3.1427 \(\int (b d+2 c d x)^3 \left (a+b x+c x^2\right )^p \, dx\)

Optimal. Leaf size=68 \[ \frac{d^3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{p+1}}{(p+1) (p+2)}+\frac{d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^{p+1}}{p+2} \]

[Out]

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

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Rubi [A]  time = 0.0744712, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{d^3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{p+1}}{(p+1) (p+2)}+\frac{d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^{p+1}}{p+2} \]

Antiderivative was successfully verified.

[In]  Int[(b*d + 2*c*d*x)^3*(a + b*x + c*x^2)^p,x]

[Out]

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

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Rubi in Sympy [A]  time = 20.2999, size = 60, normalized size = 0.88 \[ \frac{d^{3} \left (b + 2 c x\right )^{2} \left (a + b x + c x^{2}\right )^{p + 1}}{p + 2} + \frac{d^{3} \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{p + 1}}{\left (p + 1\right ) \left (p + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*c*d*x+b*d)**3*(c*x**2+b*x+a)**p,x)

[Out]

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

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Mathematica [A]  time = 0.0593114, size = 58, normalized size = 0.85 \[ \frac{d^3 (a+x (b+c x))^{p+1} \left (4 c \left (c (p+1) x^2-a\right )+b^2 (p+2)+4 b c (p+1) x\right )}{(p+1) (p+2)} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*d + 2*c*d*x)^3*(a + b*x + c*x^2)^p,x]

[Out]

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

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Maple [A]  time = 0.01, size = 74, normalized size = 1.1 \[ -{\frac{ \left ( c{x}^{2}+bx+a \right ) ^{1+p} \left ( -4\,{c}^{2}p{x}^{2}-4\,bcpx-4\,{c}^{2}{x}^{2}-{b}^{2}p-4\,bxc+4\,ac-2\,{b}^{2} \right ){d}^{3}}{{p}^{2}+3\,p+2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*c*d*x+b*d)^3*(c*x^2+b*x+a)^p,x)

[Out]

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

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Maxima [A]  time = 0.752858, size = 166, normalized size = 2.44 \[ \frac{{\left (4 \, c^{3} d^{3}{\left (p + 1\right )} x^{4} + 8 \, b c^{2} d^{3}{\left (p + 1\right )} x^{3} + a b^{2} d^{3}{\left (p + 2\right )} - 4 \, a^{2} c d^{3} +{\left (b^{2} c d^{3}{\left (5 \, p + 6\right )} + 4 \, a c^{2} d^{3} p\right )} x^{2} +{\left (b^{3} d^{3}{\left (p + 2\right )} + 4 \, a b c d^{3} p\right )} x\right )}{\left (c x^{2} + b x + a\right )}^{p}}{p^{2} + 3 \, p + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^3*(c*x^2 + b*x + a)^p,x, algorithm="maxima")

[Out]

(4*c^3*d^3*(p + 1)*x^4 + 8*b*c^2*d^3*(p + 1)*x^3 + a*b^2*d^3*(p + 2) - 4*a^2*c*d
^3 + (b^2*c*d^3*(5*p + 6) + 4*a*c^2*d^3*p)*x^2 + (b^3*d^3*(p + 2) + 4*a*b*c*d^3*
p)*x)*(c*x^2 + b*x + a)^p/(p^2 + 3*p + 2)

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Fricas [A]  time = 0.229961, size = 204, normalized size = 3. \[ \frac{{\left (a b^{2} d^{3} p + 4 \,{\left (c^{3} d^{3} p + c^{3} d^{3}\right )} x^{4} + 2 \,{\left (a b^{2} - 2 \, a^{2} c\right )} d^{3} + 8 \,{\left (b c^{2} d^{3} p + b c^{2} d^{3}\right )} x^{3} +{\left (6 \, b^{2} c d^{3} +{\left (5 \, b^{2} c + 4 \, a c^{2}\right )} d^{3} p\right )} x^{2} +{\left (2 \, b^{3} d^{3} +{\left (b^{3} + 4 \, a b c\right )} d^{3} p\right )} x\right )}{\left (c x^{2} + b x + a\right )}^{p}}{p^{2} + 3 \, p + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^3*(c*x^2 + b*x + a)^p,x, algorithm="fricas")

