3.3145 \(\int \frac{\sqrt{c+d x} (e+f x)^n}{\sqrt{a+b x}} \, dx\)

Optimal. Leaf size=121 \[ \frac{2 \sqrt{a+b x} \sqrt{c+d x} (e+f x)^n \left (\frac{b (e+f x)}{b e-a f}\right )^{-n} F_1\left (\frac{1}{2};-\frac{1}{2},-n;\frac{3}{2};-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{b \sqrt{\frac{b (c+d x)}{b c-a d}}} \]

[Out]

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

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Rubi [A]  time = 0.42387, antiderivative size = 121, 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 \[ \frac{2 \sqrt{a+b x} \sqrt{c+d x} (e+f x)^n \left (\frac{b (e+f x)}{b e-a f}\right )^{-n} F_1\left (\frac{1}{2};-\frac{1}{2},-n;\frac{3}{2};-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{b \sqrt{\frac{b (c+d x)}{b c-a d}}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 66.8471, size = 97, normalized size = 0.8 \[ \frac{2 \left (\frac{b \left (- e - f x\right )}{a f - b e}\right )^{- n} \sqrt{a + b x} \sqrt{c + d x} \left (e + f x\right )^{n} \operatorname{appellf_{1}}{\left (\frac{1}{2},- \frac{1}{2},- n,\frac{3}{2},\frac{d \left (a + b x\right )}{a d - b c},\frac{f \left (a + b x\right )}{a f - b e} \right )}}{b \sqrt{\frac{b \left (- c - d x\right )}{a d - b c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(b*(-e - f*x)/(a*f - b*e))**(-n)*sqrt(a + b*x)*sqrt(c + d*x)*(e + f*x)**n*appe
llf1(1/2, -1/2, -n, 3/2, d*(a + b*x)/(a*d - b*c), f*(a + b*x)/(a*f - b*e))/(b*sq
rt(b*(-c - d*x)/(a*d - b*c)))

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Mathematica [B]  time = 0.912902, size = 287, normalized size = 2.37 \[ \frac{6 \sqrt{a+b x} \sqrt{c+d x} (b c-a d) (b e-a f) (e+f x)^n F_1\left (\frac{1}{2};-\frac{1}{2},-n;\frac{3}{2};\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )}{b \left (3 (b c-a d) (b e-a f) F_1\left (\frac{1}{2};-\frac{1}{2},-n;\frac{3}{2};\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )-(a+b x) \left (2 f n (a d-b c) F_1\left (\frac{3}{2};-\frac{1}{2},1-n;\frac{5}{2};\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )+d (a f-b e) F_1\left (\frac{3}{2};\frac{1}{2},-n;\frac{5}{2};\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [F]  time = 0.06, size = 0, normalized size = 0. \[ \int{ \left ( fx+e \right ) ^{n}\sqrt{dx+c}{\frac{1}{\sqrt{bx+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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