Integrand size = 19, antiderivative size = 105 \[ \int \frac {(c x)^{13/2}}{\sqrt {a+b x^3}} \, dx=-\frac {a c^5 (c x)^{3/2} \sqrt {a+b x^3}}{4 b^2}+\frac {c^2 (c x)^{9/2} \sqrt {a+b x^3}}{6 b}+\frac {a^2 c^{13/2} \text {arctanh}\left (\frac {\sqrt {b} (c x)^{3/2}}{c^{3/2} \sqrt {a+b x^3}}\right )}{4 b^{5/2}} \] Output:
-1/4*a*c^5*(c*x)^(3/2)*(b*x^3+a)^(1/2)/b^2+1/6*c^2*(c*x)^(9/2)*(b*x^3+a)^( 1/2)/b+1/4*a^2*c^(13/2)*arctanh(b^(1/2)*(c*x)^(3/2)/c^(3/2)/(b*x^3+a)^(1/2 ))/b^(5/2)
Time = 0.24 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.82 \[ \int \frac {(c x)^{13/2}}{\sqrt {a+b x^3}} \, dx=\frac {c^6 \sqrt {c x} \left (\sqrt {b} x^{3/2} \sqrt {a+b x^3} \left (-3 a+2 b x^3\right )+3 a^2 \log \left (\sqrt {b} x^{3/2}+\sqrt {a+b x^3}\right )\right )}{12 b^{5/2} \sqrt {x}} \] Input:
Integrate[(c*x)^(13/2)/Sqrt[a + b*x^3],x]
Output:
(c^6*Sqrt[c*x]*(Sqrt[b]*x^(3/2)*Sqrt[a + b*x^3]*(-3*a + 2*b*x^3) + 3*a^2*L og[Sqrt[b]*x^(3/2) + Sqrt[a + b*x^3]]))/(12*b^(5/2)*Sqrt[x])
Time = 0.44 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.10, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {843, 843, 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.
\(\displaystyle \int \frac {(c x)^{13/2}}{\sqrt {a+b x^3}} \, dx\) |
\(\Big \downarrow \) 843 |
\(\displaystyle \frac {c^2 (c x)^{9/2} \sqrt {a+b x^3}}{6 b}-\frac {3 a c^3 \int \frac {(c x)^{7/2}}{\sqrt {b x^3+a}}dx}{4 b}\) |
\(\Big \downarrow \) 843 |
\(\displaystyle \frac {c^2 (c x)^{9/2} \sqrt {a+b x^3}}{6 b}-\frac {3 a c^3 \left (\frac {c^2 (c x)^{3/2} \sqrt {a+b x^3}}{3 b}-\frac {a c^3 \int \frac {\sqrt {c x}}{\sqrt {b x^3+a}}dx}{2 b}\right )}{4 b}\) |
\(\Big \downarrow \) 851 |
\(\displaystyle \frac {c^2 (c x)^{9/2} \sqrt {a+b x^3}}{6 b}-\frac {3 a c^3 \left (\frac {c^2 (c x)^{3/2} \sqrt {a+b x^3}}{3 b}-\frac {a c^2 \int \frac {c x}{\sqrt {b x^3+a}}d\sqrt {c x}}{b}\right )}{4 b}\) |
\(\Big \downarrow \) 807 |
\(\displaystyle \frac {c^2 (c x)^{9/2} \sqrt {a+b x^3}}{6 b}-\frac {3 a c^3 \left (\frac {c^2 (c x)^{3/2} \sqrt {a+b x^3}}{3 b}-\frac {a c^2 \int \frac {1}{\sqrt {a+\frac {b x}{c^2}}}d(c x)^{3/2}}{3 b}\right )}{4 b}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {c^2 (c x)^{9/2} \sqrt {a+b x^3}}{6 b}-\frac {3 a c^3 \left (\frac {c^2 (c x)^{3/2} \sqrt {a+b x^3}}{3 b}-\frac {a c^2 \int \frac {1}{1-\frac {b x}{c^2}}d\frac {(c x)^{3/2}}{\sqrt {a+\frac {b x}{c^2}}}}{3 b}\right )}{4 b}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {c^2 (c x)^{9/2} \sqrt {a+b x^3}}{6 b}-\frac {3 a c^3 \left (\frac {c^2 (c x)^{3/2} \sqrt {a+b x^3}}{3 b}-\frac {a c^{7/2} \text {arctanh}\left (\frac {\sqrt {b} (c x)^{3/2}}{c^{3/2} \sqrt {a+\frac {b x}{c^2}}}\right )}{3 b^{3/2}}\right )}{4 b}\) |
