\(\int \frac {\sqrt {a+b x} \sqrt [3]{c+d x}}{e+f x} \, dx\) [3176]

Optimal result

Integrand size = 26, antiderivative size = 100 \[ \int \frac {\sqrt {a+b x} \sqrt [3]{c+d x}}{e+f x} \, dx=\frac {2 (a+b x)^{3/2} \sqrt [3]{c+d x} \operatorname {AppellF1}\left (\frac {3}{2},-\frac {1}{3},1,\frac {5}{2},-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{3 (b e-a f) \sqrt [3]{\frac {b (c+d x)}{b c-a d}}} \]

[Out]

2/3*(b*x+a)^(3/2)*(d*x+c)^(1/3)*AppellF1(3/2,-1/3,1,5/2,-d*(b*x+a)/(-a*d+b*c),-f*(b*x+a)/(-a*f+b*e))/(-a*f+b*e
)/(b*(d*x+c)/(-a*d+b*c))^(1/3)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {142, 141} \[ \int \frac {\sqrt {a+b x} \sqrt [3]{c+d x}}{e+f x} \, dx=\frac {2 (a+b x)^{3/2} \sqrt [3]{c+d x} \operatorname {AppellF1}\left (\frac {3}{2},-\frac {1}{3},1,\frac {5}{2},-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{3 (b e-a f) \sqrt [3]{\frac {b (c+d x)}{b c-a d}}} \]

[In]

Int[(Sqrt[a + b*x]*(c + d*x)^(1/3))/(e + f*x),x]

[Out]

(2*(a + b*x)^(3/2)*(c + d*x)^(1/3)*AppellF1[3/2, -1/3, 1, 5/2, -((d*(a + b*x))/(b*c - a*d)), -((f*(a + b*x))/(
b*e - a*f))])/(3*(b*e - a*f)*((b*(c + d*x))/(b*c - a*d))^(1/3))

Rule 141

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[(b*e - a*f
)^p*((a + b*x)^(m + 1)/(b^(p + 1)*(m + 1)*(b/(b*c - a*d))^n))*AppellF1[m + 1, -n, -p, m + 2, (-d)*((a + b*x)/(
b*c - a*d)), (-f)*((a + b*x)/(b*e - a*f))], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] &&  !IntegerQ[m] &&  !Int
egerQ[n] && IntegerQ[p] && GtQ[b/(b*c - a*d), 0] &&  !(GtQ[d/(d*a - c*b), 0] && SimplerQ[c + d*x, a + b*x])

Rule 142

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Dist[(c + d*x)^
FracPart[n]/((b/(b*c - a*d))^IntPart[n]*(b*((c + d*x)/(b*c - a*d)))^FracPart[n]), Int[(a + b*x)^m*(b*(c/(b*c -
a*d)) + b*d*(x/(b*c - a*d)))^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] &&  !IntegerQ[m] &&
!IntegerQ[n] && IntegerQ[p] &&  !GtQ[b/(b*c - a*d), 0] &&  !SimplerQ[c + d*x, a + b*x]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [3]{c+d x} \int \frac {\sqrt {a+b x} \sqrt [3]{\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}}}{e+f x} \, dx}{\sqrt [3]{\frac {b (c+d x)}{b c-a d}}} \\ & = \frac {2 (a+b x)^{3/2} \sqrt [3]{c+d x} F_1\left (\frac {3}{2};-\frac {1}{3},1;\frac {5}{2};-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{3 (b e-a f) \sqrt [3]{\frac {b (c+d x)}{b c-a d}}} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(202\) vs. \(2(100)=200\).

