3.3170 \(\int \frac{(a+b x)^{4/3}}{\sqrt{c+d x} (e+f x)} \, dx\)

Optimal. Leaf size=100 \[ \frac{3 (a+b x)^{7/3} \sqrt{\frac{b (c+d x)}{b c-a d}} F_1\left (\frac{7}{3};\frac{1}{2},1;\frac{10}{3};-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{7 \sqrt{c+d x} (b e-a f)} \]

[Out]

(3*(a + b*x)^(7/3)*Sqrt[(b*(c + d*x))/(b*c - a*d)]*AppellF1[7/3, 1/2, 1, 10/3, -((d*(a + b*x))/(b*c - a*d)), -
((f*(a + b*x))/(b*e - a*f))])/(7*(b*e - a*f)*Sqrt[c + d*x])

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Rubi [A]  time = 0.0363616, 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} \[ \frac{3 (a+b x)^{7/3} \sqrt{\frac{b (c+d x)}{b c-a d}} F_1\left (\frac{7}{3};\frac{1}{2},1;\frac{10}{3};-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{7 \sqrt{c+d x} (b e-a f)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(4/3)/(Sqrt[c + d*x]*(e + f*x)),x]

[Out]

(3*(a + b*x)^(7/3)*Sqrt[(b*(c + d*x))/(b*c - a*d)]*AppellF1[7/3, 1/2, 1, 10/3, -((d*(a + b*x))/(b*c - a*d)), -
((f*(a + b*x))/(b*e - a*f))])/(7*(b*e - a*f)*Sqrt[c + d*x])

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{(a+b x)^{4/3}}{\sqrt{c+d x} (e+f x)} \, dx &=\frac{\sqrt{\frac{b (c+d x)}{b c-a d}} \int \frac{(a+b x)^{4/3}}{\sqrt{\frac{b c}{b c-a d}+\frac{b d x}{b c-a d}} (e+f x)} \, dx}{\sqrt{c+d x}}\\ &=\frac{3 (a+b x)^{7/3} \sqrt{\frac{b (c+d x)}{b c-a d}} F_1\left (\frac{7}{3};\frac{1}{2},1;\frac{10}{3};-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{7 (b e-a f) \sqrt{c+d x}}\\ \end{align*}

Mathematica [B]  time = 0.703715, size = 212, normalized size = 2.12 \[ \frac{6 \sqrt{c+d x} \left (\left (\frac{d (a+b x)}{b (c+d x)}\right )^{2/3} \left (7 b (-7 a d f+2 b c f+5 b d e) F_1\left (\frac{1}{6};\frac{2}{3},1;\frac{7}{6};\frac{b c-a d}{b c+b d x},\frac{c f-d e}{f (c+d x)}\right )-\frac{(b c-a d) (-5 a d f+2 b c f+3 b d e) F_1\left (\frac{7}{6};\frac{2}{3},1;\frac{13}{6};\frac{b c-a d}{b c+b d x},\frac{c f-d e}{f (c+d x)}\right )}{c+d x}\right )+7 b d f (a+b x)\right )}{35 d^2 f^2 (a+b x)^{2/3}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*x)^(4/3)/(Sqrt[c + d*x]*(e + f*x)),x]

[Out]

(6*Sqrt[c + d*x]*(7*b*d*f*(a + b*x) + ((d*(a + b*x))/(b*(c + d*x)))^(2/3)*(7*b*(5*b*d*e + 2*b*c*f - 7*a*d*f)*A
ppellF1[1/6, 2/3, 1, 7/6, (b*c - a*d)/(b*c + b*d*x), (-(d*e) + c*f)/(f*(c + d*x))] - ((b*c - a*d)*(3*b*d*e + 2
*b*c*f - 5*a*d*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))))/(35*d^2*f^2*(a + b*x)^(2/3))

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Maple [F]  time = 0.064, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{fx+e} \left ( bx+a \right ) ^{{\frac{4}{3}}}{\frac{1}{\sqrt{dx+c}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{\frac{4}{3}}}{\sqrt{d x + c}{\left (f x + e\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(4/3)/(f*x+e)/(d*x+c)^(1/2),x, algorithm="maxima")

[Out]

integrate((b*x + a)^(4/3)/(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)^(4/3)/(f*x+e)/(d*x+c)^(1/2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b x\right )^{\frac{4}{3}}}{\sqrt{c + d x} \left (e + f x\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**(4/3)/(f*x+e)/(d*x+c)**(1/2),x)

[Out]

Integral((a + b*x)**(4/3)/(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{4}{3}}}{\sqrt{d x + c}{\left (f x + e\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(4/3)/(f*x+e)/(d*x+c)^(1/2),x, algorithm="giac")

[Out]

integrate((b*x + a)^(4/3)/(sqrt(d*x + c)*(f*x + e)), x)