Integrand size = 19, antiderivative size = 73 \[ \int \frac {(c x)^{7/2}}{\left (a+b x^3\right )^{3/2}} \, dx=-\frac {2 c^2 (c x)^{3/2}}{3 b \sqrt {a+b x^3}}+\frac {2 c^{7/2} \text {arctanh}\left (\frac {\sqrt {b} (c x)^{3/2}}{c^{3/2} \sqrt {a+b x^3}}\right )}{3 b^{3/2}} \] Output:
-2/3*c^2*(c*x)^(3/2)/b/(b*x^3+a)^(1/2)+2/3*c^(7/2)*arctanh(b^(1/2)*(c*x)^( 3/2)/c^(3/2)/(b*x^3+a)^(1/2))/b^(3/2)
Time = 0.25 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.95 \[ \int \frac {(c x)^{7/2}}{\left (a+b x^3\right )^{3/2}} \, dx=\frac {2 (c x)^{7/2} \left (-\frac {\sqrt {b} x^{3/2}}{\sqrt {a+b x^3}}+\log \left (\sqrt {b} x^{3/2}+\sqrt {a+b x^3}\right )\right )}{3 b^{3/2} x^{7/2}} \] Input:
Integrate[(c*x)^(7/2)/(a + b*x^3)^(3/2),x]
Output:
(2*(c*x)^(7/2)*(-((Sqrt[b]*x^(3/2))/Sqrt[a + b*x^3]) + Log[Sqrt[b]*x^(3/2) + Sqrt[a + b*x^3]]))/(3*b^(3/2)*x^(7/2))
Time = 0.37 (sec) , antiderivative size = 74, 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 = {817, 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.
\(\displaystyle \int \frac {(c x)^{7/2}}{\left (a+b x^3\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 817 |
\(\displaystyle \frac {c^3 \int \frac {\sqrt {c x}}{\sqrt {b x^3+a}}dx}{b}-\frac {2 c^2 (c x)^{3/2}}{3 b \sqrt {a+b x^3}}\) |
\(\Big \downarrow \) 851 |
\(\displaystyle \frac {2 c^2 \int \frac {c x}{\sqrt {b x^3+a}}d\sqrt {c x}}{b}-\frac {2 c^2 (c x)^{3/2}}{3 b \sqrt {a+b x^3}}\) |
\(\Big \downarrow \) 807 |
\(\displaystyle \frac {2 c^2 \int \frac {1}{\sqrt {a+\frac {b x}{c^2}}}d(c x)^{3/2}}{3 b}-\frac {2 c^2 (c x)^{3/2}}{3 b \sqrt {a+b x^3}}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {2 c^2 \int \frac {1}{1-\frac {b x}{c^2}}d\frac {(c x)^{3/2}}{\sqrt {a+\frac {b x}{c^2}}}}{3 b}-\frac {2 c^2 (c x)^{3/2}}{3 b \sqrt {a+b x^3}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {2 c^{7/2} \text {arctanh}\left (\frac {\sqrt {b} (c x)^{3/2}}{c^{3/2} \sqrt {a+\frac {b x}{c^2}}}\right )}{3 b^{3/2}}-\frac {2 c^2 (c x)^{3/2}}{3 b \sqrt {a+b x^3}}\) |
Input:
Int[(c*x)^(7/2)/(a + b*x^3)^(3/2),x]
Output:
(-2*c^2*(c*x)^(3/2))/(3*b*Sqrt[a + b*x^3]) + (2*c^(7/2)*ArcTanh[(Sqrt[b]*( c*x)^(3/2))/(c^(3/2)*Sqrt[a + (b*x)/c^2])])/(3*b^(3/2))
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*n*(p + 1))), x] - Simp[c^n *((m - n + 1)/(b*n*(p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x], x ] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] && ! ILtQ[(m + n*(p + 1) + 1)/n, 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 = 0.50 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.05
method | result | size |
default | \(\frac {2 c^{3} \sqrt {c x}\, \left (-x^{2} \sqrt {b c}+\operatorname {arctanh}\left (\frac {\sqrt {c x \left (b \,x^{3}+a \right )}}{x^{2} \sqrt {b c}}\right ) \sqrt {c x \left (b \,x^{3}+a \right )}\right )}{3 x \sqrt {b \,x^{3}+a}\, b \sqrt {b c}}\) | \(77\) |
elliptic | \(\text {Expression too large to display}\) | \(1042\) |
Input:
int((c*x)^(7/2)/(b*x^3+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
2/3*c^3*(c*x)^(1/2)*(-x^2*(b*c)^(1/2)+arctanh((c*x*(b*x^3+a))^(1/2)/x^2/(b *c)^(1/2))*(c*x*(b*x^3+a))^(1/2))/x/(b*x^3+a)^(1/2)/b/(b*c)^(1/2)
