3.32.70 \(\int \frac {(a+b x)^{4/3}}{\sqrt {c+d x} (e+f x)} \, dx\) [3170]

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

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 100, normalized size of antiderivative = 1.00, 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 = {142, 141} \begin {gather*} \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)} \end {gather*}

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 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*} \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*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(212\) vs. \(2(100)=200\).
time = 20.54, size = 212, normalized size = 2.12 \begin {gather*} \frac {6 \sqrt {c+d x} \left (7 b d f (a+b x)+\left (\frac {d (a+b x)}{b (c+d x)}\right )^{2/3} \left (7 b (5 b d e+2 b c f-7 a d f) F_1\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 {(b c-a d) (3 b d e+2 b c f-5 a d f) F_1\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 )\right )}{35 d^2 f^2 (a+b x)^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[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 = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (b x +a \right )^{\frac {4}{3}}}{\left (f x +e \right ) \sqrt {d x +c}}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b x\right )^{\frac {4}{3}}}{\sqrt {c + d x} \left (e + f x\right )}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^{4/3}}{\left (e+f\,x\right )\,\sqrt {c+d\,x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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