3.349 \(\int \frac{x^m (c+d x)^3}{a+b x} \, dx\)

Optimal. Leaf size=127 \[ \frac{d x^{m+1} \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{b^3 (m+1)}+\frac{x^{m+1} (b c-a d)^3 \, _2F_1\left (1,1;1-m;\frac{a}{a+b x}\right )}{b^3 m (a+b x)}+\frac{d^2 x^{m+2} (3 b c-a d)}{b^2 (m+2)}+\frac{d^3 x^{m+3}}{b (m+3)} \]

[Out]

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

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Rubi [A]  time = 0.250571, antiderivative size = 171, normalized size of antiderivative = 1.35, number of steps used = 7, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{x^{m+1} (b c-a d)^3 \, _2F_1\left (1,m+1;m+2;-\frac{b x}{a}\right )}{a b^3 (m+1)}+\frac{d x^{m+1} (b c-a d)^2}{b^3 (m+1)}+\frac{d^2 x^{m+2} (b c-a d)}{b^2 (m+2)}+\frac{c d x^{m+1} (b c-a d)}{b^2 (m+1)}+\frac{c^2 d x^{m+1}}{b (m+1)}+\frac{2 c d^2 x^{m+2}}{b (m+2)}+\frac{d^3 x^{m+3}}{b (m+3)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 39.2944, size = 144, normalized size = 1.13 \[ \frac{c^{2} d x^{m + 1}}{b \left (m + 1\right )} + \frac{2 c d^{2} x^{m + 2}}{b \left (m + 2\right )} + \frac{d^{3} x^{m + 3}}{b \left (m + 3\right )} - \frac{c d x^{m + 1} \left (a d - b c\right )}{b^{2} \left (m + 1\right )} - \frac{d^{2} x^{m + 2} \left (a d - b c\right )}{b^{2} \left (m + 2\right )} + \frac{d x^{m + 1} \left (a d - b c\right )^{2}}{b^{3} \left (m + 1\right )} - \frac{x^{m + 1} \left (a d - b c\right )^{3}{{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{- \frac{b x}{a}} \right )}}{a b^{3} \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**m*(d*x+c)**3/(b*x+a),x)

[Out]

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

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Mathematica [A]  time = 0.227463, size = 113, normalized size = 0.89 \[ \frac{x^{m+1} \left (\frac{c^3 \, _2F_1\left (1,m+1;m+2;-\frac{b x}{a}\right )}{m+1}+d x \left (\frac{3 c^2 \, _2F_1\left (1,m+2;m+3;-\frac{b x}{a}\right )}{m+2}+d x \left (\frac{3 c \, _2F_1\left (1,m+3;m+4;-\frac{b x}{a}\right )}{m+3}+\frac{d x \, _2F_1\left (1,m+4;m+5;-\frac{b x}{a}\right )}{m+4}\right )\right )\right )}{a} \]

Antiderivative was successfully verified.

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

[Out]

(x^(1 + m)*((c^3*Hypergeometric2F1[1, 1 + m, 2 + m, -((b*x)/a)])/(1 + m) + d*x*(
(3*c^2*Hypergeometric2F1[1, 2 + m, 3 + m, -((b*x)/a)])/(2 + m) + d*x*((3*c*Hyper
geometric2F1[1, 3 + m, 4 + m, -((b*x)/a)])/(3 + m) + (d*x*Hypergeometric2F1[1, 4
+ m, 5 + m, -((b*x)/a)])/(4 + m)))))/a

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Maple [F]  time = 0.056, size = 0, normalized size = 0. \[ \int{\frac{{x}^{m} \left ( dx+c \right ) ^{3}}{bx+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m*(d*x+c)^3/(b*x+a),x)

[Out]

int(x^m*(d*x+c)^3/(b*x+a),x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((d*x + c)^3*x^m/(b*x + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3)*x^m/(b*x + a), x)

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Sympy [A]  time = 10.9081, size = 303, normalized size = 2.39 \[ \frac{c^{3} m x x^{m} \Phi \left (\frac{b x e^{i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{a \Gamma \left (m + 2\right )} + \frac{c^{3} x x^{m} \Phi \left (\frac{b x e^{i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{a \Gamma \left (m + 2\right )} + \frac{3 c^{2} d m x^{2} x^{m} \Phi \left (\frac{b x e^{i \pi }}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{a \Gamma \left (m + 3\right )} + \frac{6 c^{2} d x^{2} x^{m} \Phi \left (\frac{b x e^{i \pi }}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{a \Gamma \left (m + 3\right )} + \frac{3 c d^{2} m x^{3} x^{m} \Phi \left (\frac{b x e^{i \pi }}{a}, 1, m + 3\right ) \Gamma \left (m + 3\right )}{a \Gamma \left (m + 4\right )} + \frac{9 c d^{2} x^{3} x^{m} \Phi \left (\frac{b x e^{i \pi }}{a}, 1, m + 3\right ) \Gamma \left (m + 3\right )}{a \Gamma \left (m + 4\right )} + \frac{d^{3} m x^{4} x^{m} \Phi \left (\frac{b x e^{i \pi }}{a}, 1, m + 4\right ) \Gamma \left (m + 4\right )}{a \Gamma \left (m + 5\right )} + \frac{4 d^{3} x^{4} x^{m} \Phi \left (\frac{b x e^{i \pi }}{a}, 1, m + 4\right ) \Gamma \left (m + 4\right )}{a \Gamma \left (m + 5\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**m*(d*x+c)**3/(b*x+a),x)

[Out]

c**3*m*x*x**m*lerchphi(b*x*exp_polar(I*pi)/a, 1, m + 1)*gamma(m + 1)/(a*gamma(m
+ 2)) + c**3*x*x**m*lerchphi(b*x*exp_polar(I*pi)/a, 1, m + 1)*gamma(m + 1)/(a*ga
mma(m + 2)) + 3*c**2*d*m*x**2*x**m*lerchphi(b*x*exp_polar(I*pi)/a, 1, m + 2)*gam
ma(m + 2)/(a*gamma(m + 3)) + 6*c**2*d*x**2*x**m*lerchphi(b*x*exp_polar(I*pi)/a,
1, m + 2)*gamma(m + 2)/(a*gamma(m + 3)) + 3*c*d**2*m*x**3*x**m*lerchphi(b*x*exp_
polar(I*pi)/a, 1, m + 3)*gamma(m + 3)/(a*gamma(m + 4)) + 9*c*d**2*x**3*x**m*lerc
hphi(b*x*exp_polar(I*pi)/a, 1, m + 3)*gamma(m + 3)/(a*gamma(m + 4)) + d**3*m*x**
4*x**m*lerchphi(b*x*exp_polar(I*pi)/a, 1, m + 4)*gamma(m + 4)/(a*gamma(m + 5)) +
4*d**3*x**4*x**m*lerchphi(b*x*exp_polar(I*pi)/a, 1, m + 4)*gamma(m + 4)/(a*gamm
a(m + 5))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((d*x + c)^3*x^m/(b*x + a), x)