Integrand size = 15, antiderivative size = 78 \[ \int \frac {\sqrt [4]{a+b x}}{x^{7/4}} \, dx=-\frac {4 \sqrt [4]{a+b x}}{3 x^{3/4}}-\frac {4 \left (\frac {b x}{a+b x}\right )^{3/4} (a+b x)^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arcsin \left (\frac {\sqrt {a}}{\sqrt {a+b x}}\right ),2\right )}{3 \sqrt {a} x^{3/4}} \] Output:
-4/3*(b*x+a)^(1/4)/x^(3/4)-4/3*(b*x/(b*x+a))^(3/4)*(b*x+a)^(3/4)*InverseJa cobiAM(1/2*arcsin(a^(1/2)/(b*x+a)^(1/2)),2^(1/2))/a^(1/2)/x^(3/4)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.01 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.60 \[ \int \frac {\sqrt [4]{a+b x}}{x^{7/4}} \, dx=-\frac {4 \sqrt [4]{a+b x} \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},-\frac {1}{4},\frac {1}{4},-\frac {b x}{a}\right )}{3 x^{3/4} \sqrt [4]{1+\frac {b x}{a}}} \] Input:
Integrate[(a + b*x)^(1/4)/x^(7/4),x]
Output:
(-4*(a + b*x)^(1/4)*Hypergeometric2F1[-3/4, -1/4, 1/4, -((b*x)/a)])/(3*x^( 3/4)*(1 + (b*x)/a)^(1/4))
Time = 0.21 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {57, 73, 768, 858, 807, 229}
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.
\(\displaystyle \int \frac {\sqrt [4]{a+b x}}{x^{7/4}} \, dx\) |
\(\Big \downarrow \) 57 |
\(\displaystyle \frac {1}{3} b \int \frac {1}{x^{3/4} (a+b x)^{3/4}}dx-\frac {4 \sqrt [4]{a+b x}}{3 x^{3/4}}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {4}{3} b \int \frac {1}{(a+b x)^{3/4}}d\sqrt [4]{x}-\frac {4 \sqrt [4]{a+b x}}{3 x^{3/4}}\) |
\(\Big \downarrow \) 768 |
\(\displaystyle \frac {4 b x^{3/4} \left (\frac {a}{b x}+1\right )^{3/4} \int \frac {1}{\left (\frac {a}{b x}+1\right )^{3/4} x^{3/4}}d\sqrt [4]{x}}{3 (a+b x)^{3/4}}-\frac {4 \sqrt [4]{a+b x}}{3 x^{3/4}}\) |
\(\Big \downarrow \) 858 |
\(\displaystyle -\frac {4 b x^{3/4} \left (\frac {a}{b x}+1\right )^{3/4} \int \frac {1}{\sqrt [4]{x} \left (\frac {a x}{b}+1\right )^{3/4}}d\frac {1}{\sqrt [4]{x}}}{3 (a+b x)^{3/4}}-\frac {4 \sqrt [4]{a+b x}}{3 x^{3/4}}\) |
\(\Big \downarrow \) 807 |
\(\displaystyle -\frac {2 b x^{3/4} \left (\frac {a}{b x}+1\right )^{3/4} \int \frac {1}{\left (\frac {\sqrt {x} a}{b}+1\right )^{3/4}}d\sqrt {x}}{3 (a+b x)^{3/4}}-\frac {4 \sqrt [4]{a+b x}}{3 x^{3/4}}\) |
\(\Big \downarrow \) 229 |
\(\displaystyle -\frac {4 b^{3/2} x^{3/4} \left (\frac {a}{b x}+1\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),2\right )}{3 \sqrt {a} (a+b x)^{3/4}}-\frac {4 \sqrt [4]{a+b x}}{3 x^{3/4}}\) |
Input:
Int[(a + b*x)^(1/4)/x^(7/4),x]
Output:
(-4*(a + b*x)^(1/4))/(3*x^(3/4)) - (4*b^(3/2)*(1 + a/(b*x))^(3/4)*x^(3/4)* EllipticF[ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]]/2, 2])/(3*Sqrt[a]*(a + b*x)^(3 /4))
