\(\int \frac {\sqrt [3]{a+b x} \sqrt {c+d x}}{e+f x} \, dx\) [3177]

Optimal result

Integrand size = 26, antiderivative size = 100 \[ \int \frac {\sqrt [3]{a+b x} \sqrt {c+d x}}{e+f x} \, dx=\frac {3 (a+b x)^{4/3} \sqrt {c+d x} \operatorname {AppellF1}\left (\frac {4}{3},-\frac {1}{2},1,\frac {7}{3},-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{4 (b e-a f) \sqrt {\frac {b (c+d x)}{b c-a d}}} \]

[Out]

3/4*(b*x+a)^(4/3)*AppellF1(4/3,-1/2,1,7/3,-d*(b*x+a)/(-a*d+b*c),-f*(b*x+a)/(-a*f+b*e))*(d*x+c)^(1/2)/(-a*f+b*e
)/(b*(d*x+c)/(-a*d+b*c))^(1/2)

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} \[ \int \frac {\sqrt [3]{a+b x} \sqrt {c+d x}}{e+f x} \, dx=\frac {3 (a+b x)^{4/3} \sqrt {c+d x} \operatorname {AppellF1}\left (\frac {4}{3},-\frac {1}{2},1,\frac {7}{3},-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{4 (b e-a f) \sqrt {\frac {b (c+d x)}{b c-a d}}} \]

[In]

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

[Out]

(3*(a + b*x)^(4/3)*Sqrt[c + d*x]*AppellF1[4/3, -1/2, 1, 7/3, -((d*(a + b*x))/(b*c - a*d)), -((f*(a + b*x))/(b*
e - a*f))])/(4*(b*e - a*f)*Sqrt[(b*(c + d*x))/(b*c - a*d)])

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 {c+d x} \int \frac {\sqrt [3]{a+b x} \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}}}{e+f x} \, dx}{\sqrt {\frac {b (c+d x)}{b c-a d}}} \\ & = \frac {3 (a+b x)^{4/3} \sqrt {c+d x} F_1\left (\frac {4}{3};-\frac {1}{2},1;\frac {7}{3};-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{4 (b e-a f) \sqrt {\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. \(201\) vs. \(2(100)=200\).

Time = 21.45 (sec) , antiderivative size = 201, normalized size of antiderivative = 2.01 \[ \int \frac {\sqrt [3]{a+b x} \sqrt {c+d x}}{e+f x} \, dx=\frac {6 \sqrt {c+d x} \left (7 f (a+b x)+\frac {\left (\frac {d (a+b x)}{b (c+d x)}\right )^{2/3} \left (7 (5 b d e-3 b c f-2 a d f) \operatorname {AppellF1}\left (\frac {1}{6},\frac {2}{3},1,\frac {7}{6},\frac {b c-a d}{b c+b d x},\frac {-d e+c f}{f (c+d x)}\right )+\frac {3 (b c-a d) (-d e+c f) \operatorname {AppellF1}\left (\frac {7}{6},\frac {2}{3},1,\frac {13}{6},\frac {b c-a d}{b c+b d x},\frac {-d e+c f}{f (c+d x)}\right )}{c+d x}\right )}{d}\right )}{35 f^2 (a+b x)^{2/3}} \]

[In]

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

[Out]

(6*Sqrt[c + d*x]*(7*f*(a + b*x) + (((d*(a + b*x))/(b*(c + d*x)))^(2/3)*(7*(5*b*d*e - 3*b*c*f - 2*a*d*f)*Appell
F1[1/6, 2/3, 1, 7/6, (b*c - a*d)/(b*c + b*d*x), (-(d*e) + c*f)/(f*(c + d*x))] + (3*(b*c - a*d)*(-(d*e) + c*f)*
AppellF1[7/6, 2/3, 1, 13/6, (b*c - a*d)/(b*c + b*d*x), (-(d*e) + c*f)/(f*(c + d*x))])/(c + d*x)))/d))/(35*f^2*
(a + b*x)^(2/3))

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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