[Out]

(a*b^2*d^3*p + 4*(c^3*d^3*p + c^3*d^3)*x^4 + 2*(a*b^2 - 2*a^2*c)*d^3 + 8*(b*c^2*
d^3*p + b*c^2*d^3)*x^3 + (6*b^2*c*d^3 + (5*b^2*c + 4*a*c^2)*d^3*p)*x^2 + (2*b^3*
d^3 + (b^3 + 4*a*b*c)*d^3*p)*x)*(c*x^2 + b*x + a)^p/(p^2 + 3*p + 2)

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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((2*c*d*x+b*d)**3*(c*x**2+b*x+a)**p,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.222567, size = 452, normalized size = 6.65 \[ \frac{4 \, c^{3} d^{3} p x^{4} e^{\left (p{\rm ln}\left (c x^{2} + b x + a\right )\right )} + 8 \, b c^{2} d^{3} p x^{3} e^{\left (p{\rm ln}\left (c x^{2} + b x + a\right )\right )} + 4 \, c^{3} d^{3} x^{4} e^{\left (p{\rm ln}\left (c x^{2} + b x + a\right )\right )} + 5 \, b^{2} c d^{3} p x^{2} e^{\left (p{\rm ln}\left (c x^{2} + b x + a\right )\right )} + 4 \, a c^{2} d^{3} p x^{2} e^{\left (p{\rm ln}\left (c x^{2} + b x + a\right )\right )} + 8 \, b c^{2} d^{3} x^{3} e^{\left (p{\rm ln}\left (c x^{2} + b x + a\right )\right )} + b^{3} d^{3} p x e^{\left (p{\rm ln}\left (c x^{2} + b x + a\right )\right )} + 4 \, a b c d^{3} p x e^{\left (p{\rm ln}\left (c x^{2} + b x + a\right )\right )} + 6 \, b^{2} c d^{3} x^{2} e^{\left (p{\rm ln}\left (c x^{2} + b x + a\right )\right )} + a b^{2} d^{3} p e^{\left (p{\rm ln}\left (c x^{2} + b x + a\right )\right )} + 2 \, b^{3} d^{3} x e^{\left (p{\rm ln}\left (c x^{2} + b x + a\right )\right )} + 2 \, a b^{2} d^{3} e^{\left (p{\rm ln}\left (c x^{2} + b x + a\right )\right )} - 4 \, a^{2} c d^{3} e^{\left (p{\rm ln}\left (c x^{2} + b x + a\right )\right )}}{p^{2} + 3 \, p + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^3*(c*x^2 + b*x + a)^p,x, algorithm="giac")

[Out]

(4*c^3*d^3*p*x^4*e^(p*ln(c*x^2 + b*x + a)) + 8*b*c^2*d^3*p*x^3*e^(p*ln(c*x^2 + b
*x + a)) + 4*c^3*d^3*x^4*e^(p*ln(c*x^2 + b*x + a)) + 5*b^2*c*d^3*p*x^2*e^(p*ln(c
*x^2 + b*x + a)) + 4*a*c^2*d^3*p*x^2*e^(p*ln(c*x^2 + b*x + a)) + 8*b*c^2*d^3*x^3
*e^(p*ln(c*x^2 + b*x + a)) + b^3*d^3*p*x*e^(p*ln(c*x^2 + b*x + a)) + 4*a*b*c*d^3
*p*x*e^(p*ln(c*x^2 + b*x + a)) + 6*b^2*c*d^3*x^2*e^(p*ln(c*x^2 + b*x + a)) + a*b
^2*d^3*p*e^(p*ln(c*x^2 + b*x + a)) + 2*b^3*d^3*x*e^(p*ln(c*x^2 + b*x + a)) + 2*a
*b^2*d^3*e^(p*ln(c*x^2 + b*x + a)) - 4*a^2*c*d^3*e^(p*ln(c*x^2 + b*x + a)))/(p^2
+ 3*p + 2)