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

Optimal. Leaf size=250 \[ \frac{2 \sqrt{a+b x} (A b-a B) (c+d x)^n (e+f x)^p \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} \left (\frac{b (e+f x)}{b e-a f}\right )^{-p} F_1\left (\frac{1}{2};-n,-p;\frac{3}{2};-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{b^2}+\frac{2 B (a+b x)^{3/2} (c+d x)^n (e+f x)^p \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} \left (\frac{b (e+f x)}{b e-a f}\right )^{-p} F_1\left (\frac{3}{2};-n,-p;\frac{5}{2};-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{3 b^2} \]

[Out]

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

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Rubi [A]  time = 0.726544, antiderivative size = 250, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138 \[ \frac{2 \sqrt{a+b x} (A b-a B) (c+d x)^n (e+f x)^p \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} \left (\frac{b (e+f x)}{b e-a f}\right )^{-p} F_1\left (\frac{1}{2};-n,-p;\frac{3}{2};-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{b^2}+\frac{2 B (a+b x)^{3/2} (c+d x)^n (e+f x)^p \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} \left (\frac{b (e+f x)}{b e-a f}\right )^{-p} F_1\left (\frac{3}{2};-n,-p;\frac{5}{2};-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{3 b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 158.767, size = 199, normalized size = 0.8 \[ \frac{2 B \left (\frac{b \left (- c - d x\right )}{a d - b c}\right )^{- n} \left (\frac{b \left (- e - f x\right )}{a f - b e}\right )^{- p} \left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{n} \left (e + f x\right )^{p} \operatorname{appellf_{1}}{\left (\frac{3}{2},- n,- p,\frac{5}{2},\frac{d \left (a + b x\right )}{a d - b c},\frac{f \left (a + b x\right )}{a f - b e} \right )}}{3 b^{2}} + \frac{2 \left (\frac{b \left (- c - d x\right )}{a d - b c}\right )^{- n} \left (\frac{b \left (- e - f x\right )}{a f - b e}\right )^{- p} \sqrt{a + b x} \left (c + d x\right )^{n} \left (e + f x\right )^{p} \left (A b - B a\right ) \operatorname{appellf_{1}}{\left (\frac{1}{2},- n,- p,\frac{3}{2},\frac{d \left (a + b x\right )}{a d - b c},\frac{f \left (a + b x\right )}{a f - b e} \right )}}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 2.69672, size = 551, normalized size = 2.2 \[ \frac{2 \sqrt{a+b x} (b c-a d) (b e-a f) (c+d x)^n (e+f x)^p \left (\frac{9 (A b-a B) F_1\left (\frac{1}{2};-n,-p;\frac{3}{2};\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )}{3 (b c-a d) (b e-a f) F_1\left (\frac{1}{2};-n,-p;\frac{3}{2};\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )-2 (a+b x) \left (d n (a f-b e) F_1\left (\frac{3}{2};1-n,-p;\frac{5}{2};\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )+f p (a d-b c) F_1\left (\frac{3}{2};-n,1-p;\frac{5}{2};\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )\right )}+\frac{5 B (a+b x) F_1\left (\frac{3}{2};-n,-p;\frac{5}{2};\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )}{5 (b c-a d) (b e-a f) F_1\left (\frac{3}{2};-n,-p;\frac{5}{2};\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )-2 (a+b x) \left (d n (a f-b e) F_1\left (\frac{5}{2};1-n,-p;\frac{7}{2};\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )+f p (a d-b c) F_1\left (\frac{5}{2};-n,1-p;\frac{7}{2};\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )\right )}\right )}{3 b^2} \]

Warning: Unable to verify antiderivative.

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

[Out]

(2*(b*c - a*d)*(b*e - a*f)*Sqrt[a + b*x]*(c + d*x)^n*(e + f*x)^p*((9*(A*b - a*B)
*AppellF1[1/2, -n, -p, 3/2, (d*(a + b*x))/(-(b*c) + a*d), (f*(a + b*x))/(-(b*e)
+ a*f)])/(3*(b*c - a*d)*(b*e - a*f)*AppellF1[1/2, -n, -p, 3/2, (d*(a + b*x))/(-(
b*c) + a*d), (f*(a + b*x))/(-(b*e) + a*f)] - 2*(a + b*x)*(d*(-(b*e) + a*f)*n*App
ellF1[3/2, 1 - n, -p, 5/2, (d*(a + b*x))/(-(b*c) + a*d), (f*(a + b*x))/(-(b*e) +
a*f)] + (-(b*c) + a*d)*f*p*AppellF1[3/2, -n, 1 - p, 5/2, (d*(a + b*x))/(-(b*c)
+ a*d), (f*(a + b*x))/(-(b*e) + a*f)])) + (5*B*(a + b*x)*AppellF1[3/2, -n, -p, 5
/2, (d*(a + b*x))/(-(b*c) + a*d), (f*(a + b*x))/(-(b*e) + a*f)])/(5*(b*c - a*d)*
(b*e - a*f)*AppellF1[3/2, -n, -p, 5/2, (d*(a + b*x))/(-(b*c) + a*d), (f*(a + b*x
))/(-(b*e) + a*f)] - 2*(a + b*x)*(d*(-(b*e) + a*f)*n*AppellF1[5/2, 1 - n, -p, 7/
2, (d*(a + b*x))/(-(b*c) + a*d), (f*(a + b*x))/(-(b*e) + a*f)] + (-(b*c) + a*d)*
f*p*AppellF1[5/2, -n, 1 - p, 7/2, (d*(a + b*x))/(-(b*c) + a*d), (f*(a + b*x))/(-
(b*e) + a*f)]))))/(3*b^2)

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Maple [F]  time = 0.076, size = 0, normalized size = 0. \[ \int{ \left ( Bx+A \right ) \left ( dx+c \right ) ^{n} \left ( fx+e \right ) ^{p}{\frac{1}{\sqrt{bx+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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