3.3.68 \(\int (A+B x) (a+c x^2)^3 \, dx\) [268]

Optimal. Leaf size=56 \[ a^3 A x+a^2 A c x^3+\frac {3}{5} a A c^2 x^5+\frac {1}{7} A c^3 x^7+\frac {B \left (a+c x^2\right )^4}{8 c} \]

[Out]

a^3*A*x+a^2*A*c*x^3+3/5*a*A*c^2*x^5+1/7*A*c^3*x^7+1/8*B*(c*x^2+a)^4/c

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Rubi [A]
time = 0.03, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {655, 200} \begin {gather*} a^3 A x+a^2 A c x^3+\frac {3}{5} a A c^2 x^5+\frac {B \left (a+c x^2\right )^4}{8 c}+\frac {1}{7} A c^3 x^7 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x)*(a + c*x^2)^3,x]

[Out]

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

Rule 200

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rule 655

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[e*((a + c*x^2)^(p + 1)/(2*c*(p + 1))),
x] + Dist[d, Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[p, -1]

Rubi steps

\begin {align*} \int (A+B x) \left (a+c x^2\right )^3 \, dx &=\frac {B \left (a+c x^2\right )^4}{8 c}+A \int \left (a+c x^2\right )^3 \, dx\\ &=\frac {B \left (a+c x^2\right )^4}{8 c}+A \int \left (a^3+3 a^2 c x^2+3 a c^2 x^4+c^3 x^6\right ) \, dx\\ &=a^3 A x+a^2 A c x^3+\frac {3}{5} a A c^2 x^5+\frac {1}{7} A c^3 x^7+\frac {B \left (a+c x^2\right )^4}{8 c}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 85, normalized size = 1.52 \begin {gather*} a^3 A x+\frac {1}{2} a^3 B x^2+a^2 A c x^3+\frac {3}{4} a^2 B c x^4+\frac {3}{5} a A c^2 x^5+\frac {1}{2} a B c^2 x^6+\frac {1}{7} A c^3 x^7+\frac {1}{8} B c^3 x^8 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)*(a + c*x^2)^3,x]

[Out]

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

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Maple [A]
time = 0.56, size = 74, normalized size = 1.32

method	result	size
gosper	\(\frac {1}{8} B \,c^{3} x^{8}+\frac {1}{7} A \,c^{3} x^{7}+\frac {1}{2} B a \,c^{2} x^{6}+\frac {3}{5} a A \,c^{2} x^{5}+\frac {3}{4} B \,a^{2} c \,x^{4}+a^{2} A c \,x^{3}+\frac {1}{2} B \,a^{3} x^{2}+a^{3} A x\)	\(74\)
default	\(\frac {1}{8} B \,c^{3} x^{8}+\frac {1}{7} A \,c^{3} x^{7}+\frac {1}{2} B a \,c^{2} x^{6}+\frac {3}{5} a A \,c^{2} x^{5}+\frac {3}{4} B \,a^{2} c \,x^{4}+a^{2} A c \,x^{3}+\frac {1}{2} B \,a^{3} x^{2}+a^{3} A x\)	\(74\)
norman	\(\frac {1}{8} B \,c^{3} x^{8}+\frac {1}{7} A \,c^{3} x^{7}+\frac {1}{2} B a \,c^{2} x^{6}+\frac {3}{5} a A \,c^{2} x^{5}+\frac {3}{4} B \,a^{2} c \,x^{4}+a^{2} A c \,x^{3}+\frac {1}{2} B \,a^{3} x^{2}+a^{3} A x\)	\(74\)
risch	\(\frac {1}{8} B \,c^{3} x^{8}+\frac {1}{7} A \,c^{3} x^{7}+\frac {1}{2} B a \,c^{2} x^{6}+\frac {3}{5} a A \,c^{2} x^{5}+\frac {3}{4} B \,a^{2} c \,x^{4}+a^{2} A c \,x^{3}+\frac {1}{2} B \,a^{3} x^{2}+a^{3} A x\)	\(74\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+a)^3,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.29, size = 73, normalized size = 1.30 \begin {gather*} \frac {1}{8} \, B c^{3} x^{8} + \frac {1}{7} \, A c^{3} x^{7} + \frac {1}{2} \, B a c^{2} x^{6} + \frac {3}{5} \, A a c^{2} x^{5} + \frac {3}{4} \, B a^{2} c x^{4} + A a^{2} c x^{3} + \frac {1}{2} \, B a^{3} x^{2} + A a^{3} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 2.63, size = 73, normalized size = 1.30 \begin {gather*} \frac {1}{8} \, B c^{3} x^{8} + \frac {1}{7} \, A c^{3} x^{7} + \frac {1}{2} \, B a c^{2} x^{6} + \frac {3}{5} \, A a c^{2} x^{5} + \frac {3}{4} \, B a^{2} c x^{4} + A a^{2} c x^{3} + \frac {1}{2} \, B a^{3} x^{2} + A a^{3} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.01, size = 85, normalized size = 1.52 \begin {gather*} A a^{3} x + A a^{2} c x^{3} + \frac {3 A a c^{2} x^{5}}{5} + \frac {A c^{3} x^{7}}{7} + \frac {B a^{3} x^{2}}{2} + \frac {3 B a^{2} c x^{4}}{4} + \frac {B a c^{2} x^{6}}{2} + \frac {B c^{3} x^{8}}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+a)**3,x)

[Out]

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

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Giac [A]
time = 0.74, size = 73, normalized size = 1.30 \begin {gather*} \frac {1}{8} \, B c^{3} x^{8} + \frac {1}{7} \, A c^{3} x^{7} + \frac {1}{2} \, B a c^{2} x^{6} + \frac {3}{5} \, A a c^{2} x^{5} + \frac {3}{4} \, B a^{2} c x^{4} + A a^{2} c x^{3} + \frac {1}{2} \, B a^{3} x^{2} + A a^{3} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^3,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.03, size = 73, normalized size = 1.30 \begin {gather*} \frac {B\,a^3\,x^2}{2}+A\,a^3\,x+\frac {3\,B\,a^2\,c\,x^4}{4}+A\,a^2\,c\,x^3+\frac {B\,a\,c^2\,x^6}{2}+\frac {3\,A\,a\,c^2\,x^5}{5}+\frac {B\,c^3\,x^8}{8}+\frac {A\,c^3\,x^7}{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^2)^3*(A + B*x),x)

[Out]

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

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