3.2903 \(\int (c+d x)^3 \left (a+b (c+d x)^4\right )^p \, dx\)

Optimal. Leaf size=30 \[ \frac{\left (a+b (c+d x)^4\right )^{p+1}}{4 b d (p+1)} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0272146, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048 \[ \frac{\left (a+b (c+d x)^4\right )^{p+1}}{4 b d (p+1)} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^3*(a + b*(c + d*x)^4)^p,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 4.9432, size = 20, normalized size = 0.67 \[ \frac{\left (a + b \left (c + d x\right )^{4}\right )^{p + 1}}{4 b d \left (p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**3*(a+b*(d*x+c)**4)**p,x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0328594, size = 32, normalized size = 1.07 \[ \frac{\left (a+b (c+d x)^4\right )^{p+1}}{2 d (2 b p+2 b)} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^3*(a + b*(c + d*x)^4)^p,x]

[Out]

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

_______________________________________________________________________________________

Maple [B]  time = 0.01, size = 63, normalized size = 2.1 \[{\frac{ \left ( b{d}^{4}{x}^{4}+4\,bc{d}^{3}{x}^{3}+6\,b{c}^{2}{d}^{2}{x}^{2}+4\,b{c}^{3}dx+b{c}^{4}+a \right ) ^{1+p}}{4\,bd \left ( 1+p \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^3*(a+b*(d*x+c)^4)^p,x)

[Out]

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

_______________________________________________________________________________________

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((d*x + c)^3*((d*x + c)^4*b + a)^p,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.224008, size = 140, normalized size = 4.67 \[ \frac{{\left (b d^{4} x^{4} + 4 \, b c d^{3} x^{3} + 6 \, b c^{2} d^{2} x^{2} + 4 \, b c^{3} d x + b c^{4} + a\right )}{\left (b d^{4} x^{4} + 4 \, b c d^{3} x^{3} + 6 \, b c^{2} d^{2} x^{2} + 4 \, b c^{3} d x + b c^{4} + a\right )}^{p}}{4 \,{\left (b d p + b d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^3*((d*x + c)^4*b + a)^p,x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

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)**4)**p,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.217122, size = 38, normalized size = 1.27 \[ \frac{{\left ({\left (d x + c\right )}^{4} b + a\right )}^{p + 1}}{4 \, b d{\left (p + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^3*((d*x + c)^4*b + a)^p,x, algorithm="giac")

[Out]

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