Integrand size = 15, antiderivative size = 81 \[ \int \frac {x^{3/4}}{(a+b x)^{3/4}} \, dx=\frac {x^{3/4} \sqrt [4]{a+b x}}{b}+\frac {3 a \arctan \left (\frac {\sqrt [4]{b} \sqrt [4]{x}}{\sqrt [4]{a+b x}}\right )}{2 b^{7/4}}-\frac {3 a \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt [4]{x}}{\sqrt [4]{a+b x}}\right )}{2 b^{7/4}} \] Output:
x^(3/4)*(b*x+a)^(1/4)/b+3/2*a*arctan(b^(1/4)*x^(1/4)/(b*x+a)^(1/4))/b^(7/4 )-3/2*a*arctanh(b^(1/4)*x^(1/4)/(b*x+a)^(1/4))/b^(7/4)
Time = 0.29 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00 \[ \int \frac {x^{3/4}}{(a+b x)^{3/4}} \, dx=\frac {x^{3/4} \sqrt [4]{a+b x}}{b}+\frac {3 a \arctan \left (\frac {\sqrt [4]{b} \sqrt [4]{x}}{\sqrt [4]{a+b x}}\right )}{2 b^{7/4}}-\frac {3 a \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt [4]{x}}{\sqrt [4]{a+b x}}\right )}{2 b^{7/4}} \] Input:
Integrate[x^(3/4)/(a + b*x)^(3/4),x]
Output:
(x^(3/4)*(a + b*x)^(1/4))/b + (3*a*ArcTan[(b^(1/4)*x^(1/4))/(a + b*x)^(1/4 )])/(2*b^(7/4)) - (3*a*ArcTanh[(b^(1/4)*x^(1/4))/(a + b*x)^(1/4)])/(2*b^(7 /4))
Time = 0.19 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.06, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {60, 73, 854, 827, 216, 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.
\(\displaystyle \int \frac {x^{3/4}}{(a+b x)^{3/4}} \, dx\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {x^{3/4} \sqrt [4]{a+b x}}{b}-\frac {3 a \int \frac {1}{\sqrt [4]{x} (a+b x)^{3/4}}dx}{4 b}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {x^{3/4} \sqrt [4]{a+b x}}{b}-\frac {3 a \int \frac {\sqrt {x}}{(a+b x)^{3/4}}d\sqrt [4]{x}}{b}\) |
\(\Big \downarrow \) 854 |
\(\displaystyle \frac {x^{3/4} \sqrt [4]{a+b x}}{b}-\frac {3 a \int \frac {\sqrt {x}}{1-b x}d\frac {\sqrt [4]{x}}{\sqrt [4]{a+b x}}}{b}\) |
\(\Big \downarrow \) 827 |
\(\displaystyle \frac {x^{3/4} \sqrt [4]{a+b x}}{b}-\frac {3 a \left (\frac {\int \frac {1}{1-\sqrt {b} \sqrt {x}}d\frac {\sqrt [4]{x}}{\sqrt [4]{a+b x}}}{2 \sqrt {b}}-\frac {\int \frac {1}{\sqrt {b} \sqrt {x}+1}d\frac {\sqrt [4]{x}}{\sqrt [4]{a+b x}}}{2 \sqrt {b}}\right )}{b}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {x^{3/4} \sqrt [4]{a+b x}}{b}-\frac {3 a \left (\frac {\int \frac {1}{1-\sqrt {b} \sqrt {x}}d\frac {\sqrt [4]{x}}{\sqrt [4]{a+b x}}}{2 \sqrt {b}}-\frac {\arctan \left (\frac {\sqrt [4]{b} \sqrt [4]{x}}{\sqrt [4]{a+b x}}\right )}{2 b^{3/4}}\right )}{b}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {x^{3/4} \sqrt [4]{a+b x}}{b}-\frac {3 a \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt [4]{x}}{\sqrt [4]{a+b x}}\right )}{2 b^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{b} \sqrt [4]{x}}{\sqrt [4]{a+b x}}\right )}{2 b^{3/4}}\right )}{b}\) |
Input:
Int[x^(3/4)/(a + b*x)^(3/4),x]
Output:
(x^(3/4)*(a + b*x)^(1/4))/b - (3*a*(-1/2*ArcTan[(b^(1/4)*x^(1/4))/(a + b*x )^(1/4)]/b^(3/4) + ArcTanh[(b^(1/4)*x^(1/4))/(a + b*x)^(1/4)]/(2*b^(3/4))) )/b
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
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[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b) Int[1/(r + s*x^2), x], x] - Simp[s/(2*b) Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ [a/b, 0]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + (m + 1)/n) Subst[Int[x^m/(1 - b*x^n)^(p + (m + 1)/n + 1), x], x, x/(a + b*x^n )^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, - 2^(-1)] && IntegersQ[m, p + (m + 1)/n]
\[\int \frac {x^{\frac {3}{4}}}{\left (b x +a \right )^{\frac {3}{4}}}d x\]
Input:
int(x^(3/4)/(b*x+a)^(3/4),x)
Output:
int(x^(3/4)/(b*x+a)^(3/4),x)
Result contains complex when optimal does not.
