3.350 \(\int \frac{x^m (c+d x)^2}{a+b x} \, dx\)

Optimal. Leaf size=99 \[ \frac{x^{m+1} (b c-a d)^2 \, _2F_1\left (1,m+1;m+2;-\frac{b x}{a}\right )}{a b^2 (m+1)}+\frac{d x^{m+1} (b c-a d)}{b^2 (m+1)}+\frac{c d x^{m+1}}{b (m+1)}+\frac{d^2 x^{m+2}}{b (m+2)} \]

[Out]

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

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Rubi [A]  time = 0.138152, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{x^{m+1} (b c-a d)^2 \, _2F_1\left (1,m+1;m+2;-\frac{b x}{a}\right )}{a b^2 (m+1)}+\frac{d x^{m+1} (b c-a d)}{b^2 (m+1)}+\frac{c d x^{m+1}}{b (m+1)}+\frac{d^2 x^{m+2}}{b (m+2)} \]

Antiderivative was successfully verified.

[In]  Int[(x^m*(c + d*x)^2)/(a + b*x),x]

[Out]

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

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Rubi in Sympy [A]  time = 22.2243, size = 80, normalized size = 0.81 \[ \frac{c d x^{m + 1}}{b \left (m + 1\right )} + \frac{d^{2} x^{m + 2}}{b \left (m + 2\right )} - \frac{d x^{m + 1} \left (a d - b c\right )}{b^{2} \left (m + 1\right )} + \frac{x^{m + 1} \left (a d - b c\right )^{2}{{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{- \frac{b x}{a}} \right )}}{a b^{2} \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**m*(d*x+c)**2/(b*x+a),x)

[Out]

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

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Mathematica [A]  time = 0.130349, size = 84, normalized size = 0.85 \[ \frac{x^{m+1} \left (\frac{c^2 \, _2F_1\left (1,m+1;m+2;-\frac{b x}{a}\right )}{m+1}+d x \left (\frac{2 c \, _2F_1\left (1,m+2;m+3;-\frac{b x}{a}\right )}{m+2}+\frac{d x \, _2F_1\left (1,m+3;m+4;-\frac{b x}{a}\right )}{m+3}\right )\right )}{a} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^m*(c + d*x)^2)/(a + b*x),x]

[Out]

(x^(1 + m)*((c^2*Hypergeometric2F1[1, 1 + m, 2 + m, -((b*x)/a)])/(1 + m) + d*x*(
(2*c*Hypergeometric2F1[1, 2 + m, 3 + m, -((b*x)/a)])/(2 + m) + (d*x*Hypergeometr
ic2F1[1, 3 + m, 4 + m, -((b*x)/a)])/(3 + m))))/a

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Maple [F]  time = 0.053, size = 0, normalized size = 0. \[ \int{\frac{{x}^{m} \left ( dx+c \right ) ^{2}}{bx+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m*(d*x+c)^2/(b*x+a),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^2*x^m/(b*x + a),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^2*x^m/(b*x + a),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 8.1215, size = 219, normalized size = 2.21 \[ \frac{c^{2} m x x^{m} \Phi \left (\frac{b x e^{i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{a \Gamma \left (m + 2\right )} + \frac{c^{2} x x^{m} \Phi \left (\frac{b x e^{i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{a \Gamma \left (m + 2\right )} + \frac{2 c d m x^{2} x^{m} \Phi \left (\frac{b x e^{i \pi }}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{a \Gamma \left (m + 3\right )} + \frac{4 c d x^{2} x^{m} \Phi \left (\frac{b x e^{i \pi }}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{a \Gamma \left (m + 3\right )} + \frac{d^{2} m x^{3} x^{m} \Phi \left (\frac{b x e^{i \pi }}{a}, 1, m + 3\right ) \Gamma \left (m + 3\right )}{a \Gamma \left (m + 4\right )} + \frac{3 d^{2} x^{3} x^{m} \Phi \left (\frac{b x e^{i \pi }}{a}, 1, m + 3\right ) \Gamma \left (m + 3\right )}{a \Gamma \left (m + 4\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**m*(d*x+c)**2/(b*x+a),x)

[Out]

c**2*m*x*x**m*lerchphi(b*x*exp_polar(I*pi)/a, 1, m + 1)*gamma(m + 1)/(a*gamma(m
+ 2)) + c**2*x*x**m*lerchphi(b*x*exp_polar(I*pi)/a, 1, m + 1)*gamma(m + 1)/(a*ga
mma(m + 2)) + 2*c*d*m*x**2*x**m*lerchphi(b*x*exp_polar(I*pi)/a, 1, m + 2)*gamma(
m + 2)/(a*gamma(m + 3)) + 4*c*d*x**2*x**m*lerchphi(b*x*exp_polar(I*pi)/a, 1, m +
2)*gamma(m + 2)/(a*gamma(m + 3)) + 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)) + 3*d**2*x**3*x**m*lerchphi(b*x*exp
_polar(I*pi)/a, 1, m + 3)*gamma(m + 3)/(a*gamma(m + 4))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^2*x^m/(b*x + a),x, algorithm="giac")

[Out]

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