3.486 \(\int (c+d x) \left (a+b x^2\right )^4 \, dx\)

Optimal. Leaf size=73 \[ a^4 c x+\frac{4}{3} a^3 b c x^3+\frac{6}{5} a^2 b^2 c x^5+\frac{4}{7} a b^3 c x^7+\frac{d \left (a+b x^2\right )^5}{10 b}+\frac{1}{9} b^4 c x^9 \]

[Out]

a^4*c*x + (4*a^3*b*c*x^3)/3 + (6*a^2*b^2*c*x^5)/5 + (4*a*b^3*c*x^7)/7 + (b^4*c*x
^9)/9 + (d*(a + b*x^2)^5)/(10*b)

_______________________________________________________________________________________

Rubi [A]  time = 0.0894868, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ a^4 c x+\frac{4}{3} a^3 b c x^3+\frac{6}{5} a^2 b^2 c x^5+\frac{4}{7} a b^3 c x^7+\frac{d \left (a+b x^2\right )^5}{10 b}+\frac{1}{9} b^4 c x^9 \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)*(a + b*x^2)^4,x]

[Out]

a^4*c*x + (4*a^3*b*c*x^3)/3 + (6*a^2*b^2*c*x^5)/5 + (4*a*b^3*c*x^7)/7 + (b^4*c*x
^9)/9 + (d*(a + b*x^2)^5)/(10*b)

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{4 a^{3} b c x^{3}}{3} + \frac{6 a^{2} b^{2} c x^{5}}{5} + \frac{4 a b^{3} c x^{7}}{7} + \frac{b^{4} c x^{9}}{9} + c \int a^{4}\, dx + \frac{d \left (a + b x^{2}\right )^{5}}{10 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)*(b*x**2+a)**4,x)

[Out]

4*a**3*b*c*x**3/3 + 6*a**2*b**2*c*x**5/5 + 4*a*b**3*c*x**7/7 + b**4*c*x**9/9 + c
*Integral(a**4, x) + d*(a + b*x**2)**5/(10*b)

_______________________________________________________________________________________

Mathematica [A]  time = 0.00627615, size = 110, normalized size = 1.51 \[ a^4 c x+\frac{1}{2} a^4 d x^2+\frac{4}{3} a^3 b c x^3+a^3 b d x^4+\frac{6}{5} a^2 b^2 c x^5+a^2 b^2 d x^6+\frac{4}{7} a b^3 c x^7+\frac{1}{2} a b^3 d x^8+\frac{1}{9} b^4 c x^9+\frac{1}{10} b^4 d x^{10} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)*(a + b*x^2)^4,x]

[Out]

a^4*c*x + (a^4*d*x^2)/2 + (4*a^3*b*c*x^3)/3 + a^3*b*d*x^4 + (6*a^2*b^2*c*x^5)/5
+ a^2*b^2*d*x^6 + (4*a*b^3*c*x^7)/7 + (a*b^3*d*x^8)/2 + (b^4*c*x^9)/9 + (b^4*d*x
^10)/10

_______________________________________________________________________________________

Maple [A]  time = 0.002, size = 97, normalized size = 1.3 \[{\frac{d{b}^{4}{x}^{10}}{10}}+{\frac{{b}^{4}c{x}^{9}}{9}}+{\frac{ad{b}^{3}{x}^{8}}{2}}+{\frac{4\,a{b}^{3}c{x}^{7}}{7}}+{a}^{2}d{b}^{2}{x}^{6}+{\frac{6\,{a}^{2}{b}^{2}c{x}^{5}}{5}}+{a}^{3}bd{x}^{4}+{\frac{4\,{a}^{3}bc{x}^{3}}{3}}+{\frac{{a}^{4}d{x}^{2}}{2}}+{a}^{4}cx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)*(b*x^2+a)^4,x)

[Out]

1/10*d*b^4*x^10+1/9*b^4*c*x^9+1/2*a*d*b^3*x^8+4/7*a*b^3*c*x^7+a^2*d*b^2*x^6+6/5*
a^2*b^2*c*x^5+a^3*b*d*x^4+4/3*a^3*b*c*x^3+1/2*a^4*d*x^2+a^4*c*x