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] & & GtQ[n, 0] && LtQ[m, -1] && !(IntegerQ[n] && !IntegerQ[m]) && !(ILeQ[m + n + 2, 0] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[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)^(-3/4), x_Symbol] :> Simp[(2/(a^(3/4)*Rt[b/a, 2]) )*EllipticF[(1/2)*ArcTan[Rt[b/a, 2]*x], 2], x] /; FreeQ[{a, b}, x] && GtQ[a , 0] && PosQ[b/a]
Int[((a_) + (b_.)*(x_)^4)^(-3/4), x_Symbol] :> Simp[x^3*((1 + a/(b*x^4))^(3 /4)/(a + b*x^4)^(3/4)) Int[1/(x^3*(1 + a/(b*x^4))^(3/4)), x], x] /; FreeQ [{a, b}, x]
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[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x] /; FreeQ[{a, b, p}, x] && ILtQ[n, 0] && Int egerQ[m]
\[\int \frac {\left (b x +a \right )^{\frac {1}{4}}}{x^{\frac {7}{4}}}d x\]
Input:
int((b*x+a)^(1/4)/x^(7/4),x)
Output:
int((b*x+a)^(1/4)/x^(7/4),x)
\[ \int \frac {\sqrt [4]{a+b x}}{x^{7/4}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {1}{4}}}{x^{\frac {7}{4}}} \,d x } \] Input:
integrate((b*x+a)^(1/4)/x^(7/4),x, algorithm="fricas")
Output:
integral((b*x + a)^(1/4)/x^(7/4), x)
Result contains complex when optimal does not.
Time = 1.46 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.40 \[ \int \frac {\sqrt [4]{a+b x}}{x^{7/4}} \, dx=- \frac {2 \sqrt [4]{b} {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {1}{2} \\ \frac {3}{2} \end {matrix}\middle | {\frac {a e^{i \pi }}{b x}} \right )}}{\sqrt {x}} \] Input:
integrate((b*x+a)**(1/4)/x**(7/4),x)
Output:
-2*b**(1/4)*hyper((-1/4, 1/2), (3/2,), a*exp_polar(I*pi)/(b*x))/sqrt(x)
\[ \int \frac {\sqrt [4]{a+b x}}{x^{7/4}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {1}{4}}}{x^{\frac {7}{4}}} \,d x } \] Input:
integrate((b*x+a)^(1/4)/x^(7/4),x, algorithm="maxima")
Output:
integrate((b*x + a)^(1/4)/x^(7/4), x)
\[ \int \frac {\sqrt [4]{a+b x}}{x^{7/4}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {1}{4}}}{x^{\frac {7}{4}}} \,d x } \] Input:
integrate((b*x+a)^(1/4)/x^(7/4),x, algorithm="giac")
Output:
integrate((b*x + a)^(1/4)/x^(7/4), x)
Timed out. \[ \int \frac {\sqrt [4]{a+b x}}{x^{7/4}} \, dx=\int \frac {{\left (a+b\,x\right )}^{1/4}}{x^{7/4}} \,d x \] Input:
int((a + b*x)^(1/4)/x^(7/4),x)
Output:
int((a + b*x)^(1/4)/x^(7/4), x)
\[ \int \frac {\sqrt [4]{a+b x}}{x^{7/4}} \, dx=\frac {-4 \left (b x +a \right )^{\frac {1}{4}}-x^{\frac {3}{4}} \left (\int \frac {\left (b x +a \right )^{\frac {1}{4}}}{x^{\frac {7}{4}} a +x^{\frac {11}{4}} b}d x \right ) a}{2 x^{\frac {3}{4}}} \] Input:
int((b*x+a)^(1/4)/x^(7/4),x)
Output:
( - 4*(a + b*x)**(1/4) - x**(3/4)*int((a + b*x)**(1/4)/(x**(3/4)*a*x + x** (3/4)*b*x**2),x)*a)/(2*x**(3/4))