∖intxm(A+Bx) (a+bx)3 ∖,dx

3.347 \(\int \frac{x^m (A+B x)}{(a+b x)^3} \, dx\)

Optimal. Leaf size=81 \[ \frac{x^{m+1} (a B (m+1)+A b (1-m)) \, _2F_1\left (2,m+1;m+2;-\frac{b x}{a}\right )}{2 a^3 b (m+1)}+\frac{x^{m+1} (A b-a B)}{2 a b (a+b x)^2} \]

[Out]

((A*b - a*B)*x^(1 + m))/(2*a*b*(a + b*x)^2) + ((A*b*(1 - m) + a*B*(1 + m))*x^(1

+ m)*Hypergeometric2F1[2, 1 + m, 2 + m, -((b*x)/a)])/(2*a^3*b*(1 + m))

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Rubi [A] time = 0.100127, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{x^{m+1} (a B (m+1)+A (b-b m)) \, _2F_1\left (2,m+1;m+2;-\frac{b x}{a}\right )}{2 a^3 b (m+1)}+\frac{x^{m+1} (A b-a B)}{2 a b (a+b x)^2} \]

Antiderivative was successfully veriﬁed.

[In] Int[(x^m*(A + B*x))/(a + b*x)^3,x]

[Out]

((A*b - a*B)*x^(1 + m))/(2*a*b*(a + b*x)^2) + ((a*B*(1 + m) + A*(b - b*m))*x^(1

+ m)*Hypergeometric2F1[2, 1 + m, 2 + m, -((b*x)/a)])/(2*a^3*b*(1 + m))

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Rubi in Sympy [A] time = 8.91922, size = 63, normalized size = 0.78 \[ \frac{x^{m + 1} \left (A b - B a\right )}{2 a b \left (a + b x\right )^{2}} + \frac{x^{m + 1} \left (A b \left (- m + 1\right ) + B a \left (m + 1\right )\right ){{}_{2}F_{1}\left (\begin{matrix} 2, m + 1 \\ m + 2 \end{matrix}\middle |{- \frac{b x}{a}} \right )}}{2 a^{3} b \left (m + 1\right )} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] rubi_integrate(x**m*(B*x+A)/(b*x+a)**3,x)

[Out]

x**(m + 1)*(A*b - B*a)/(2*a*b*(a + b*x)**2) + x**(m + 1)*(A*b*(-m + 1) + B*a*(m

+ 1))*hyper((2, m + 1), (m + 2,), -b*x/a)/(2*a**3*b*(m + 1))

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Mathematica [A] time = 0.0702465, size = 60, normalized size = 0.74 \[ \frac{x^{m+1} \left ((A b-a B) \, _2F_1\left (3,m+1;m+2;-\frac{b x}{a}\right )+a B \, _2F_1\left (2,m+1;m+2;-\frac{b x}{a}\right )\right )}{a^3 b (m+1)} \]

Antiderivative was successfully veriﬁed.

[In] Integrate[(x^m*(A + B*x))/(a + b*x)^3,x]

[Out]

(x^(1 + m)*(a*B*Hypergeometric2F1[2, 1 + m, 2 + m, -((b*x)/a)] + (A*b - a*B)*Hyp

ergeometric2F1[3, 1 + m, 2 + m, -((b*x)/a)]))/(a^3*b*(1 + m))

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Maple [F] time = 0.067, size = 0, normalized size = 0. \[ \int{\frac{{x}^{m} \left ( Bx+A \right ) }{ \left ( bx+a \right ) ^{3}}}\, dx \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] int(x^m*(B*x+A)/(b*x+a)^3,x)

[Out]

int(x^m*(B*x+A)/(b*x+a)^3,x)

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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x + A\right )} x^{m}}{{\left (b x + a\right )}^{3}}\,{d x} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate((B*x + A)*x^m/(b*x + a)^3,x, algorithm="maxima")

