3.7 \(\int \frac {(d+e x^2) (a+c x^4)^5}{x^3} \, dx\)

Optimal. Leaf size=142 \[ -\frac {a^5 d}{2 x^2}+a^5 e \log (x)+\frac {5}{2} a^4 c d x^2+\frac {5}{4} a^4 c e x^4+\frac {5}{3} a^3 c^2 d x^6+\frac {5}{4} a^3 c^2 e x^8+a^2 c^3 d x^{10}+\frac {5}{6} a^2 c^3 e x^{12}+\frac {5}{14} a c^4 d x^{14}+\frac {5}{16} a c^4 e x^{16}+\frac {1}{18} c^5 d x^{18}+\frac {1}{20} c^5 e x^{20} \]

[Out]

-1/2*a^5*d/x^2+5/2*a^4*c*d*x^2+5/4*a^4*c*e*x^4+5/3*a^3*c^2*d*x^6+5/4*a^3*c^2*e*x^8+a^2*c^3*d*x^10+5/6*a^2*c^3*
e*x^12+5/14*a*c^4*d*x^14+5/16*a*c^4*e*x^16+1/18*c^5*d*x^18+1/20*c^5*e*x^20+a^5*e*ln(x)

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Rubi [A]  time = 0.12, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1252, 766} \[ a^2 c^3 d x^{10}+\frac {5}{3} a^3 c^2 d x^6+\frac {5}{6} a^2 c^3 e x^{12}+\frac {5}{4} a^3 c^2 e x^8+\frac {5}{2} a^4 c d x^2+\frac {5}{4} a^4 c e x^4-\frac {a^5 d}{2 x^2}+a^5 e \log (x)+\frac {5}{14} a c^4 d x^{14}+\frac {5}{16} a c^4 e x^{16}+\frac {1}{18} c^5 d x^{18}+\frac {1}{20} c^5 e x^{20} \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x^2)*(a + c*x^4)^5)/x^3,x]

[Out]

-(a^5*d)/(2*x^2) + (5*a^4*c*d*x^2)/2 + (5*a^4*c*e*x^4)/4 + (5*a^3*c^2*d*x^6)/3 + (5*a^3*c^2*e*x^8)/4 + a^2*c^3
*d*x^10 + (5*a^2*c^3*e*x^12)/6 + (5*a*c^4*d*x^14)/14 + (5*a*c^4*e*x^16)/16 + (c^5*d*x^18)/18 + (c^5*e*x^20)/20
+ a^5*e*Log[x]

Rule 766

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(e*x
)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, e, f, g, m}, x] && IGtQ[p, 0]

Rule 1252

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m
- 1)/2)*(d + e*x)^q*(a + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, c, d, e, p, q}, x] && IntegerQ[(m + 1)/2]

Rubi steps

\begin {align*} \int \frac {\left (d+e x^2\right ) \left (a+c x^4\right )^5}{x^3} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(d+e x) \left (a+c x^2\right )^5}{x^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (5 a^4 c d+\frac {a^5 d}{x^2}+\frac {a^5 e}{x}+5 a^4 c e x+10 a^3 c^2 d x^2+10 a^3 c^2 e x^3+10 a^2 c^3 d x^4+10 a^2 c^3 e x^5+5 a c^4 d x^6+5 a c^4 e x^7+c^5 d x^8+c^5 e x^9\right ) \, dx,x,x^2\right )\\ &=-\frac {a^5 d}{2 x^2}+\frac {5}{2} a^4 c d x^2+\frac {5}{4} a^4 c e x^4+\frac {5}{3} a^3 c^2 d x^6+\frac {5}{4} a^3 c^2 e x^8+a^2 c^3 d x^{10}+\frac {5}{6} a^2 c^3 e x^{12}+\frac {5}{14} a c^4 d x^{14}+\frac {5}{16} a c^4 e x^{16}+\frac {1}{18} c^5 d x^{18}+\frac {1}{20} c^5 e x^{20}+a^5 e \log (x)\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 142, normalized size = 1.00 \[ -\frac {a^5 d}{2 x^2}+a^5 e \log (x)+\frac {5}{2} a^4 c d x^2+\frac {5}{4} a^4 c e x^4+\frac {5}{3} a^3 c^2 d x^6+\frac {5}{4} a^3 c^2 e x^8+a^2 c^3 d x^{10}+\frac {5}{6} a^2 c^3 e x^{12}+\frac {5}{14} a c^4 d x^{14}+\frac {5}{16} a c^4 e x^{16}+\frac {1}{18} c^5 d x^{18}+\frac {1}{20} c^5 e x^{20} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x^2)*(a + c*x^4)^5)/x^3,x]

[Out]

-1/2*(a^5*d)/x^2 + (5*a^4*c*d*x^2)/2 + (5*a^4*c*e*x^4)/4 + (5*a^3*c^2*d*x^6)/3 + (5*a^3*c^2*e*x^8)/4 + a^2*c^3
*d*x^10 + (5*a^2*c^3*e*x^12)/6 + (5*a*c^4*d*x^14)/14 + (5*a*c^4*e*x^16)/16 + (c^5*d*x^18)/18 + (c^5*e*x^20)/20
+ a^5*e*Log[x]

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fricas [A]  time = 0.69, size = 129, normalized size = 0.91 \[ \frac {252 \, c^{5} e x^{22} + 280 \, c^{5} d x^{20} + 1575 \, a c^{4} e x^{18} + 1800 \, a c^{4} d x^{16} + 4200 \, a^{2} c^{3} e x^{14} + 5040 \, a^{2} c^{3} d x^{12} + 6300 \, a^{3} c^{2} e x^{10} + 8400 \, a^{3} c^{2} d x^{8} + 6300 \, a^{4} c e x^{6} + 12600 \, a^{4} c d x^{4} + 5040 \, a^{5} e x^{2} \log \relax (x) - 2520 \, a^{5} d}{5040 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)*(c*x^4+a)^5/x^3,x, algorithm="fricas")