Input:
Int[(c*x)^(13/2)/Sqrt[a + b*x^3],x]
Output:
(c^2*(c*x)^(9/2)*Sqrt[a + b*x^3])/(6*b) - (3*a*c^3*((c^2*(c*x)^(3/2)*Sqrt[ a + b*x^3])/(3*b) - (a*c^(7/2)*ArcTanh[(Sqrt[b]*(c*x)^(3/2))/(c^(3/2)*Sqrt [a + (b*x)/c^2])])/(3*b^(3/2))))/(4*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^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*(m + n*p + 1))), x] - Simp[ a*c^n*((m - n + 1)/(b*(m + n*p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^p, x] , x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && 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 = 1.04 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.96
method | result | size |
risch | \(-\frac {x^{2} \left (-2 b \,x^{3}+3 a \right ) \sqrt {b \,x^{3}+a}\, c^{7}}{12 b^{2} \sqrt {c x}}+\frac {a^{2} \operatorname {arctanh}\left (\frac {\sqrt {c x \left (b \,x^{3}+a \right )}}{x^{2} \sqrt {b c}}\right ) c^{7} \sqrt {c x \left (b \,x^{3}+a \right )}}{4 b^{2} \sqrt {b c}\, \sqrt {c x}\, \sqrt {b \,x^{3}+a}}\) | \(101\) |
default | \(\frac {c^{6} \sqrt {c x}\, \sqrt {b \,x^{3}+a}\, \left (2 \sqrt {c x \left (b \,x^{3}+a \right )}\, \sqrt {b c}\, b \,x^{4}+3 \,\operatorname {arctanh}\left (\frac {\sqrt {c x \left (b \,x^{3}+a \right )}}{x^{2} \sqrt {b c}}\right ) c \,a^{2}-3 a x \sqrt {c x \left (b \,x^{3}+a \right )}\, \sqrt {b c}\right )}{12 \sqrt {c x \left (b \,x^{3}+a \right )}\, b^{2} \sqrt {b c}}\) | \(113\) |
elliptic | \(\text {Expression too large to display}\) | \(1069\) |
Input:
int((c*x)^(13/2)/(b*x^3+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/12*x^2*(-2*b*x^3+3*a)*(b*x^3+a)^(1/2)/b^2*c^7/(c*x)^(1/2)+1/4*a^2/b^2/( b*c)^(1/2)*arctanh((c*x*(b*x^3+a))^(1/2)/x^2/(b*c)^(1/2))*c^7*(c*x*(b*x^3+ a))^(1/2)/(c*x)^(1/2)/(b*x^3+a)^(1/2)
Time = 0.17 (sec) , antiderivative size = 211, normalized size of antiderivative = 2.01 \[ \int \frac {(c x)^{13/2}}{\sqrt {a+b x^3}} \, dx=\left [\frac {3 \, a^{2} c^{6} \sqrt {\frac {c}{b}} \log \left (-8 \, b^{2} c x^{6} - 8 \, a b c x^{3} - a^{2} c - 4 \, {\left (2 \, b^{2} x^{4} + a b x\right )} \sqrt {b x^{3} + a} \sqrt {c x} \sqrt {\frac {c}{b}}\right ) + 4 \, {\left (2 \, b c^{6} x^{4} - 3 \, a c^{6} x\right )} \sqrt {b x^{3} + a} \sqrt {c x}}{48 \, b^{2}}, -\frac {3 \, a^{2} c^{6} \sqrt {-\frac {c}{b}} \arctan \left (\frac {2 \, \sqrt {b x^{3} + a} \sqrt {c x} b x \sqrt {-\frac {c}{b}}}{2 \, b c x^{3} + a c}\right ) - 2 \, {\left (2 \, b c^{6} x^{4} - 3 \, a c^{6} x\right )} \sqrt {b x^{3} + a} \sqrt {c x}}{24 \, b^{2}}\right ] \] Input:
