3.1414 \(\int (b d+2 c d x)^m \left (a+b x+c x^2\right ) \, dx\)

Optimal. Leaf size=65 \[ \frac{(b d+2 c d x)^{m+3}}{8 c^2 d^3 (m+3)}-\frac{\left (b^2-4 a c\right ) (b d+2 c d x)^{m+1}}{8 c^2 d (m+1)} \]

[Out]

-((b^2 - 4*a*c)*(b*d + 2*c*d*x)^(1 + m))/(8*c^2*d*(1 + m)) + (b*d + 2*c*d*x)^(3
+ m)/(8*c^2*d^3*(3 + m))

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Rubi [A]  time = 0.0767089, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ \frac{(b d+2 c d x)^{m+3}}{8 c^2 d^3 (m+3)}-\frac{\left (b^2-4 a c\right ) (b d+2 c d x)^{m+1}}{8 c^2 d (m+1)} \]

Antiderivative was successfully verified.

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

[Out]

-((b^2 - 4*a*c)*(b*d + 2*c*d*x)^(1 + m))/(8*c^2*d*(1 + m)) + (b*d + 2*c*d*x)^(3
+ m)/(8*c^2*d^3*(3 + m))

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Rubi in Sympy [A]  time = 19.028, size = 56, normalized size = 0.86 \[ - \frac{\left (- a c + \frac{b^{2}}{4}\right ) \left (b d + 2 c d x\right )^{m + 1}}{2 c^{2} d \left (m + 1\right )} + \frac{\left (b d + 2 c d x\right )^{m + 3}}{8 c^{2} d^{3} \left (m + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-a*c + b**2/4)*(b*d + 2*c*d*x)**(m + 1)/(2*c**2*d*(m + 1)) + (b*d + 2*c*d*x)**
(m + 3)/(8*c**2*d**3*(m + 3))

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Mathematica [A]  time = 0.0581982, size = 64, normalized size = 0.98 \[ \frac{(b+2 c x) \left (2 c \left (a (m+3)+c (m+1) x^2\right )-b^2+2 b c (m+1) x\right ) (d (b+2 c x))^m}{4 c^2 (m+1) (m+3)} \]

Antiderivative was successfully verified.

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

[Out]

((b + 2*c*x)*(d*(b + 2*c*x))^m*(-b^2 + 2*b*c*(1 + m)*x + 2*c*(a*(3 + m) + c*(1 +
m)*x^2)))/(4*c^2*(1 + m)*(3 + m))