[Out]

integrate((B*x + A)*x^m/(b*x + a)^3, x)

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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (B x + A\right )} x^{m}}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}, x\right ) \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate((B*x + A)*x^m/(b*x + a)^3,x, algorithm="fricas")

[Out]

integral((B*x + A)*x^m/(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a^3), x)

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Sympy [A] time = 16.4864, size = 1680, normalized size = 20.74 \[ \text{result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate(x**m*(B*x+A)/(b*x+a)**3,x)

[Out]

A*(a**2*m**3*x*x**m*lerchphi(b*x*exp_polar(I*pi)/a, 1, m + 1)*gamma(m + 1)/(2*a*

*5*gamma(m + 2) + 4*a**4*b*x*gamma(m + 2) + 2*a**3*b**2*x**2*gamma(m + 2)) - a**

2*m**2*x*x**m*gamma(m + 1)/(2*a**5*gamma(m + 2) + 4*a**4*b*x*gamma(m + 2) + 2*a*

*3*b**2*x**2*gamma(m + 2)) - a**2*m*x*x**m*lerchphi(b*x*exp_polar(I*pi)/a, 1, m

+ 1)*gamma(m + 1)/(2*a**5*gamma(m + 2) + 4*a**4*b*x*gamma(m + 2) + 2*a**3*b**2*x

**2*gamma(m + 2)) + a**2*m*x*x**m*gamma(m + 1)/(2*a**5*gamma(m + 2) + 4*a**4*b*x

*gamma(m + 2) + 2*a**3*b**2*x**2*gamma(m + 2)) + 2*a**2*x*x**m*gamma(m + 1)/(2*a

**5*gamma(m + 2) + 4*a**4*b*x*gamma(m + 2) + 2*a**3*b**2*x**2*gamma(m + 2)) + 2*

a*b*m**3*x**2*x**m*lerchphi(b*x*exp_polar(I*pi)/a, 1, m + 1)*gamma(m + 1)/(2*a**

5*gamma(m + 2) + 4*a**4*b*x*gamma(m + 2) + 2*a**3*b**2*x**2*gamma(m + 2)) - a*b*

m**2*x**2*x**m*gamma(m + 1)/(2*a**5*gamma(m + 2) + 4*a**4*b*x*gamma(m + 2) + 2*a

**3*b**2*x**2*gamma(m + 2)) - 2*a*b*m*x**2*x**m*lerchphi(b*x*exp_polar(I*pi)/a,

1, m + 1)*gamma(m + 1)/(2*a**5*gamma(m + 2) + 4*a**4*b*x*gamma(m + 2) + 2*a**3*b

**2*x**2*gamma(m + 2)) + a*b*x**2*x**m*gamma(m + 1)/(2*a**5*gamma(m + 2) + 4*a**

4*b*x*gamma(m + 2) + 2*a**3*b**2*x**2*gamma(m + 2)) + b**2*m**3*x**3*x**m*lerchp

hi(b*x*exp_polar(I*pi)/a, 1, m + 1)*gamma(m + 1)/(2*a**5*gamma(m + 2) + 4*a**4*b

*x*gamma(m + 2) + 2*a**3*b**2*x**2*gamma(m + 2)) - b**2*m*x**3*x**m*lerchphi(b*x

*exp_polar(I*pi)/a, 1, m + 1)*gamma(m + 1)/(2*a**5*gamma(m + 2) + 4*a**4*b*x*gam

ma(m + 2) + 2*a**3*b**2*x**2*gamma(m + 2))) + B*(a**2*m**3*x**2*x**m*lerchphi(b*

x*exp_polar(I*pi)/a, 1, m + 2)*gamma(m + 2)/(2*a**5*gamma(m + 3) + 4*a**4*b*x*ga