Time = 21.45 (sec) , antiderivative size = 202, normalized size of antiderivative = 2.02 \[ \int \frac {\sqrt {a+b x} \sqrt [3]{c+d x}}{e+f x} \, dx=\frac {6 \sqrt {a+b x} \left (7 f (c+d x)-\frac {\left (\frac {b (c+d x)}{d (a+b x)}\right )^{2/3} \left (-7 (5 b d e-2 b c f-3 a d f) \operatorname {AppellF1}\left (\frac {1}{6},\frac {2}{3},1,\frac {7}{6},\frac {-b c+a d}{d (a+b x)},\frac {-b e+a f}{f (a+b x)}\right )-\frac {3 (b c-a d) (b e-a f) \operatorname {AppellF1}\left (\frac {7}{6},\frac {2}{3},1,\frac {13}{6},\frac {-b c+a d}{d (a+b x)},\frac {-b e+a f}{f (a+b x)}\right )}{a+b x}\right )}{b}\right )}{35 f^2 (c+d x)^{2/3}} \]

[In]

Integrate[(Sqrt[a + b*x]*(c + d*x)^(1/3))/(e + f*x),x]

[Out]

(6*Sqrt[a + b*x]*(7*f*(c + d*x) - (((b*(c + d*x))/(d*(a + b*x)))^(2/3)*(-7*(5*b*d*e - 2*b*c*f - 3*a*d*f)*Appel
lF1[1/6, 2/3, 1, 7/6, (-(b*c) + a*d)/(d*(a + b*x)), (-(b*e) + a*f)/(f*(a + b*x))] - (3*(b*c - a*d)*(b*e - a*f)
*AppellF1[7/6, 2/3, 1, 13/6, (-(b*c) + a*d)/(d*(a + b*x)), (-(b*e) + a*f)/(f*(a + b*x))])/(a + b*x)))/b))/(35*
f^2*(c + d*x)^(2/3))

Maple [F]

\[\int \frac {\sqrt {b x +a}\, \left (d x +c \right )^{\frac {1}{3}}}{f x +e}d x\]

[In]

int((b*x+a)^(1/2)*(d*x+c)^(1/3)/(f*x+e),x)

[Out]

int((b*x+a)^(1/2)*(d*x+c)^(1/3)/(f*x+e),x)

Fricas [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x} \sqrt [3]{c+d x}}{e+f x} \, dx=\text {Timed out} \]

[In]

integrate((b*x+a)^(1/2)*(d*x+c)^(1/3)/(f*x+e),x, algorithm="fricas")

[Out]

Timed out

Sympy [F]

\[ \int \frac {\sqrt {a+b x} \sqrt [3]{c+d x}}{e+f x} \, dx=\int \frac {\sqrt {a + b x} \sqrt [3]{c + d x}}{e + f x}\, dx \]

[In]

integrate((b*x+a)**(1/2)*(d*x+c)**(1/3)/(f*x+e),x)

[Out]

Integral(sqrt(a + b*x)*(c + d*x)**(1/3)/(e + f*x), x)

Maxima [F]

\[ \int \frac {\sqrt {a+b x} \sqrt [3]{c+d x}}{e+f x} \, dx=\int { \frac {\sqrt {b x + a} {\left (d x + c\right )}^{\frac {1}{3}}}{f x + e} \,d x } \]

[In]

integrate((b*x+a)^(1/2)*(d*x+c)^(1/3)/(f*x+e),x, algorithm="maxima")

[Out]

integrate(sqrt(b*x + a)*(d*x + c)^(1/3)/(f*x + e), x)

Giac [F]

\[ \int \frac {\sqrt {a+b x} \sqrt [3]{c+d x}}{e+f x} \, dx=\int { \frac {\sqrt {b x + a} {\left (d x + c\right )}^{\frac {1}{3}}}{f x + e} \,d x } \]

[In]

integrate((b*x+a)^(1/2)*(d*x+c)^(1/3)/(f*x+e),x, algorithm="giac")

[Out]

integrate(sqrt(b*x + a)*(d*x + c)^(1/3)/(f*x + e), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x} \sqrt [3]{c+d x}}{e+f x} \, dx=\int \frac {\sqrt {a+b\,x}\,{\left (c+d\,x\right )}^{1/3}}{e+f\,x} \,d x \]

[In]

int(((a + b*x)^(1/2)*(c + d*x)^(1/3))/(e + f*x),x)

[Out]

int(((a + b*x)^(1/2)*(c + d*x)^(1/3))/(e + f*x), x)