3.3147 \(\int \frac{(e+f x)^n}{(a+b x)^{3/2} \sqrt{c+d x}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.427611, 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 \[ -\frac{2 (e+f x)^n \sqrt{\frac{b (c+d x)}{b c-a d}} \left (\frac{b (e+f x)}{b e-a f}\right )^{-n} F_1\left (-\frac{1}{2};\frac{1}{2},-n;\frac{1}{2};-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{b \sqrt{a+b x} \sqrt{c+d x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 66.2878, size = 102, normalized size = 0.84 \[ \frac{2 \left (\frac{b \left (- e - f x\right )}{a f - b e}\right )^{- n} \sqrt{c + d x} \left (e + f x\right )^{n} \operatorname{appellf_{1}}{\left (- \frac{1}{2},\frac{1}{2},- n,\frac{1}{2},\frac{d \left (a + b x\right )}{a d - b c},\frac{f \left (a + b x\right )}{a f - b e} \right )}}{\sqrt{\frac{b \left (- c - d x\right )}{a d - b c}} \sqrt{a + b x} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 4.01153, size = 825, normalized size = 6.82 \[ \frac{2 (b e-a f) (e+f x)^n \left (\frac{3 (c+d x) F_1\left (-\frac{1}{2};-\frac{1}{2},-n;\frac{1}{2};\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right ) (b c-a d)^2}{(a d-b c) \left ((b c-a d) (b e-a f) F_1\left (-\frac{1}{2};-\frac{1}{2},-n;\frac{1}{2};\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )-(a+b x) \left (2 (a d-b c) f n F_1\left (\frac{1}{2};-\frac{1}{2},1-n;\frac{3}{2};\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )+d (a f-b e) F_1\left (\frac{1}{2};\frac{1}{2},-n;\frac{3}{2};\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )\right )\right )}-\frac{9 d (a+b x) (c+d x) F_1\left (\frac{1}{2};-\frac{1}{2},-n;\frac{3}{2};\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )}{3 (b c-a d) (b e-a f) F_1\left (\frac{1}{2};-\frac{1}{2},-n;\frac{3}{2};\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )-(a+b x) \left (2 (a d-b c) f n F_1\left (\frac{3}{2};-\frac{1}{2},1-n;\frac{5}{2};\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )+d (a f-b e) F_1\left (\frac{3}{2};\frac{1}{2},-n;\frac{5}{2};\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )\right )}+\frac{5 d^2 (a+b x)^2 F_1\left (\frac{3}{2};\frac{1}{2},-n;\frac{5}{2};\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )}{b \left (5 (b c-a d) (b e-a f) F_1\left (\frac{3}{2};\frac{1}{2},-n;\frac{5}{2};\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )-(a+b x) \left (2 (a d-b c) f n F_1\left (\frac{5}{2};\frac{1}{2},1-n;\frac{7}{2};\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )+d (b e-a f) F_1\left (\frac{5}{2};\frac{3}{2},-n;\frac{7}{2};\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )\right )\right )}\right )}{3 (b c-a d) \sqrt{a+b x} \sqrt{c+d x}} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [F]  time = 0.061, size = 0, normalized size = 0. \[ \int{ \left ( fx+e \right ) ^{n} \left ( bx+a \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{dx+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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