3.3171 \(\int \frac{(c+d x)^{2/5} (e+f x)^{3/5}}{\sqrt{a+b x}} \, dx\)

Optimal. Leaf size=123 \[ \frac{2 \sqrt{a+b x} (c+d x)^{2/5} (e+f x)^{3/5} F_1\left (\frac{1}{2};-\frac{2}{5},-\frac{3}{5};\frac{3}{2};-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{b \left (\frac{b (c+d x)}{b c-a d}\right )^{2/5} \left (\frac{b (e+f x)}{b e-a f}\right )^{3/5}} \]

[Out]

(2*Sqrt[a + b*x]*(c + d*x)^(2/5)*(e + f*x)^(3/5)*AppellF1[1/2, -2/5, -3/5, 3/2, -((d*(a + b*x))/(b*c - a*d)),
-((f*(a + b*x))/(b*e - a*f))])/(b*((b*(c + d*x))/(b*c - a*d))^(2/5)*((b*(e + f*x))/(b*e - a*f))^(3/5))

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Rubi [A]  time = 0.0724332, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107, Rules used = {140, 139, 138} \[ \frac{2 \sqrt{a+b x} (c+d x)^{2/5} (e+f x)^{3/5} F_1\left (\frac{1}{2};-\frac{2}{5},-\frac{3}{5};\frac{3}{2};-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{b \left (\frac{b (c+d x)}{b c-a d}\right )^{2/5} \left (\frac{b (e+f x)}{b e-a f}\right )^{3/5}} \]

Antiderivative was successfully verified.

[In]

Int[((c + d*x)^(2/5)*(e + f*x)^(3/5))/Sqrt[a + b*x],x]

[Out]

(2*Sqrt[a + b*x]*(c + d*x)^(2/5)*(e + f*x)^(3/5)*AppellF1[1/2, -2/5, -3/5, 3/2, -((d*(a + b*x))/(b*c - a*d)),
-((f*(a + b*x))/(b*e - a*f))])/(b*((b*(c + d*x))/(b*c - a*d))^(2/5)*((b*(e + f*x))/(b*e - a*f))^(3/5))

Rule 140

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, p}, x] &&  !IntegerQ[m]
&&  !IntegerQ[n] &&  !IntegerQ[p] &&  !GtQ[b/(b*c - a*d), 0] &&  !SimplerQ[c + d*x, a + b*x] &&  !SimplerQ[e +
f*x, a + b*x]

Rule 139

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Dist[(e + f*x)^
FracPart[p]/((b/(b*e - a*f))^IntPart[p]*((b*(e + f*x))/(b*e - a*f))^FracPart[p]), Int[(a + b*x)^m*(c + d*x)^n*
((b*e)/(b*e - a*f) + (b*f*x)/(b*e - a*f))^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] &&  !IntegerQ[m]
&&  !IntegerQ[n] &&  !IntegerQ[p] && GtQ[b/(b*c - a*d), 0] &&  !GtQ[b/(b*e - a*f), 0]

Rule 138

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[((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*(m + 1
)*(b/(b*c - a*d))^n*(b/(b*e - a*f))^p), x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] &&  !IntegerQ[m] &&  !Inte
gerQ[n] &&  !IntegerQ[p] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] &&  !(GtQ[d/(d*a - c*b), 0] && GtQ[
d/(d*e - c*f), 0] && SimplerQ[c + d*x, a + b*x]) &&  !(GtQ[f/(f*a - e*b), 0] && GtQ[f/(f*c - e*d), 0] && Simpl
erQ[e + f*x, a + b*x])

Rubi steps

\begin{align*} \int \frac{(c+d x)^{2/5} (e+f x)^{3/5}}{\sqrt{a+b x}} \, dx &=\frac{(c+d x)^{2/5} \int \frac{\left (\frac{b c}{b c-a d}+\frac{b d x}{b c-a d}\right )^{2/5} (e+f x)^{3/5}}{\sqrt{a+b x}} \, dx}{\left (\frac{b (c+d x)}{b c-a d}\right )^{2/5}}\\ &=\frac{\left ((c+d x)^{2/5} (e+f x)^{3/5}\right ) \int \frac{\left (\frac{b c}{b c-a d}+\frac{b d x}{b c-a d}\right )^{2/5} \left (\frac{b e}{b e-a f}+\frac{b f x}{b e-a f}\right )^{3/5}}{\sqrt{a+b x}} \, dx}{\left (\frac{b (c+d x)}{b c-a d}\right )^{2/5} \left (\frac{b (e+f x)}{b e-a f}\right )^{3/5}}\\ &=\frac{2 \sqrt{a+b x} (c+d x)^{2/5} (e+f x)^{3/5} F_1\left (\frac{1}{2};-\frac{2}{5},-\frac{3}{5};\frac{3}{2};-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{b \left (\frac{b (c+d x)}{b c-a d}\right )^{2/5} \left (\frac{b (e+f x)}{b e-a f}\right )^{3/5}}\\ \end{align*}