Time = 0.10 (sec) , antiderivative size = 203, normalized size of antiderivative = 2.51 \[ \int \frac {x^{3/4}}{(a+b x)^{3/4}} \, dx=-\frac {3 \, b \left (\frac {a^{4}}{b^{7}}\right )^{\frac {1}{4}} \log \left (\frac {3 \, {\left (b^{2} x \left (\frac {a^{4}}{b^{7}}\right )^{\frac {1}{4}} + {\left (b x + a\right )}^{\frac {1}{4}} a x^{\frac {3}{4}}\right )}}{x}\right ) - 3 \, b \left (\frac {a^{4}}{b^{7}}\right )^{\frac {1}{4}} \log \left (-\frac {3 \, {\left (b^{2} x \left (\frac {a^{4}}{b^{7}}\right )^{\frac {1}{4}} - {\left (b x + a\right )}^{\frac {1}{4}} a x^{\frac {3}{4}}\right )}}{x}\right ) - 3 i \, b \left (\frac {a^{4}}{b^{7}}\right )^{\frac {1}{4}} \log \left (-\frac {3 \, {\left (i \, b^{2} x \left (\frac {a^{4}}{b^{7}}\right )^{\frac {1}{4}} - {\left (b x + a\right )}^{\frac {1}{4}} a x^{\frac {3}{4}}\right )}}{x}\right ) + 3 i \, b \left (\frac {a^{4}}{b^{7}}\right )^{\frac {1}{4}} \log \left (-\frac {3 \, {\left (-i \, b^{2} x \left (\frac {a^{4}}{b^{7}}\right )^{\frac {1}{4}} - {\left (b x + a\right )}^{\frac {1}{4}} a x^{\frac {3}{4}}\right )}}{x}\right ) - 4 \, {\left (b x + a\right )}^{\frac {1}{4}} x^{\frac {3}{4}}}{4 \, b} \] Input:
integrate(x^(3/4)/(b*x+a)^(3/4),x, algorithm="fricas")
Output:
-1/4*(3*b*(a^4/b^7)^(1/4)*log(3*(b^2*x*(a^4/b^7)^(1/4) + (b*x + a)^(1/4)*a *x^(3/4))/x) - 3*b*(a^4/b^7)^(1/4)*log(-3*(b^2*x*(a^4/b^7)^(1/4) - (b*x + a)^(1/4)*a*x^(3/4))/x) - 3*I*b*(a^4/b^7)^(1/4)*log(-3*(I*b^2*x*(a^4/b^7)^( 1/4) - (b*x + a)^(1/4)*a*x^(3/4))/x) + 3*I*b*(a^4/b^7)^(1/4)*log(-3*(-I*b^ 2*x*(a^4/b^7)^(1/4) - (b*x + a)^(1/4)*a*x^(3/4))/x) - 4*(b*x + a)^(1/4)*x^ (3/4))/b
Result contains complex when optimal does not.
Time = 1.33 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.44 \[ \int \frac {x^{3/4}}{(a+b x)^{3/4}} \, dx=\frac {x^{\frac {7}{4}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {b x e^{i \pi }}{a}} \right )}}{a^{\frac {3}{4}} \Gamma \left (\frac {11}{4}\right )} \] Input:
integrate(x**(3/4)/(b*x+a)**(3/4),x)
Output:
x**(7/4)*gamma(7/4)*hyper((3/4, 7/4), (11/4,), b*x*exp_polar(I*pi)/a)/(a** (3/4)*gamma(11/4))
Time = 0.11 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.23 \[ \int \frac {x^{3/4}}{(a+b x)^{3/4}} \, dx=-\frac {3 \, {\left (\frac {2 \, a \arctan \left (\frac {{\left (b x + a\right )}^{\frac {1}{4}}}{b^{\frac {1}{4}} x^{\frac {1}{4}}}\right )}{b^{\frac {3}{4}}} - \frac {a \log \left (-\frac {b^{\frac {1}{4}} - \frac {{\left (b x + a\right )}^{\frac {1}{4}}}{x^{\frac {1}{4}}}}{b^{\frac {1}{4}} + \frac {{\left (b x + a\right )}^{\frac {1}{4}}}{x^{\frac {1}{4}}}}\right )}{b^{\frac {3}{4}}}\right )}}{4 \, b} - \frac {{\left (b x + a\right )}^{\frac {1}{4}} a}{{\left (b^{2} - \frac {{\left (b x + a\right )} b}{x}\right )} x^{\frac {1}{4}}} \] Input:
integrate(x^(3/4)/(b*x+a)^(3/4),x, algorithm="maxima")
Output:
-3/4*(2*a*arctan((b*x + a)^(1/4)/(b^(1/4)*x^(1/4)))/b^(3/4) - a*log(-(b^(1 /4) - (b*x + a)^(1/4)/x^(1/4))/(b^(1/4) + (b*x + a)^(1/4)/x^(1/4)))/b^(3/4 ))/b - (b*x + a)^(1/4)*a/((b^2 - (b*x + a)*b/x)*x^(1/4))
\[ \int \frac {x^{3/4}}{(a+b x)^{3/4}} \, dx=\int { \frac {x^{\frac {3}{4}}}{{\left (b x + a\right )}^{\frac {3}{4}}} \,d x } \] Input:
integrate(x^(3/4)/(b*x+a)^(3/4),x, algorithm="giac")
Output:
integrate(x^(3/4)/(b*x + a)^(3/4), x)
Timed out. \[ \int \frac {x^{3/4}}{(a+b x)^{3/4}} \, dx=\int \frac {x^{3/4}}{{\left (a+b\,x\right )}^{3/4}} \,d x \] Input:
int(x^(3/4)/(a + b*x)^(3/4),x)
Output:
int(x^(3/4)/(a + b*x)^(3/4), x)
\[ \int \frac {x^{3/4}}{(a+b x)^{3/4}} \, dx=\int \frac {x^{\frac {3}{4}}}{\left (b x +a \right )^{\frac {3}{4}}}d x \] Input:
int(x^(3/4)/(b*x+a)^(3/4),x)
Output:
int(x**(3/4)/(a + b*x)**(3/4),x)