3.3146 \(\int \frac{(e+f x)^n}{\sqrt{a+b x} (c+d x)^{3/2}} \, dx\)

Optimal. Leaf size=128 \[ \frac{2 \sqrt{a+b x} (e+f x)^n \sqrt{\frac{b (c+d x)}{b c-a d}} \left (\frac{b (e+f x)}{b e-a f}\right )^{-n} F_1\left (\frac{1}{2};\frac{3}{2},-n;\frac{3}{2};-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{\sqrt{c+d x} (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.432806, antiderivative size = 128, 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 \[ \frac{2 \sqrt{a+b x} (e+f x)^n \sqrt{\frac{b (c+d x)}{b c-a d}} \left (\frac{b (e+f x)}{b e-a f}\right )^{-n} F_1\left (\frac{1}{2};\frac{3}{2},-n;\frac{3}{2};-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{\sqrt{c+d x} (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 66.9227, size = 104, normalized size = 0.81 \[ \frac{2 b \left (\frac{b \left (- e - f x\right )}{a f - b e}\right )^{- n} \sqrt{a + b x} \sqrt{c + d x} \left (e + f x\right )^{n} \operatorname{appellf_{1}}{\left (\frac{1}{2},\frac{3}{2},- n,\frac{3}{2},\frac{d \left (a + b x\right )}{a d - b c},\frac{f \left (a + b x\right )}{a f - b e} \right )}}{\sqrt{\frac{b \left (- c - d x\right )}{a d - b c}} \left (a d - b c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((f*x+e)**n/(d*x+c)**(3/2)/(b*x+a)**(1/2),x)

[Out]

2*b*(b*(-e - f*x)/(a*f - b*e))**(-n)*sqrt(a + b*x)*sqrt(c + d*x)*(e + f*x)**n*ap
pellf1(1/2, 3/2, -n, 3/2, d*(a + b*x)/(a*d - b*c), f*(a + b*x)/(a*f - b*e))/(sqr
t(b*(-c - d*x)/(a*d - b*c))*(a*d - b*c)**2)

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Mathematica [B]  time = 5.56215, size = 816, normalized size = 6.38 \[ \frac{2 (b e-a f) \sqrt{a+b x} (e+f x)^n \left (\frac{9 b F_1\left (\frac{1}{2};-\frac{1}{2},-n;\frac{3}{2};\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right ) (c+d x)^2}{(b c-a d) \left (3 (b c-a d) (b e-a f) F_1\left (\frac{1}{2};-\frac{1}{2},-n;\frac{3}{2};\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )-(a+b x) \left (2 (a d-b c) f n F_1\left (\frac{3}{2};-\frac{1}{2},1-n;\frac{5}{2};\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )+d (a f-b e) F_1\left (\frac{3}{2};\frac{1}{2},-n;\frac{5}{2};\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )\right )\right )}-\frac{5 d (a+b x) F_1\left (\frac{3}{2};\frac{1}{2},-n;\frac{5}{2};\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right ) (c+d x)}{(b c-a d) \left (5 (b c-a d) (b e-a f) F_1\left (\frac{3}{2};\frac{1}{2},-n;\frac{5}{2};\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )-(a+b x) \left (2 (a d-b c) f n F_1\left (\frac{5}{2};\frac{1}{2},1-n;\frac{7}{2};\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )+d (b e-a f) F_1\left (\frac{5}{2};\frac{3}{2},-n;\frac{7}{2};\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )\right )\right )}-\frac{5 d (a+b x) F_1\left (\frac{3}{2};\frac{3}{2},-n;\frac{5}{2};\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )}{b \left (5 (b c-a d) (b e-a f) F_1\left (\frac{3}{2};\frac{3}{2},-n;\frac{5}{2};\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )-(a+b x) \left (2 (a d-b c) f n F_1\left (\frac{5}{2};\frac{3}{2},1-n;\frac{7}{2};\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )+3 d (b e-a f) F_1\left (\frac{5}{2};\frac{5}{2},-n;\frac{7}{2};\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )\right )\right )}\right )}{3 (c+d x)^{3/2}} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [F]  time = 0.06, size = 0, normalized size = 0. \[ \int{ \left ( fx+e \right ) ^{n} \left ( dx+c \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{bx+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x + e)^n/(sqrt(b*x + a)*(d*x + c)^(3/2)),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (f x + e\right )}^{n}}{\sqrt{b x + a}{\left (d x + c\right )}^{\frac{3}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x + e)^n/(sqrt(b*x + a)*(d*x + c)^(3/2)),x, algorithm="fricas")

[Out]

integral((f*x + e)^n/(sqrt(b*x + a)*(d*x + c)^(3/2)), 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)**(3/2)/(b*x+a)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (f x + e\right )}^{n}}{\sqrt{b x + a}{\left (d x + c\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x + e)^n/(sqrt(b*x + a)*(d*x + c)^(3/2)),x, algorithm="giac")

[Out]

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