_______________________________________________________________________________________

Maxima [A]  time = 0.698636, size = 130, normalized size = 1.78 \[ \frac{1}{10} \, b^{4} d x^{10} + \frac{1}{9} \, b^{4} c x^{9} + \frac{1}{2} \, a b^{3} d x^{8} + \frac{4}{7} \, a b^{3} c x^{7} + a^{2} b^{2} d x^{6} + \frac{6}{5} \, a^{2} b^{2} c x^{5} + a^{3} b d x^{4} + \frac{4}{3} \, a^{3} b c x^{3} + \frac{1}{2} \, a^{4} d x^{2} + a^{4} c x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^4*(d*x + c),x, algorithm="maxima")

[Out]

1/10*b^4*d*x^10 + 1/9*b^4*c*x^9 + 1/2*a*b^3*d*x^8 + 4/7*a*b^3*c*x^7 + a^2*b^2*d*
x^6 + 6/5*a^2*b^2*c*x^5 + a^3*b*d*x^4 + 4/3*a^3*b*c*x^3 + 1/2*a^4*d*x^2 + a^4*c*
x

_______________________________________________________________________________________

Fricas [A]  time = 0.189352, size = 1, normalized size = 0.01 \[ \frac{1}{10} x^{10} d b^{4} + \frac{1}{9} x^{9} c b^{4} + \frac{1}{2} x^{8} d b^{3} a + \frac{4}{7} x^{7} c b^{3} a + x^{6} d b^{2} a^{2} + \frac{6}{5} x^{5} c b^{2} a^{2} + x^{4} d b a^{3} + \frac{4}{3} x^{3} c b a^{3} + \frac{1}{2} x^{2} d a^{4} + x c a^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^4*(d*x + c),x, algorithm="fricas")

[Out]

1/10*x^10*d*b^4 + 1/9*x^9*c*b^4 + 1/2*x^8*d*b^3*a + 4/7*x^7*c*b^3*a + x^6*d*b^2*
a^2 + 6/5*x^5*c*b^2*a^2 + x^4*d*b*a^3 + 4/3*x^3*c*b*a^3 + 1/2*x^2*d*a^4 + x*c*a^
4

_______________________________________________________________________________________

Sympy [A]  time = 0.144977, size = 112, normalized size = 1.53 \[ a^{4} c x + \frac{a^{4} d x^{2}}{2} + \frac{4 a^{3} b c x^{3}}{3} + a^{3} b d x^{4} + \frac{6 a^{2} b^{2} c x^{5}}{5} + a^{2} b^{2} d x^{6} + \frac{4 a b^{3} c x^{7}}{7} + \frac{a b^{3} d x^{8}}{2} + \frac{b^{4} c x^{9}}{9} + \frac{b^{4} d x^{10}}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x+c)*(b*x**2+a)**4,x)

[Out]

a**4*c*x + a**4*d*x**2/2 + 4*a**3*b*c*x**3/3 + a**3*b*d*x**4 + 6*a**2*b**2*c*x**
5/5 + a**2*b**2*d*x**6 + 4*a*b**3*c*x**7/7 + a*b**3*d*x**8/2 + b**4*c*x**9/9 + b
**4*d*x**10/10

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.211148, size = 130, normalized size = 1.78 \[ \frac{1}{10} \, b^{4} d x^{10} + \frac{1}{9} \, b^{4} c x^{9} + \frac{1}{2} \, a b^{3} d x^{8} + \frac{4}{7} \, a b^{3} c x^{7} + a^{2} b^{2} d x^{6} + \frac{6}{5} \, a^{2} b^{2} c x^{5} + a^{3} b d x^{4} + \frac{4}{3} \, a^{3} b c x^{3} + \frac{1}{2} \, a^{4} d x^{2} + a^{4} c x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^4*(d*x + c),x, algorithm="giac")

[Out]

1/10*b^4*d*x^10 + 1/9*b^4*c*x^9 + 1/2*a*b^3*d*x^8 + 4/7*a*b^3*c*x^7 + a^2*b^2*d*
x^6 + 6/5*a^2*b^2*c*x^5 + a^3*b*d*x^4 + 4/3*a^3*b*c*x^3 + 1/2*a^4*d*x^2 + a^4*c*
x