Time = 0.17 (sec) , antiderivative size = 220, normalized size of antiderivative = 3.01 \[ \int \frac {(c x)^{7/2}}{\left (a+b x^3\right )^{3/2}} \, dx=\left [-\frac {4 \, \sqrt {b x^{3} + a} \sqrt {c x} c^{3} x - {\left (b c^{3} x^{3} + a c^{3}\right )} \sqrt {\frac {c}{b}} \log \left (-8 \, b^{2} c x^{6} - 8 \, a b c x^{3} - a^{2} c - 4 \, {\left (2 \, b^{2} x^{4} + a b x\right )} \sqrt {b x^{3} + a} \sqrt {c x} \sqrt {\frac {c}{b}}\right )}{6 \, {\left (b^{2} x^{3} + a b\right )}}, -\frac {2 \, \sqrt {b x^{3} + a} \sqrt {c x} c^{3} x + {\left (b c^{3} x^{3} + a c^{3}\right )} \sqrt {-\frac {c}{b}} \arctan \left (\frac {2 \, \sqrt {b x^{3} + a} \sqrt {c x} b x \sqrt {-\frac {c}{b}}}{2 \, b c x^{3} + a c}\right )}{3 \, {\left (b^{2} x^{3} + a b\right )}}\right ] \] Input:
integrate((c*x)^(7/2)/(b*x^3+a)^(3/2),x, algorithm="fricas")
Output:
[-1/6*(4*sqrt(b*x^3 + a)*sqrt(c*x)*c^3*x - (b*c^3*x^3 + a*c^3)*sqrt(c/b)*l og(-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)))/(b^2*x^3 + a*b), -1/3*(2*sqrt(b*x^3 + a)*sqrt(c* x)*c^3*x + (b*c^3*x^3 + a*c^3)*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)))/(b^2*x^3 + a*b)]
Time = 12.94 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.84 \[ \int \frac {(c x)^{7/2}}{\left (a+b x^3\right )^{3/2}} \, dx=\frac {2 c^{\frac {7}{2}} \operatorname {asinh}{\left (\frac {\sqrt {b} x^{\frac {3}{2}}}{\sqrt {a}} \right )}}{3 b^{\frac {3}{2}}} - \frac {2 c^{\frac {7}{2}} x^{\frac {3}{2}}}{3 \sqrt {a} b \sqrt {1 + \frac {b x^{3}}{a}}} \] Input:
integrate((c*x)**(7/2)/(b*x**3+a)**(3/2),x)
Output:
2*c**(7/2)*asinh(sqrt(b)*x**(3/2)/sqrt(a))/(3*b**(3/2)) - 2*c**(7/2)*x**(3 /2)/(3*sqrt(a)*b*sqrt(1 + b*x**3/a))
\[ \int \frac {(c x)^{7/2}}{\left (a+b x^3\right )^{3/2}} \, dx=\int { \frac {\left (c x\right )^{\frac {7}{2}}}{{\left (b x^{3} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((c*x)^(7/2)/(b*x^3+a)^(3/2),x, algorithm="maxima")
Output:
integrate((c*x)^(7/2)/(b*x^3 + a)^(3/2), x)
Time = 0.17 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.11 \[ \int \frac {(c x)^{7/2}}{\left (a+b x^3\right )^{3/2}} \, dx=-\frac {2 \, \sqrt {c x} c^{5} x}{3 \, \sqrt {b c^{4} x^{3} + a c^{4}} b} - \frac {2 \, c^{6} \log \left ({\left | -\sqrt {b c} \sqrt {c x} c x + \sqrt {b c^{4} x^{3} + a c^{4}} \right |}\right )}{3 \, \sqrt {b c} b {\left | c \right |}^{2}} \] Input:
integrate((c*x)^(7/2)/(b*x^3+a)^(3/2),x, algorithm="giac")
Output:
-2/3*sqrt(c*x)*c^5*x/(sqrt(b*c^4*x^3 + a*c^4)*b) - 2/3*c^6*log(abs(-sqrt(b *c)*sqrt(c*x)*c*x + sqrt(b*c^4*x^3 + a*c^4)))/(sqrt(b*c)*b*abs(c)^2)
Timed out. \[ \int \frac {(c x)^{7/2}}{\left (a+b x^3\right )^{3/2}} \, dx=\int \frac {{\left (c\,x\right )}^{7/2}}{{\left (b\,x^3+a\right )}^{3/2}} \,d x \] Input:
int((c*x)^(7/2)/(a + b*x^3)^(3/2),x)
Output:
int((c*x)^(7/2)/(a + b*x^3)^(3/2), x)
Time = 0.27 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.70 \[ \int \frac {(c x)^{7/2}}{\left (a+b x^3\right )^{3/2}} \, dx=\frac {\sqrt {c}\, c^{3} \left (-2 \sqrt {x}\, \sqrt {b \,x^{3}+a}\, b x -\sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{3}+a}-\sqrt {x}\, \sqrt {b}\, x \right ) a -\sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{3}+a}-\sqrt {x}\, \sqrt {b}\, x \right ) b \,x^{3}+\sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{3}+a}+\sqrt {x}\, \sqrt {b}\, x \right ) a +\sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{3}+a}+\sqrt {x}\, \sqrt {b}\, x \right ) b \,x^{3}\right )}{3 b^{2} \left (b \,x^{3}+a \right )} \] Input:
int((c*x)^(7/2)/(b*x^3+a)^(3/2),x)
Output:
(sqrt(c)*c**3*( - 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)*b*x**3 + 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)*b*x**3))/(3*b**2*(a + b*x**3 ))