3.3177 \(\int \frac{\sqrt [3]{a+b x} \sqrt{c+d x}}{e+f x} \, dx\)

Optimal. Leaf size=100 \[ \frac{3 (a+b x)^{4/3} \sqrt{c+d x} F_1\left (\frac{4}{3};-\frac{1}{2},1;\frac{7}{3};-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{4 (b e-a f) \sqrt{\frac{b (c+d x)}{b c-a d}}} \]

[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)])

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Rubi [A]  time = 0.0355959, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {137, 136} \[ \frac{3 (a+b x)^{4/3} \sqrt{c+d x} F_1\left (\frac{4}{3};-\frac{1}{2},1;\frac{7}{3};-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{4 (b e-a f) \sqrt{\frac{b (c+d x)}{b c-a d}}} \]

Antiderivative was successfully verified.

[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 137

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]

Rule 136

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

Rubi steps

\begin{align*} \int \frac{\sqrt [3]{a+b x} \sqrt{c+d x}}{e+f x} \, dx &=\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]  time = 0.574925, size = 201, normalized size = 2.01 \[ \frac{6 \sqrt{c+d x} \left (\frac{\left (\frac{d (a+b x)}{b (c+d x)}\right )^{2/3} \left (7 (-2 a d f-3 b c f+5 b d e) F_1\left (\frac{1}{6};\frac{2}{3},1;\frac{7}{6};\frac{b c-a d}{b c+b d x},\frac{c f-d e}{f (c+d x)}\right )+\frac{3 (b c-a d) (c f-d e) F_1\left (\frac{7}{6};\frac{2}{3},1;\frac{13}{6};\frac{b c-a d}{b c+b d x},\frac{c f-d e}{f (c+d x)}\right )}{c+d x}\right )}{d}+7 f (a+b x)\right )}{35 f^2 (a+b x)^{2/3}} \]

Warning: Unable to verify antiderivative.

[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))

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Maple [F]  time = 0.043, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{fx+e}\sqrt [3]{bx+a}\sqrt{dx+c}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{\frac{1}{3}} \sqrt{d x + c}}{f x + e}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt [3]{a + b x} \sqrt{c + d x}}{e + f x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{\frac{1}{3}} \sqrt{d x + c}}{f x + e}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)