integrate((c*x)^(13/2)/(b*x^3+a)^(1/2),x, algorithm="fricas")
Output:
[1/48*(3*a^2*c^6*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)) + 4*(2*b*c^6*x^4 - 3* a*c^6*x)*sqrt(b*x^3 + a)*sqrt(c*x))/b^2, -1/24*(3*a^2*c^6*sqrt(-c/b)*arcta n(2*sqrt(b*x^3 + a)*sqrt(c*x)*b*x*sqrt(-c/b)/(2*b*c*x^3 + a*c)) - 2*(2*b*c ^6*x^4 - 3*a*c^6*x)*sqrt(b*x^3 + a)*sqrt(c*x))/b^2]
Timed out. \[ \int \frac {(c x)^{13/2}}{\sqrt {a+b x^3}} \, dx=\text {Timed out} \] Input:
integrate((c*x)**(13/2)/(b*x**3+a)**(1/2),x)
Output:
Timed out
\[ \int \frac {(c x)^{13/2}}{\sqrt {a+b x^3}} \, dx=\int { \frac {\left (c x\right )^{\frac {13}{2}}}{\sqrt {b x^{3} + a}} \,d x } \] Input:
integrate((c*x)^(13/2)/(b*x^3+a)^(1/2),x, algorithm="maxima")
Output:
integrate((c*x)^(13/2)/sqrt(b*x^3 + a), x)
Time = 0.16 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.01 \[ \int \frac {(c x)^{13/2}}{\sqrt {a+b x^3}} \, dx=\frac {\sqrt {b c^{4} x^{3} + a c^{4}} \sqrt {c x} c^{11} x {\left (\frac {2 \, x^{3}}{b c^{5}} - \frac {3 \, a}{b^{2} c^{5}}\right )}}{12 \, {\left | c \right |}^{2}} - \frac {a^{2} c^{11} \log \left ({\left | -\sqrt {b c} \sqrt {c x} c x + \sqrt {b c^{4} x^{3} + a c^{4}} \right |}\right )}{4 \, \sqrt {b c} b^{2} {\left | c \right |}^{4}} \] Input:
integrate((c*x)^(13/2)/(b*x^3+a)^(1/2),x, algorithm="giac")
Output:
1/12*sqrt(b*c^4*x^3 + a*c^4)*sqrt(c*x)*c^11*x*(2*x^3/(b*c^5) - 3*a/(b^2*c^ 5))/abs(c)^2 - 1/4*a^2*c^11*log(abs(-sqrt(b*c)*sqrt(c*x)*c*x + sqrt(b*c^4* x^3 + a*c^4)))/(sqrt(b*c)*b^2*abs(c)^4)
Timed out. \[ \int \frac {(c x)^{13/2}}{\sqrt {a+b x^3}} \, dx=\int \frac {{\left (c\,x\right )}^{13/2}}{\sqrt {b\,x^3+a}} \,d x \] Input:
int((c*x)^(13/2)/(a + b*x^3)^(1/2),x)
Output:
int((c*x)^(13/2)/(a + b*x^3)^(1/2), x)
Time = 0.27 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.87 \[ \int \frac {(c x)^{13/2}}{\sqrt {a+b x^3}} \, dx=\frac {\sqrt {c}\, c^{6} \left (-6 \sqrt {x}\, \sqrt {b \,x^{3}+a}\, a b x +4 \sqrt {x}\, \sqrt {b \,x^{3}+a}\, b^{2} x^{4}-3 \sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{3}+a}-\sqrt {x}\, \sqrt {b}\, x \right ) a^{2}+3 \sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{3}+a}+\sqrt {x}\, \sqrt {b}\, x \right ) a^{2}\right )}{24 b^{3}} \] Input:
int((c*x)^(13/2)/(b*x^3+a)^(1/2),x)
Output:
(sqrt(c)*c**6*( - 6*sqrt(x)*sqrt(a + b*x**3)*a*b*x + 4*sqrt(x)*sqrt(a + b* x**3)*b**2*x**4 - 3*sqrt(b)*log(sqrt(a + b*x**3) - sqrt(x)*sqrt(b)*x)*a**2 + 3*sqrt(b)*log(sqrt(a + b*x**3) + sqrt(x)*sqrt(b)*x)*a**2))/(24*b**3)