\(\int (c+d x) (a+b x^2)^4 \, dx\) [497]

Optimal result

Integrand size = 15, antiderivative size = 73 \[ \int (c+d x) \left (a+b x^2\right )^4 \, dx=a^4 c x+\frac {4}{3} a^3 b c x^3+\frac {6}{5} a^2 b^2 c x^5+\frac {4}{7} a b^3 c x^7+\frac {1}{9} b^4 c x^9+\frac {d \left (a+b x^2\right )^5}{10 b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {655, 200} \[ \int (c+d x) \left (a+b x^2\right )^4 \, dx=a^4 c x+\frac {4}{3} a^3 b c x^3+\frac {6}{5} a^2 b^2 c x^5+\frac {4}{7} a b^3 c x^7+\frac {d \left (a+b x^2\right )^5}{10 b}+\frac {1}{9} b^4 c x^9 \]

[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)

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*} \text {integral}& = \frac {d \left (a+b x^2\right )^5}{10 b}+c \int \left (a+b x^2\right )^4 \, dx \\ & = \frac {d \left (a+b x^2\right )^5}{10 b}+c \int \left (a^4+4 a^3 b x^2+6 a^2 b^2 x^4+4 a b^3 x^6+b^4 x^8\right ) \, dx \\ & = 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 {1}{9} b^4 c x^9+\frac {d \left (a+b x^2\right )^5}{10 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.51 \[ \int (c+d x) \left (a+b x^2\right )^4 \, dx=a^4 c x+\frac {1}{2} a^4 d x^2+\frac {4}{3} a^3 b c x^3+a^3 b d x^4+\frac {6}{5} a^2 b^2 c x^5+a^2 b^2 d x^6+\frac {4}{7} a b^3 c x^7+\frac {1}{2} a b^3 d x^8+\frac {1}{9} b^4 c x^9+\frac {1}{10} b^4 d x^{10} \]

[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] (verified)

Time = 2.13 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.33

method	result	size
gosper	\(\frac {1}{10} b^{4} d \,x^{10}+\frac {1}{9} b^{4} c \,x^{9}+\frac {1}{2} a \,b^{3} d \,x^{8}+\frac {4}{7} a \,b^{3} c \,x^{7}+a^{2} b^{2} d \,x^{6}+\frac {6}{5} a^{2} b^{2} c \,x^{5}+a^{3} b d \,x^{4}+\frac {4}{3} a^{3} b c \,x^{3}+\frac {1}{2} a^{4} d \,x^{2}+a^{4} c x\)	\(97\)
default	\(\frac {1}{10} b^{4} d \,x^{10}+\frac {1}{9} b^{4} c \,x^{9}+\frac {1}{2} a \,b^{3} d \,x^{8}+\frac {4}{7} a \,b^{3} c \,x^{7}+a^{2} b^{2} d \,x^{6}+\frac {6}{5} a^{2} b^{2} c \,x^{5}+a^{3} b d \,x^{4}+\frac {4}{3} a^{3} b c \,x^{3}+\frac {1}{2} a^{4} d \,x^{2}+a^{4} c x\)	\(97\)
norman	\(\frac {1}{10} b^{4} d \,x^{10}+\frac {1}{9} b^{4} c \,x^{9}+\frac {1}{2} a \,b^{3} d \,x^{8}+\frac {4}{7} a \,b^{3} c \,x^{7}+a^{2} b^{2} d \,x^{6}+\frac {6}{5} a^{2} b^{2} c \,x^{5}+a^{3} b d \,x^{4}+\frac {4}{3} a^{3} b c \,x^{3}+\frac {1}{2} a^{4} d \,x^{2}+a^{4} c x\)	\(97\)
risch	\(\frac {1}{10} b^{4} d \,x^{10}+\frac {1}{9} b^{4} c \,x^{9}+\frac {1}{2} a \,b^{3} d \,x^{8}+\frac {4}{7} a \,b^{3} c \,x^{7}+a^{2} b^{2} d \,x^{6}+\frac {6}{5} a^{2} b^{2} c \,x^{5}+a^{3} b d \,x^{4}+\frac {4}{3} a^{3} b c \,x^{3}+\frac {1}{2} a^{4} d \,x^{2}+a^{4} c x\)	\(97\)
parallelrisch	\(\frac {1}{10} b^{4} d \,x^{10}+\frac {1}{9} b^{4} c \,x^{9}+\frac {1}{2} a \,b^{3} d \,x^{8}+\frac {4}{7} a \,b^{3} c \,x^{7}+a^{2} b^{2} d \,x^{6}+\frac {6}{5} a^{2} b^{2} c \,x^{5}+a^{3} b d \,x^{4}+\frac {4}{3} a^{3} b c \,x^{3}+\frac {1}{2} a^{4} d \,x^{2}+a^{4} c x\)	\(97\)