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Maple [A]  time = 0.005, size = 76, normalized size = 1.2 \[{\frac{ \left ( 2\,cdx+bd \right ) ^{m} \left ( 2\,{c}^{2}m{x}^{2}+2\,bcmx+2\,{c}^{2}{x}^{2}+2\,acm+2\,bxc+6\,ac-{b}^{2} \right ) \left ( 2\,cx+b \right ) }{4\,{c}^{2} \left ({m}^{2}+4\,m+3 \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.22771, size = 143, normalized size = 2.2 \[ \frac{{\left (2 \, a b c m + 4 \,{\left (c^{3} m + c^{3}\right )} x^{3} - b^{3} + 6 \, a b c + 6 \,{\left (b c^{2} m + b c^{2}\right )} x^{2} + 2 \,{\left (6 \, a c^{2} +{\left (b^{2} c + 2 \, a c^{2}\right )} m\right )} x\right )}{\left (2 \, c d x + b d\right )}^{m}}{4 \,{\left (c^{2} m^{2} + 4 \, c^{2} m + 3 \, c^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.48799, size = 707, normalized size = 10.88 \[ \begin{cases} \left (b d\right )^{m} \left (a x + \frac{b x^{2}}{2}\right ) & \text{for}\: c = 0 \\- \frac{4 a c}{16 b^{2} c^{2} d^{3} + 64 b c^{3} d^{3} x + 64 c^{4} d^{3} x^{2}} + \frac{2 b^{2} \log{\left (\frac{b}{2 c} + x \right )}}{16 b^{2} c^{2} d^{3} + 64 b c^{3} d^{3} x + 64 c^{4} d^{3} x^{2}} + \frac{b^{2}}{16 b^{2} c^{2} d^{3} + 64 b c^{3} d^{3} x + 64 c^{4} d^{3} x^{2}} + \frac{8 b c x \log{\left (\frac{b}{2 c} + x \right )}}{16 b^{2} c^{2} d^{3} + 64 b c^{3} d^{3} x + 64 c^{4} d^{3} x^{2}} + \frac{8 c^{2} x^{2} \log{\left (\frac{b}{2 c} + x \right )}}{16 b^{2} c^{2} d^{3} + 64 b c^{3} d^{3} x + 64 c^{4} d^{3} x^{2}} & \text{for}\: m = -3 \\\frac{a \log{\left (\frac{b}{2 c} + x \right )}}{2 c d} - \frac{b^{2} \log{\left (\frac{b}{2 c} + x \right )}}{8 c^{2} d} + \frac{b x}{4 c d} + \frac{x^{2}}{4 d} & \text{for}\: m = -1 \\\frac{2 a b c m \left (b d + 2 c d x\right )^{m}}{4 c^{2} m^{2} + 16 c^{2} m + 12 c^{2}} + \frac{6 a b c \left (b d + 2 c d x\right )^{m}}{4 c^{2} m^{2} + 16 c^{2} m + 12 c^{2}} + \frac{4 a c^{2} m x \left (b d + 2 c d x\right )^{m}}{4 c^{2} m^{2} + 16 c^{2} m + 12 c^{2}} + \frac{12 a c^{2} x \left (b d + 2 c d x\right )^{m}}{4 c^{2} m^{2} + 16 c^{2} m + 12 c^{2}} - \frac{b^{3} \left (b d + 2 c d x\right )^{m}}{4 c^{2} m^{2} + 16 c^{2} m + 12 c^{2}} + \frac{2 b^{2} c m x \left (b d + 2 c d x\right )^{m}}{4 c^{2} m^{2} + 16 c^{2} m + 12 c^{2}} + \frac{6 b c^{2} m x^{2} \left (b d + 2 c d x\right )^{m}}{4 c^{2} m^{2} + 16 c^{2} m + 12 c^{2}} + \frac{6 b c^{2} x^{2} \left (b d + 2 c d x\right )^{m}}{4 c^{2} m^{2} + 16 c^{2} m + 12 c^{2}} + \frac{4 c^{3} m x^{3} \left (b d + 2 c d x\right )^{m}}{4 c^{2} m^{2} + 16 c^{2} m + 12 c^{2}} + \frac{4 c^{3} x^{3} \left (b d + 2 c d x\right )^{m}}{4 c^{2} m^{2} + 16 c^{2} m + 12 c^{2}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise(((b*d)**m*(a*x + b*x**2/2), Eq(c, 0)), (-4*a*c/(16*b**2*c**2*d**3 + 64
*b*c**3*d**3*x + 64*c**4*d**3*x**2) + 2*b**2*log(b/(2*c) + x)/(16*b**2*c**2*d**3
+ 64*b*c**3*d**3*x + 64*c**4*d**3*x**2) + b**2/(16*b**2*c**2*d**3 + 64*b*c**3*d
**3*x + 64*c**4*d**3*x**2) + 8*b*c*x*log(b/(2*c) + x)/(16*b**2*c**2*d**3 + 64*b*
c**3*d**3*x + 64*c**4*d**3*x**2) + 8*c**2*x**2*log(b/(2*c) + x)/(16*b**2*c**2*d*
*3 + 64*b*c**3*d**3*x + 64*c**4*d**3*x**2), Eq(m, -3)), (a*log(b/(2*c) + x)/(2*c
*d) - b**2*log(b/(2*c) + x)/(8*c**2*d) + b*x/(4*c*d) + x**2/(4*d), Eq(m, -1)), (
2*a*b*c*m*(b*d + 2*c*d*x)**m/(4*c**2*m**2 + 16*c**2*m + 12*c**2) + 6*a*b*c*(b*d
+ 2*c*d*x)**m/(4*c**2*m**2 + 16*c**2*m + 12*c**2) + 4*a*c**2*m*x*(b*d + 2*c*d*x)
**m/(4*c**2*m**2 + 16*c**2*m + 12*c**2) + 12*a*c**2*x*(b*d + 2*c*d*x)**m/(4*c**2
*m**2 + 16*c**2*m + 12*c**2) - b**3*(b*d + 2*c*d*x)**m/(4*c**2*m**2 + 16*c**2*m
+ 12*c**2) + 2*b**2*c*m*x*(b*d + 2*c*d*x)**m/(4*c**2*m**2 + 16*c**2*m + 12*c**2)
+ 6*b*c**2*m*x**2*(b*d + 2*c*d*x)**m/(4*c**2*m**2 + 16*c**2*m + 12*c**2) + 6*b*
c**2*x**2*(b*d + 2*c*d*x)**m/(4*c**2*m**2 + 16*c**2*m + 12*c**2) + 4*c**3*m*x**3
*(b*d + 2*c*d*x)**m/(4*c**2*m**2 + 16*c**2*m + 12*c**2) + 4*c**3*x**3*(b*d + 2*c
*d*x)**m/(4*c**2*m**2 + 16*c**2*m + 12*c**2), True))

