3.3144 \(\int \frac{\sqrt{a+b x} (e+f x)^n}{\sqrt{c+d x}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.430036, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{2 (a+b x)^{3/2} (e+f x)^n \sqrt{\frac{b (c+d x)}{b c-a d}} \left (\frac{b (e+f x)}{b e-a f}\right )^{-n} F_1\left (\frac{3}{2};\frac{1}{2},-n;\frac{5}{2};-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{3 b \sqrt{c+d x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 67.5214, size = 104, normalized size = 0.85 \[ - \frac{2 \left (\frac{b \left (- e - f x\right )}{a f - b e}\right )^{- n} \left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (e + f x\right )^{n} \operatorname{appellf_{1}}{\left (\frac{3}{2},\frac{1}{2},- n,\frac{5}{2},\frac{d \left (a + b x\right )}{a d - b c},\frac{f \left (a + b x\right )}{a f - b e} \right )}}{3 \sqrt{\frac{b \left (- c - d x\right )}{a d - b c}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((f*x+e)**n*(b*x+a)**(1/2)/(d*x+c)**(1/2),x)

[Out]

-2*(b*(-e - f*x)/(a*f - b*e))**(-n)*(a + b*x)**(3/2)*sqrt(c + d*x)*(e + f*x)**n*
appellf1(3/2, 1/2, -n, 5/2, d*(a + b*x)/(a*d - b*c), f*(a + b*x)/(a*f - b*e))/(3
*sqrt(b*(-c - d*x)/(a*d - b*c))*(a*d - b*c))

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Mathematica [B]  time = 1.1406, size = 289, normalized size = 2.35 \[ \frac{10 (a+b x)^{3/2} (b c-a d) (b e-a f) (e+f x)^n F_1\left (\frac{3}{2};\frac{1}{2},-n;\frac{5}{2};\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )}{3 b \sqrt{c+d x} \left (5 (b c-a d) (b e-a f) F_1\left (\frac{3}{2};\frac{1}{2},-n;\frac{5}{2};\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )-(a+b x) \left (2 f n (a d-b c) F_1\left (\frac{5}{2};\frac{1}{2},1-n;\frac{7}{2};\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )+d (b e-a f) F_1\left (\frac{5}{2};\frac{3}{2},-n;\frac{7}{2};\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(10*(b*c - a*d)*(b*e - a*f)*(a + b*x)^(3/2)*(e + f*x)^n*AppellF1[3/2, 1/2, -n, 5
/2, (d*(a + b*x))/(-(b*c) + a*d), (f*(a + b*x))/(-(b*e) + a*f)])/(3*b*Sqrt[c + d
*x]*(5*(b*c - a*d)*(b*e - a*f)*AppellF1[3/2, 1/2, -n, 5/2, (d*(a + b*x))/(-(b*c)
+ a*d), (f*(a + b*x))/(-(b*e) + a*f)] - (a + b*x)*(2*(-(b*c) + a*d)*f*n*AppellF
1[5/2, 1/2, 1 - n, 7/2, (d*(a + b*x))/(-(b*c) + a*d), (f*(a + b*x))/(-(b*e) + a*
f)] + d*(b*e - a*f)*AppellF1[5/2, 3/2, -n, 7/2, (d*(a + b*x))/(-(b*c) + a*d), (f
*(a + b*x))/(-(b*e) + a*f)])))

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Maple [F]  time = 0.061, size = 0, normalized size = 0. \[ \int{ \left ( fx+e \right ) ^{n}\sqrt{bx+a}{\frac{1}{\sqrt{dx+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((f*x+e)^n*(b*x+a)^(1/2)/(d*x+c)^(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x + a)*(f*x + e)^n/sqrt(d*x + c),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{b x + a}{\left (f x + e\right )}^{n}}{\sqrt{d x + c}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x + a)*(f*x + e)^n/sqrt(d*x + c),x, algorithm="fricas")

[Out]

integral(sqrt(b*x + a)*(f*x + e)^n/sqrt(d*x + c), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x+e)**n*(b*x+a)**(1/2)/(d*x+c)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{b x + a}{\left (f x + e\right )}^{n}}{\sqrt{d x + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x + a)*(f*x + e)^n/sqrt(d*x + c),x, algorithm="giac")

[Out]

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