3.2844 \(\int (c+d x) \left (a+b (c+d x)^2\right )^p \, dx\)

Optimal. Leaf size=30 \[ \frac{\left (a+b (c+d x)^2\right )^{p+1}}{2 b d (p+1)} \]

[Out]

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

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Rubi [A]  time = 0.024824, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053 \[ \frac{\left (a+b (c+d x)^2\right )^{p+1}}{2 b d (p+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 4.86639, size = 20, normalized size = 0.67 \[ \frac{\left (a + b \left (c + d x\right )^{2}\right )^{p + 1}}{2 b d \left (p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0140591, size = 29, normalized size = 0.97 \[ \frac{\left (a+b (c+d x)^2\right )^{p+1}}{d (2 b p+2 b)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 39, normalized size = 1.3 \[{\frac{ \left ( b{d}^{2}{x}^{2}+2\,bcdx+b{c}^{2}+a \right ) ^{1+p}}{2\,bd \left ( 1+p \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)*(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/(1+p)

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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((d*x + c)*((d*x + c)^2*b + a)^p,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.229411, size = 76, normalized size = 2.53 \[ \frac{{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )}{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )}^{p}}{2 \,{\left (b d p + b d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.22003, size = 181, normalized size = 6.03 \[ \frac{b d^{2} x^{2} e^{\left (p{\rm ln}\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )\right )} + 2 \, b c d x e^{\left (p{\rm ln}\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )\right )} + b c^{2} e^{\left (p{\rm ln}\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )\right )} + a e^{\left (p{\rm ln}\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )\right )}}{2 \,{\left (b d p + b d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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