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GIAC/XCAS [A]  time = 0.214019, size = 309, normalized size = 4.75 \[ \frac{4 \, c^{3} m x^{3} e^{\left (m{\rm ln}\left (2 \, c d x + b d\right )\right )} + 6 \, b c^{2} m x^{2} e^{\left (m{\rm ln}\left (2 \, c d x + b d\right )\right )} + 4 \, c^{3} x^{3} e^{\left (m{\rm ln}\left (2 \, c d x + b d\right )\right )} + 2 \, b^{2} c m x e^{\left (m{\rm ln}\left (2 \, c d x + b d\right )\right )} + 4 \, a c^{2} m x e^{\left (m{\rm ln}\left (2 \, c d x + b d\right )\right )} + 6 \, b c^{2} x^{2} e^{\left (m{\rm ln}\left (2 \, c d x + b d\right )\right )} + 2 \, a b c m e^{\left (m{\rm ln}\left (2 \, c d x + b d\right )\right )} + 12 \, a c^{2} x e^{\left (m{\rm ln}\left (2 \, c d x + b d\right )\right )} - b^{3} e^{\left (m{\rm ln}\left (2 \, c d x + b d\right )\right )} + 6 \, a b c e^{\left (m{\rm ln}\left (2 \, c d x + b d\right )\right )}}{4 \,{\left (c^{2} m^{2} + 4 \, c^{2} m + 3 \, c^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(4*c^3*m*x^3*e^(m*ln(2*c*d*x + b*d)) + 6*b*c^2*m*x^2*e^(m*ln(2*c*d*x + b*d))
+ 4*c^3*x^3*e^(m*ln(2*c*d*x + b*d)) + 2*b^2*c*m*x*e^(m*ln(2*c*d*x + b*d)) + 4*a
*c^2*m*x*e^(m*ln(2*c*d*x + b*d)) + 6*b*c^2*x^2*e^(m*ln(2*c*d*x + b*d)) + 2*a*b*c
*m*e^(m*ln(2*c*d*x + b*d)) + 12*a*c^2*x*e^(m*ln(2*c*d*x + b*d)) - b^3*e^(m*ln(2*
c*d*x + b*d)) + 6*a*b*c*e^(m*ln(2*c*d*x + b*d)))/(c^2*m^2 + 4*c^2*m + 3*c^2)