3.184 \(\int \left (b+3 d x^2\right ) \left (b x+d x^3\right )^n \, dx\)

Optimal. Leaf size=19 \[ \frac{\left (b x+d x^3\right )^{n+1}}{n+1} \]

[Out]

(b*x + d*x^3)^(1 + n)/(1 + n)

_______________________________________________________________________________________

Rubi [A]  time = 0.0133596, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ \frac{\left (b x+d x^3\right )^{n+1}}{n+1} \]

Antiderivative was successfully verified.

[In]  Int[(b + 3*d*x^2)*(b*x + d*x^3)^n,x]

[Out]

(b*x + d*x^3)^(1 + n)/(1 + n)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 4.02258, size = 14, normalized size = 0.74 \[ \frac{\left (b x + d x^{3}\right )^{n + 1}}{n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((3*d*x**2+b)*(d*x**3+b*x)**n,x)

[Out]

(b*x + d*x**3)**(n + 1)/(n + 1)

_______________________________________________________________________________________

Mathematica [A]  time = 0.039293, size = 19, normalized size = 1. \[ \frac{\left (x \left (b+d x^2\right )\right )^{n+1}}{n+1} \]

Antiderivative was successfully verified.

[In]  Integrate[(b + 3*d*x^2)*(b*x + d*x^3)^n,x]

[Out]

(x*(b + d*x^2))^(1 + n)/(1 + n)

_______________________________________________________________________________________

Maple [A]  time = 0.005, size = 26, normalized size = 1.4 \[{\frac{x \left ( d{x}^{2}+b \right ) \left ( d{x}^{3}+bx \right ) ^{n}}{1+n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((3*d*x^2+b)*(d*x^3+b*x)^n,x)

[Out]

x*(d*x^2+b)/(1+n)*(d*x^3+b*x)^n

_______________________________________________________________________________________

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((3*d*x^2 + b)*(d*x^3 + b*x)^n,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.271494, size = 35, normalized size = 1.84 \[ \frac{{\left (d x^{3} + b x\right )}{\left (d x^{3} + b x\right )}^{n}}{n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(d*x^3 + b*x)*(d*x^3 + b*x)^n/(n + 1)

_______________________________________________________________________________________

Sympy [A]  time = 42.4005, size = 73, normalized size = 3.84 \[ \begin{cases} \frac{b x \left (b x + d x^{3}\right )^{n}}{n + 1} + \frac{d x^{3} \left (b x + d x^{3}\right )^{n}}{n + 1} & \text{for}\: n \neq -1 \\\log{\left (x \right )} + \log{\left (- i \sqrt{b} \sqrt{\frac{1}{d}} + x \right )} + \log{\left (i \sqrt{b} \sqrt{\frac{1}{d}} + x \right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*d*x**2+b)*(d*x**3+b*x)**n,x)

[Out]

Piecewise((b*x*(b*x + d*x**3)**n/(n + 1) + d*x**3*(b*x + d*x**3)**n/(n + 1), Ne(
n, -1)), (log(x) + log(-I*sqrt(b)*sqrt(1/d) + x) + log(I*sqrt(b)*sqrt(1/d) + x),
True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.271794, size = 55, normalized size = 2.89 \[ \frac{d x^{3} e^{\left (n{\rm ln}\left (d x^{3} + b x\right )\right )} + b x e^{\left (n{\rm ln}\left (d x^{3} + b x\right )\right )}}{n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(d*x^3*e^(n*ln(d*x^3 + b*x)) + b*x*e^(n*ln(d*x^3 + b*x)))/(n + 1)