c \sqrt {c x \left (b \,x^{3}+a \right )}}{3 \sqrt {b c}\, \sqrt {c x}\, \sqrt {b \,x^{3}+a}}\) | \(79\) |
elliptic | \(\text {Expression too large to display}\) | \(1031\) |
Input:
int((c*x)^(1/2)*(b*x^3+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/3*(c*x)^(1/2)*(b*x^3+a)^(1/2)/(c*x*(b*x^3+a))^(1/2)*((c*x*(b*x^3+a))^(1/ 2)*x*(b*c)^(1/2)+arctanh((c*x*(b*x^3+a))^(1/2)/x^2/(b*c)^(1/2))*a*c)/(b*c) ^(1/2)
Time = 0.17 (sec) , antiderivative size = 159, normalized size of antiderivative = 2.24 \[ \int \sqrt {c x} \sqrt {a+b x^3} \, dx=\left [\frac {1}{12} \, a \sqrt {\frac {c}{b}} \log \left (-8 \, b^{2} c x^{6} - 8 \, a b c x^{3} - a^{2} c - 4 \, {\left (2 \, b^{2} x^{4} + a b x\right )} \sqrt {b x^{3} + a} \sqrt {c x} \sqrt {\frac {c}{b}}\right ) + \frac {1}{3} \, \sqrt {b x^{3} + a} \sqrt {c x} x, -\frac {1}{6} \, a \sqrt {-\frac {c}{b}} \arctan \left (\frac {2 \, \sqrt {b x^{3} + a} \sqrt {c x} b x \sqrt {-\frac {c}{b}}}{2 \, b c x^{3} + a c}\right ) + \frac {1}{3} \, \sqrt {b x^{3} + a} \sqrt {c x} x\right ] \] Input:
integrate((c*x)^(1/2)*(b*x^3+a)^(1/2),x, algorithm="fricas")
Output:
[1/12*a*sqrt(c/b)*log(-8*b^2*c*x^6 - 8*a*b*c*x^3 - a^2*c - 4*(2*b^2*x^4 + a*b*x)*sqrt(b*x^3 + a)*sqrt(c*x)*sqrt(c/b)) + 1/3*sqrt(b*x^3 + a)*sqrt(c*x )*x, -1/6*a*sqrt(-c/b)*arctan(2*sqrt(b*x^3 + a)*sqrt(c*x)*b*x*sqrt(-c/b)/( 2*b*c*x^3 + a*c)) + 1/3*sqrt(b*x^3 + a)*sqrt(c*x)*x]
Time = 1.17 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.82 \[ \int \sqrt {c x} \sqrt {a+b x^3} \, dx=\frac {\sqrt {a} \sqrt {c} x^{\frac {3}{2}} \sqrt {1 + \frac {b x^{3}}{a}}}{3} + \frac {a \sqrt {c} \operatorname {asinh}{\left (\frac {\sqrt {b} x^{\frac {3}{2}}}{\sqrt {a}} \right )}}{3 \sqrt {b}} \] Input:
integrate((c*x)**(1/2)*(b*x**3+a)**(1/2),x)
Output:
sqrt(a)*sqrt(c)*x**(3/2)*sqrt(1 + b*x**3/a)/3 + a*sqrt(c)*asinh(sqrt(b)*x* *(3/2)/sqrt(a))/(3*sqrt(b))
\[ \int \sqrt {c x} \sqrt {a+b x^3} \, dx=\int { \sqrt {b x^{3} + a} \sqrt {c x} \,d x } \] Input:
integrate((c*x)^(1/2)*(b*x^3+a)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(b*x^3 + a)*sqrt(c*x), x)
Time = 0.13 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.10 \[ \int \sqrt {c x} \sqrt {a+b x^3} \, dx=-\frac {{\left (\frac {a c^{4} \log \left ({\left | -\sqrt {b c} \sqrt {c x} c x + \sqrt {b c^{4} x^{3} + a c^{4}} \right |}\right )}{\sqrt {b c}} - \sqrt {b c^{4} x^{3} + a c^{4}} \sqrt {c x} c x\right )} {\left | c \right |}^{2}}{3 \, c^{5}} \] Input:
integrate((c*x)^(1/2)*(b*x^3+a)^(1/2),x, algorithm="giac")
Output:
-1/3*(a*c^4*log(abs(-sqrt(b*c)*sqrt(c*x)*c*x + sqrt(b*c^4*x^3 + a*c^4)))/s qrt(b*c) - sqrt(b*c^4*x^3 + a*c^4)*sqrt(c*x)*c*x)*abs(c)^2/c^5
Timed out. \[ \int \sqrt {c x} \sqrt {a+b x^3} \, dx=\int \sqrt {c\,x}\,\sqrt {b\,x^3+a} \,d x \] Input:
int((c*x)^(1/2)*(a + b*x^3)^(1/2),x)
Output:
int((c*x)^(1/2)*(a + b*x^3)^(1/2), x)
Time = 0.24 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.90 \[ \int \sqrt {c x} \sqrt {a+b x^3} \, dx=\frac {\sqrt {c}\, \left (2 \sqrt {x}\, \sqrt {b \,x^{3}+a}\, b x -\sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{3}+a}-\sqrt {x}\, \sqrt {b}\, x \right ) a +\sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{3}+a}+\sqrt {x}\, \sqrt {b}\, x \right ) a \right )}{6 b} \] Input:
int((c*x)^(1/2)*(b*x^3+a)^(1/2),x)
Output:
(sqrt(c)*(2*sqrt(x)*sqrt(a + b*x**3)*b*x - sqrt(b)*log(sqrt(a + b*x**3) - sqrt(x)*sqrt(b)*x)*a + sqrt(b)*log(sqrt(a + b*x**3) + sqrt(x)*sqrt(b)*x)*a ))/(6*b)