3.3176 \(\int \frac{\sqrt{a+b x} \sqrt [3]{c+d x}}{e+f x} \, dx\)

Optimal. Leaf size=100 \[ \frac{2 (a+b x)^{3/2} \sqrt [3]{c+d x} F_1\left (\frac{3}{2};-\frac{1}{3},1;\frac{5}{2};-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{3 (b e-a f) \sqrt [3]{\frac{b (c+d x)}{b c-a d}}} \]

[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))

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Rubi [A]  time = 0.0372164, 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} \[ \frac{2 (a+b x)^{3/2} \sqrt [3]{c+d x} F_1\left (\frac{3}{2};-\frac{1}{3},1;\frac{5}{2};-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{3 (b e-a f) \sqrt [3]{\frac{b (c+d x)}{b c-a d}}} \]

Antiderivative was successfully verified.

[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 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{a+b x} \sqrt [3]{c+d x}}{e+f x} \, dx &=\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]  time = 0.553191, size = 202, normalized size = 2.02 \[ \frac{6 \sqrt{a+b x} \left (7 f (c+d x)-\frac{\left (\frac{b (c+d x)}{d (a+b x)}\right )^{2/3} \left (-7 (-3 a d f-2 b c f+5 b d e) F_1\left (\frac{1}{6};\frac{2}{3},1;\frac{7}{6};\frac{a d-b c}{d (a+b x)},\frac{a f-b e}{f (a+b x)}\right )-\frac{3 (b c-a d) (b e-a f) F_1\left (\frac{7}{6};\frac{2}{3},1;\frac{13}{6};\frac{a d-b c}{d (a+b x)},\frac{a f-b e}{f (a+b x)}\right )}{a+b x}\right )}{b}\right )}{35 f^2 (c+d x)^{2/3}} \]

Warning: Unable to verify antiderivative.

[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))

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Maple [F]  time = 0.052, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{fx+e}\sqrt{bx+a}\sqrt [3]{dx+c}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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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)^(1/2)*(d*x+c)^(1/3)/(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{a + b x} \sqrt [3]{c + d x}}{e + f x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[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)