3.2842 \(\int (c+d x)^3 \left (a+b (c+d x)^2\right )^p \, dx\)

Optimal. Leaf size=62 \[ \frac{\left (a+b (c+d x)^2\right )^{p+2}}{2 b^2 d (p+2)}-\frac{a \left (a+b (c+d x)^2\right )^{p+1}}{2 b^2 d (p+1)} \]

[Out]

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

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Rubi [A]  time = 0.138679, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{\left (a+b (c+d x)^2\right )^{p+2}}{2 b^2 d (p+2)}-\frac{a \left (a+b (c+d x)^2\right )^{p+1}}{2 b^2 d (p+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 16.7844, size = 48, normalized size = 0.77 \[ - \frac{a \left (a + b \left (c + d x\right )^{2}\right )^{p + 1}}{2 b^{2} d \left (p + 1\right )} + \frac{\left (a + b \left (c + d x\right )^{2}\right )^{p + 2}}{2 b^{2} d \left (p + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0519326, size = 51, normalized size = 0.82 \[ \frac{\left (a+b (c+d x)^2\right )^{p+1} \left (b (p+1) (c+d x)^2-a\right )}{2 b^2 d (p+1) (p+2)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.012, size = 91, normalized size = 1.5 \[ -{\frac{ \left ( b{d}^{2}{x}^{2}+2\,bcdx+b{c}^{2}+a \right ) ^{1+p} \left ( -b{d}^{2}p{x}^{2}-2\,bcdpx-b{d}^{2}{x}^{2}-b{c}^{2}p-2\,bcdx-b{c}^{2}+a \right ) }{2\,{b}^{2}d \left ({p}^{2}+3\,p+2 \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.47367, size = 189, normalized size = 3.05 \[ \frac{{\left (b^{2} d^{4}{\left (p + 1\right )} x^{4} + 4 \, b^{2} c d^{3}{\left (p + 1\right )} x^{3} + b^{2} c^{4}{\left (p + 1\right )} + a b c^{2} p +{\left (6 \, b^{2} c^{2} d^{2}{\left (p + 1\right )} + a b d^{2} p\right )} x^{2} - a^{2} + 2 \,{\left (2 \, b^{2} c^{3} d{\left (p + 1\right )} + a b c d p\right )} x\right )}{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )}^{p}}{2 \,{\left (p^{2} + 3 \, p + 2\right )} b^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.230273, size = 247, normalized size = 3.98 \[ \frac{{\left (b^{2} c^{4} +{\left (b^{2} d^{4} p + b^{2} d^{4}\right )} x^{4} + 4 \,{\left (b^{2} c d^{3} p + b^{2} c d^{3}\right )} x^{3} +{\left (6 \, b^{2} c^{2} d^{2} +{\left (6 \, b^{2} c^{2} + a b\right )} d^{2} p\right )} x^{2} - a^{2} +{\left (b^{2} c^{4} + a b c^{2}\right )} p + 2 \,{\left (2 \, b^{2} c^{3} d +{\left (2 \, b^{2} c^{3} + a b c\right )} d p\right )} x\right )}{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )}^{p}}{2 \,{\left (b^{2} d p^{2} + 3 \, b^{2} d p + 2 \, b^{2} d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.221825, size = 699, normalized size = 11.27 \[ \frac{b^{2} d^{4} p x^{4} e^{\left (p{\rm ln}\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )\right )} + 4 \, b^{2} c d^{3} p x^{3} e^{\left (p{\rm ln}\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )\right )} + b^{2} d^{4} x^{4} e^{\left (p{\rm ln}\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )\right )} + 6 \, b^{2} c^{2} d^{2} p x^{2} e^{\left (p{\rm ln}\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )\right )} + 4 \, b^{2} c d^{3} x^{3} e^{\left (p{\rm ln}\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )\right )} + 4 \, b^{2} c^{3} d p x e^{\left (p{\rm ln}\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )\right )} + 6 \, b^{2} c^{2} d^{2} x^{2} e^{\left (p{\rm ln}\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )\right )} + b^{2} c^{4} p e^{\left (p{\rm ln}\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )\right )} + 4 \, b^{2} c^{3} d x e^{\left (p{\rm ln}\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )\right )} + a b d^{2} p x^{2} e^{\left (p{\rm ln}\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )\right )} + b^{2} c^{4} e^{\left (p{\rm ln}\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )\right )} + 2 \, a b c d p x e^{\left (p{\rm ln}\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )\right )} + a b c^{2} p e^{\left (p{\rm ln}\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )\right )} - a^{2} e^{\left (p{\rm ln}\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )\right )}}{2 \,{\left (b^{2} d p^{2} + 3 \, b^{2} d p + 2 \, b^{2} d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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