[In]

int((d*x+c)*(b*x^2+a)^4,x,method=_RETURNVERBOSE)

[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] (verification not implemented)

none

Time = 0.40 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.32 \[ \int (c+d x) \left (a+b x^2\right )^4 \, dx=\frac {1}{10} \, b^{4} d x^{10} + \frac {1}{9} \, b^{4} c x^{9} + \frac {1}{2} \, a b^{3} d x^{8} + \frac {4}{7} \, a b^{3} c x^{7} + a^{2} b^{2} d x^{6} + \frac {6}{5} \, a^{2} b^{2} c x^{5} + a^{3} b d x^{4} + \frac {4}{3} \, a^{3} b c x^{3} + \frac {1}{2} \, a^{4} d x^{2} + a^{4} c x \]

[In]

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

[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

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.53 \[ \int (c+d x) \left (a+b x^2\right )^4 \, dx=a^{4} c x + \frac {a^{4} d x^{2}}{2} + \frac {4 a^{3} b c x^{3}}{3} + a^{3} b d x^{4} + \frac {6 a^{2} b^{2} c x^{5}}{5} + a^{2} b^{2} d x^{6} + \frac {4 a b^{3} c x^{7}}{7} + \frac {a b^{3} d x^{8}}{2} + \frac {b^{4} c x^{9}}{9} + \frac {b^{4} d x^{10}}{10} \]

[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

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.32 \[ \int (c+d x) \left (a+b x^2\right )^4 \, dx=\frac {1}{10} \, b^{4} d x^{10} + \frac {1}{9} \, b^{4} c x^{9} + \frac {1}{2} \, a b^{3} d x^{8} + \frac {4}{7} \, a b^{3} c x^{7} + a^{2} b^{2} d x^{6} + \frac {6}{5} \, a^{2} b^{2} c x^{5} + a^{3} b d x^{4} + \frac {4}{3} \, a^{3} b c x^{3} + \frac {1}{2} \, a^{4} d x^{2} + a^{4} c x \]

[In]

integrate((d*x+c)*(b*x^2+a)^4,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

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.32 \[ \int (c+d x) \left (a+b x^2\right )^4 \, dx=\frac {1}{10} \, b^{4} d x^{10} + \frac {1}{9} \, b^{4} c x^{9} + \frac {1}{2} \, a b^{3} d x^{8} + \frac {4}{7} \, a b^{3} c x^{7} + a^{2} b^{2} d x^{6} + \frac {6}{5} \, a^{2} b^{2} c x^{5} + a^{3} b d x^{4} + \frac {4}{3} \, a^{3} b c x^{3} + \frac {1}{2} \, a^{4} d x^{2} + a^{4} c x \]

[In]

integrate((d*x+c)*(b*x^2+a)^4,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

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.32 \[ \int (c+d x) \left (a+b x^2\right )^4 \, dx=\frac {d\,a^4\,x^2}{2}+c\,a^4\,x+d\,a^3\,b\,x^4+\frac {4\,c\,a^3\,b\,x^3}{3}+d\,a^2\,b^2\,x^6+\frac {6\,c\,a^2\,b^2\,x^5}{5}+\frac {d\,a\,b^3\,x^8}{2}+\frac {4\,c\,a\,b^3\,x^7}{7}+\frac {d\,b^4\,x^{10}}{10}+\frac {c\,b^4\,x^9}{9} \]

[In]

int((a + b*x^2)^4*(c + d*x),x)

[Out]

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