3.275 \(\int (A+B x) \left (a+c x^2\right )^4 \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0696689, 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 \[ a^4 A x+\frac{4}{3} a^3 A c x^3+\frac{6}{5} a^2 A c^2 x^5+\frac{4}{7} a A c^3 x^7+\frac{B \left (a+c x^2\right )^5}{10 c}+\frac{1}{9} A c^4 x^9 \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)*(a + c*x^2)^4,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+a)**4,x)

[Out]

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

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Mathematica [A]  time = 0.00523428, size = 110, normalized size = 1.51 \[ a^4 A x+\frac{1}{2} a^4 B x^2+\frac{4}{3} a^3 A c x^3+a^3 B c x^4+\frac{6}{5} a^2 A c^2 x^5+a^2 B c^2 x^6+\frac{4}{7} a A c^3 x^7+\frac{1}{2} a B c^3 x^8+\frac{1}{9} A c^4 x^9+\frac{1}{10} B c^4 x^{10} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)*(a + c*x^2)^4,x]

[Out]

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

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Maple [A]  time = 0.002, size = 97, normalized size = 1.3 \[{\frac{B{c}^{4}{x}^{10}}{10}}+{\frac{A{c}^{4}{x}^{9}}{9}}+{\frac{aB{c}^{3}{x}^{8}}{2}}+{\frac{4\,aA{c}^{3}{x}^{7}}{7}}+{a}^{2}B{c}^{2}{x}^{6}+{\frac{6\,{a}^{2}A{c}^{2}{x}^{5}}{5}}+{a}^{3}Bc{x}^{4}+{\frac{4\,{a}^{3}Ac{x}^{3}}{3}}+{\frac{{a}^{4}B{x}^{2}}{2}}+{a}^{4}Ax \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+a)^4,x)

[Out]

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

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Maxima [A]  time = 0.682402, size = 130, normalized size = 1.78 \[ \frac{1}{10} \, B c^{4} x^{10} + \frac{1}{9} \, A c^{4} x^{9} + \frac{1}{2} \, B a c^{3} x^{8} + \frac{4}{7} \, A a c^{3} x^{7} + B a^{2} c^{2} x^{6} + \frac{6}{5} \, A a^{2} c^{2} x^{5} + B a^{3} c x^{4} + \frac{4}{3} \, A a^{3} c x^{3} + \frac{1}{2} \, B a^{4} x^{2} + A a^{4} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.2512, size = 1, normalized size = 0.01 \[ \frac{1}{10} x^{10} c^{4} B + \frac{1}{9} x^{9} c^{4} A + \frac{1}{2} x^{8} c^{3} a B + \frac{4}{7} x^{7} c^{3} a A + x^{6} c^{2} a^{2} B + \frac{6}{5} x^{5} c^{2} a^{2} A + x^{4} c a^{3} B + \frac{4}{3} x^{3} c a^{3} A + \frac{1}{2} x^{2} a^{4} B + x a^{4} A \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.140251, size = 112, normalized size = 1.53 \[ A a^{4} x + \frac{4 A a^{3} c x^{3}}{3} + \frac{6 A a^{2} c^{2} x^{5}}{5} + \frac{4 A a c^{3} x^{7}}{7} + \frac{A c^{4} x^{9}}{9} + \frac{B a^{4} x^{2}}{2} + B a^{3} c x^{4} + B a^{2} c^{2} x^{6} + \frac{B a c^{3} x^{8}}{2} + \frac{B c^{4} x^{10}}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+a)**4,x)

[Out]

A*a**4*x + 4*A*a**3*c*x**3/3 + 6*A*a**2*c**2*x**5/5 + 4*A*a*c**3*x**7/7 + A*c**4
*x**9/9 + B*a**4*x**2/2 + B*a**3*c*x**4 + B*a**2*c**2*x**6 + B*a*c**3*x**8/2 + B
*c**4*x**10/10

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GIAC/XCAS [A]  time = 0.268798, size = 130, normalized size = 1.78 \[ \frac{1}{10} \, B c^{4} x^{10} + \frac{1}{9} \, A c^{4} x^{9} + \frac{1}{2} \, B a c^{3} x^{8} + \frac{4}{7} \, A a c^{3} x^{7} + B a^{2} c^{2} x^{6} + \frac{6}{5} \, A a^{2} c^{2} x^{5} + B a^{3} c x^{4} + \frac{4}{3} \, A a^{3} c x^{3} + \frac{1}{2} \, B a^{4} x^{2} + A a^{4} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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