mma(m + 3) + 2*a**3*b**2*x**2*gamma(m + 3)) + 3*a**2*m**2*x**2*x**m*lerchphi(b*x

*exp_polar(I*pi)/a, 1, m + 2)*gamma(m + 2)/(2*a**5*gamma(m + 3) + 4*a**4*b*x*gam

ma(m + 3) + 2*a**3*b**2*x**2*gamma(m + 3)) - a**2*m**2*x**2*x**m*gamma(m + 2)/(2

*a**5*gamma(m + 3) + 4*a**4*b*x*gamma(m + 3) + 2*a**3*b**2*x**2*gamma(m + 3)) +

2*a**2*m*x**2*x**m*lerchphi(b*x*exp_polar(I*pi)/a, 1, m + 2)*gamma(m + 2)/(2*a**

5*gamma(m + 3) + 4*a**4*b*x*gamma(m + 3) + 2*a**3*b**2*x**2*gamma(m + 3)) - a**2

*m*x**2*x**m*gamma(m + 2)/(2*a**5*gamma(m + 3) + 4*a**4*b*x*gamma(m + 3) + 2*a**

3*b**2*x**2*gamma(m + 3)) + 2*a**2*x**2*x**m*gamma(m + 2)/(2*a**5*gamma(m + 3) +

4*a**4*b*x*gamma(m + 3) + 2*a**3*b**2*x**2*gamma(m + 3)) + 2*a*b*m**3*x**3*x**m

*lerchphi(b*x*exp_polar(I*pi)/a, 1, m + 2)*gamma(m + 2)/(2*a**5*gamma(m + 3) + 4

*a**4*b*x*gamma(m + 3) + 2*a**3*b**2*x**2*gamma(m + 3)) + 6*a*b*m**2*x**3*x**m*l

erchphi(b*x*exp_polar(I*pi)/a, 1, m + 2)*gamma(m + 2)/(2*a**5*gamma(m + 3) + 4*a

**4*b*x*gamma(m + 3) + 2*a**3*b**2*x**2*gamma(m + 3)) - a*b*m**2*x**3*x**m*gamma

(m + 2)/(2*a**5*gamma(m + 3) + 4*a**4*b*x*gamma(m + 3) + 2*a**3*b**2*x**2*gamma(

m + 3)) + 4*a*b*m*x**3*x**m*lerchphi(b*x*exp_polar(I*pi)/a, 1, m + 2)*gamma(m +

2)/(2*a**5*gamma(m + 3) + 4*a**4*b*x*gamma(m + 3) + 2*a**3*b**2*x**2*gamma(m + 3

)) - 2*a*b*m*x**3*x**m*gamma(m + 2)/(2*a**5*gamma(m + 3) + 4*a**4*b*x*gamma(m +

3) + 2*a**3*b**2*x**2*gamma(m + 3)) + b**2*m**3*x**4*x**m*lerchphi(b*x*exp_polar

(I*pi)/a, 1, m + 2)*gamma(m + 2)/(2*a**5*gamma(m + 3) + 4*a**4*b*x*gamma(m + 3)

+ 2*a**3*b**2*x**2*gamma(m + 3)) + 3*b**2*m**2*x**4*x**m*lerchphi(b*x*exp_polar(

I*pi)/a, 1, m + 2)*gamma(m + 2)/(2*a**5*gamma(m + 3) + 4*a**4*b*x*gamma(m + 3) +

2*a**3*b**2*x**2*gamma(m + 3)) + 2*b**2*m*x**4*x**m*lerchphi(b*x*exp_polar(I*pi

)/a, 1, m + 2)*gamma(m + 2)/(2*a**5*gamma(m + 3) + 4*a**4*b*x*gamma(m + 3) + 2*a

**3*b**2*x**2*gamma(m + 3)))

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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x + A\right )} x^{m}}{{\left (b x + a\right )}^{3}}\,{d x} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate((B*x + A)*x^m/(b*x + a)^3,x, algorithm="giac")

[Out]

integrate((B*x + A)*x^m/(b*x + a)^3, x)

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