3.2169 \(\int (a+b x) (a c+b c x)^m \left (a^2+2 a b x+b^2 x^2\right )^p \, dx\)

Optimal. Leaf size=45 \[ \frac{\left (a^2+2 a b x+b^2 x^2\right )^p (a c+b c x)^{m+2}}{b c^2 (m+2 p+2)} \]

[Out]

((a*c + b*c*x)^(2 + m)*(a^2 + 2*a*b*x + b^2*x^2)^p)/(b*c^2*(2 + m + 2*p))

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Rubi [A]  time = 0.0592449, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.088 \[ \frac{\left (a^2+2 a b x+b^2 x^2\right )^p (a c+b c x)^{m+2}}{b c^2 (m+2 p+2)} \]

Antiderivative was successfully verified.

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

[Out]

((a*c + b*c*x)^(2 + m)*(a^2 + 2*a*b*x + b^2*x^2)^p)/(b*c^2*(2 + m + 2*p))

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Rubi in Sympy [A]  time = 20.591, size = 41, normalized size = 0.91 \[ \frac{\left (a c + b c x\right )^{m + 2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{p}}{b c^{2} \left (m + 2 p + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a*c + b*c*x)**(m + 2)*(a**2 + 2*a*b*x + b**2*x**2)**p/(b*c**2*(m + 2*p + 2))

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Mathematica [A]  time = 0.0340644, size = 32, normalized size = 0.71 \[ \frac{\left ((a+b x)^2\right )^{p+1} (c (a+b x))^m}{b (m+2 p+2)} \]

Antiderivative was successfully verified.

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

[Out]

((c*(a + b*x))^m*((a + b*x)^2)^(1 + p))/(b*(2 + m + 2*p))

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Maple [A]  time = 0.005, size = 48, normalized size = 1.1 \[{\frac{ \left ( bx+a \right ) ^{2} \left ( bxc+ac \right ) ^{m} \left ({b}^{2}{x}^{2}+2\,abx+{a}^{2} \right ) ^{p}}{b \left ( 2+m+2\,p \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(b*c*x+a*c)^m*(b^2*x^2+2*a*b*x+a^2)^p,x)

[Out]

(b*x+a)^2/b/(2+m+2*p)*(b*c*x+a*c)^m*(b^2*x^2+2*a*b*x+a^2)^p

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Maxima [A]  time = 0.738109, size = 171, normalized size = 3.8 \[ \frac{{\left (b c^{m} x + a c^{m}\right )} a e^{\left (m \log \left (b x + a\right ) + 2 \, p \log \left (b x + a\right )\right )}}{b{\left (m + 2 \, p + 1\right )}} + \frac{{\left (b^{2} c^{m}{\left (m + 2 \, p + 1\right )} x^{2} + a b c^{m}{\left (m + 2 \, p\right )} x - a^{2} c^{m}\right )} e^{\left (m \log \left (b x + a\right ) + 2 \, p \log \left (b x + a\right )\right )}}{{\left (m^{2} + m{\left (4 \, p + 3\right )} + 4 \, p^{2} + 6 \, p + 2\right )} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b*c^m*x + a*c^m)*a*e^(m*log(b*x + a) + 2*p*log(b*x + a))/(b*(m + 2*p + 1)) + (b
^2*c^m*(m + 2*p + 1)*x^2 + a*b*c^m*(m + 2*p)*x - a^2*c^m)*e^(m*log(b*x + a) + 2*
p*log(b*x + a))/((m^2 + m*(4*p + 3) + 4*p^2 + 6*p + 2)*b)

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Fricas [A]  time = 0.300943, size = 81, normalized size = 1.8 \[ \frac{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}{\left (b c x + a c\right )}^{m} e^{\left (2 \, p \log \left (b c x + a c\right ) + p \log \left (\frac{1}{c^{2}}\right )\right )}}{b m + 2 \, b p + 2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b^2*x^2 + 2*a*b*x + a^2)*(b*c*x + a*c)^m*e^(2*p*log(b*c*x + a*c) + p*log(c^(-2)
))/(b*m + 2*b*p + 2*b)

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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((b*x+a)*(b*c*x+a*c)**m*(b**2*x**2+2*a*b*x+a**2)**p,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.290468, size = 135, normalized size = 3. \[ \frac{b^{2} x^{2} e^{\left (m{\rm ln}\left (b x + a\right ) + 2 \, p{\rm ln}\left (b x + a\right ) + m{\rm ln}\left (c\right )\right )} + 2 \, a b x e^{\left (m{\rm ln}\left (b x + a\right ) + 2 \, p{\rm ln}\left (b x + a\right ) + m{\rm ln}\left (c\right )\right )} + a^{2} e^{\left (m{\rm ln}\left (b x + a\right ) + 2 \, p{\rm ln}\left (b x + a\right ) + m{\rm ln}\left (c\right )\right )}}{b m + 2 \, b p + 2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b^2*x^2*e^(m*ln(b*x + a) + 2*p*ln(b*x + a) + m*ln(c)) + 2*a*b*x*e^(m*ln(b*x + a
) + 2*p*ln(b*x + a) + m*ln(c)) + a^2*e^(m*ln(b*x + a) + 2*p*ln(b*x + a) + m*ln(c
)))/(b*m + 2*b*p + 2*b)