Mathematica [B]  time = 3.08353, size = 536, normalized size = 4.36 \[ \frac{2 \sqrt{a+b x} \left (15 b^2 (c+d x) (e+f x)-2 (a+b x) \left (\frac{9 \left (25 a^2 d^2 f^2-10 a b d f (2 c f+3 d e)+b^2 \left (-2 c^2 f^2+24 c d e f+3 d^2 e^2\right )\right ) F_1\left (\frac{1}{2};\frac{3}{5},\frac{2}{5};\frac{3}{2};\frac{a d-b c}{d (a+b x)},\frac{a f-b e}{f (a+b x)}\right )}{15 d f (a+b x) F_1\left (\frac{1}{2};\frac{3}{5},\frac{2}{5};\frac{3}{2};\frac{a d-b c}{d (a+b x)},\frac{a f-b e}{f (a+b x)}\right )+(4 a d f-4 b d e) F_1\left (\frac{3}{2};\frac{3}{5},\frac{7}{5};\frac{5}{2};\frac{a d-b c}{d (a+b x)},\frac{a f-b e}{f (a+b x)}\right )+6 f (a d-b c) F_1\left (\frac{3}{2};\frac{8}{5},\frac{2}{5};\frac{5}{2};\frac{a d-b c}{d (a+b x)},\frac{a f-b e}{f (a+b x)}\right )}+\frac{(b c-a d) (b e-a f) \left (\frac{b (c+d x)}{d (a+b x)}\right )^{3/5} \left (\frac{b (e+f x)}{f (a+b x)}\right )^{2/5} (-5 a d f+2 b c f+3 b d e) F_1\left (\frac{3}{2};\frac{3}{5},\frac{2}{5};\frac{5}{2};\frac{a d-b c}{d (a+b x)},\frac{a f-b e}{f (a+b x)}\right )}{d f (a+b x)^2}-\frac{3 b^2 (c+d x) (e+f x) (-5 a d f+2 b c f+3 b d e)}{d f (a+b x)^2}\right )\right )}{45 b^3 (c+d x)^{3/5} (e+f x)^{2/5}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[((c + d*x)^(2/5)*(e + f*x)^(3/5))/Sqrt[a + b*x],x]

[Out]

(2*Sqrt[a + b*x]*(15*b^2*(c + d*x)*(e + f*x) - 2*(a + b*x)*((-3*b^2*(3*b*d*e + 2*b*c*f - 5*a*d*f)*(c + d*x)*(e
+ f*x))/(d*f*(a + b*x)^2) + ((b*c - a*d)*(b*e - a*f)*(3*b*d*e + 2*b*c*f - 5*a*d*f)*((b*(c + d*x))/(d*(a + b*x
)))^(3/5)*((b*(e + f*x))/(f*(a + b*x)))^(2/5)*AppellF1[3/2, 3/5, 2/5, 5/2, (-(b*c) + a*d)/(d*(a + b*x)), (-(b*
e) + a*f)/(f*(a + b*x))])/(d*f*(a + b*x)^2) + (9*(25*a^2*d^2*f^2 - 10*a*b*d*f*(3*d*e + 2*c*f) + b^2*(3*d^2*e^2
+ 24*c*d*e*f - 2*c^2*f^2))*AppellF1[1/2, 3/5, 2/5, 3/2, (-(b*c) + a*d)/(d*(a + b*x)), (-(b*e) + a*f)/(f*(a +
b*x))])/(15*d*f*(a + b*x)*AppellF1[1/2, 3/5, 2/5, 3/2, (-(b*c) + a*d)/(d*(a + b*x)), (-(b*e) + a*f)/(f*(a + b*
x))] + (-4*b*d*e + 4*a*d*f)*AppellF1[3/2, 3/5, 7/5, 5/2, (-(b*c) + a*d)/(d*(a + b*x)), (-(b*e) + a*f)/(f*(a +
b*x))] + 6*(-(b*c) + a*d)*f*AppellF1[3/2, 8/5, 2/5, 5/2, (-(b*c) + a*d)/(d*(a + b*x)), (-(b*e) + a*f)/(f*(a +
b*x))]))))/(45*b^3*(c + d*x)^(3/5)*(e + f*x)^(2/5))

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Maple [F]  time = 0.054, size = 0, normalized size = 0. \begin{align*} \int{ \left ( dx+c \right ) ^{{\frac{2}{5}}} \left ( fx+e \right ) ^{{\frac{3}{5}}}{\frac{1}{\sqrt{bx+a}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^(2/5)*(f*x+e)^(3/5)/(b*x+a)^(1/2),x)

[Out]

int((d*x+c)^(2/5)*(f*x+e)^(3/5)/(b*x+a)^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x + c\right )}^{\frac{2}{5}}{\left (f x + e\right )}^{\frac{3}{5}}}{\sqrt{b x + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(2/5)*(f*x+e)^(3/5)/(b*x+a)^(1/2),x, algorithm="maxima")

[Out]

integrate((d*x + c)^(2/5)*(f*x + e)^(3/5)/sqrt(b*x + a), 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((d*x+c)^(2/5)*(f*x+e)^(3/5)/(b*x+a)^(1/2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [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((d*x+c)**(2/5)*(f*x+e)**(3/5)/(b*x+a)**(1/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x + c\right )}^{\frac{2}{5}}{\left (f x + e\right )}^{\frac{3}{5}}}{\sqrt{b x + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(2/5)*(f*x+e)^(3/5)/(b*x+a)^(1/2),x, algorithm="giac")

[Out]

integrate((d*x + c)^(2/5)*(f*x + e)^(3/5)/sqrt(b*x + a), x)