[Out]

1/5040*(252*c^5*e*x^22 + 280*c^5*d*x^20 + 1575*a*c^4*e*x^18 + 1800*a*c^4*d*x^16 + 4200*a^2*c^3*e*x^14 + 5040*a
^2*c^3*d*x^12 + 6300*a^3*c^2*e*x^10 + 8400*a^3*c^2*d*x^8 + 6300*a^4*c*e*x^6 + 12600*a^4*c*d*x^4 + 5040*a^5*e*x
^2*log(x) - 2520*a^5*d)/x^2

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giac [A]  time = 0.21, size = 142, normalized size = 1.00 \[ \frac {1}{20} \, c^{5} x^{20} e + \frac {1}{18} \, c^{5} d x^{18} + \frac {5}{16} \, a c^{4} x^{16} e + \frac {5}{14} \, a c^{4} d x^{14} + \frac {5}{6} \, a^{2} c^{3} x^{12} e + a^{2} c^{3} d x^{10} + \frac {5}{4} \, a^{3} c^{2} x^{8} e + \frac {5}{3} \, a^{3} c^{2} d x^{6} + \frac {5}{4} \, a^{4} c x^{4} e + \frac {5}{2} \, a^{4} c d x^{2} + \frac {1}{2} \, a^{5} e \log \left (x^{2}\right ) - \frac {a^{5} x^{2} e + a^{5} d}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)*(c*x^4+a)^5/x^3,x, algorithm="giac")

[Out]

1/20*c^5*x^20*e + 1/18*c^5*d*x^18 + 5/16*a*c^4*x^16*e + 5/14*a*c^4*d*x^14 + 5/6*a^2*c^3*x^12*e + a^2*c^3*d*x^1
0 + 5/4*a^3*c^2*x^8*e + 5/3*a^3*c^2*d*x^6 + 5/4*a^4*c*x^4*e + 5/2*a^4*c*d*x^2 + 1/2*a^5*e*log(x^2) - 1/2*(a^5*
x^2*e + a^5*d)/x^2

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maple [A]  time = 0.01, size = 123, normalized size = 0.87 \[ \frac {c^{5} e \,x^{20}}{20}+\frac {c^{5} d \,x^{18}}{18}+\frac {5 a \,c^{4} e \,x^{16}}{16}+\frac {5 a \,c^{4} d \,x^{14}}{14}+\frac {5 a^{2} c^{3} e \,x^{12}}{6}+a^{2} c^{3} d \,x^{10}+\frac {5 a^{3} c^{2} e \,x^{8}}{4}+\frac {5 a^{3} c^{2} d \,x^{6}}{3}+\frac {5 a^{4} c e \,x^{4}}{4}+\frac {5 a^{4} c d \,x^{2}}{2}+a^{5} e \ln \relax (x )-\frac {a^{5} d}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d)*(c*x^4+a)^5/x^3,x)

[Out]

-1/2*a^5*d/x^2+5/2*a^4*c*d*x^2+5/4*a^4*c*e*x^4+5/3*a^3*c^2*d*x^6+5/4*a^3*c^2*e*x^8+a^2*c^3*d*x^10+5/6*a^2*c^3*
e*x^12+5/14*a*c^4*d*x^14+5/16*a*c^4*e*x^16+1/18*c^5*d*x^18+1/20*c^5*e*x^20+a^5*e*ln(x)

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maxima [A]  time = 0.50, size = 125, normalized size = 0.88 \[ \frac {1}{20} \, c^{5} e x^{20} + \frac {1}{18} \, c^{5} d x^{18} + \frac {5}{16} \, a c^{4} e x^{16} + \frac {5}{14} \, a c^{4} d x^{14} + \frac {5}{6} \, a^{2} c^{3} e x^{12} + a^{2} c^{3} d x^{10} + \frac {5}{4} \, a^{3} c^{2} e x^{8} + \frac {5}{3} \, a^{3} c^{2} d x^{6} + \frac {5}{4} \, a^{4} c e x^{4} + \frac {5}{2} \, a^{4} c d x^{2} + \frac {1}{2} \, a^{5} e \log \left (x^{2}\right ) - \frac {a^{5} d}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)*(c*x^4+a)^5/x^3,x, algorithm="maxima")

[Out]

1/20*c^5*e*x^20 + 1/18*c^5*d*x^18 + 5/16*a*c^4*e*x^16 + 5/14*a*c^4*d*x^14 + 5/6*a^2*c^3*e*x^12 + a^2*c^3*d*x^1
0 + 5/4*a^3*c^2*e*x^8 + 5/3*a^3*c^2*d*x^6 + 5/4*a^4*c*e*x^4 + 5/2*a^4*c*d*x^2 + 1/2*a^5*e*log(x^2) - 1/2*a^5*d
/x^2

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mupad [B]  time = 0.07, size = 122, normalized size = 0.86 \[ \frac {c^5\,d\,x^{18}}{18}-\frac {a^5\,d}{2\,x^2}+\frac {c^5\,e\,x^{20}}{20}+a^5\,e\,\ln \relax (x)+\frac {5\,a^3\,c^2\,d\,x^6}{3}+a^2\,c^3\,d\,x^{10}+\frac {5\,a^3\,c^2\,e\,x^8}{4}+\frac {5\,a^2\,c^3\,e\,x^{12}}{6}+\frac {5\,a^4\,c\,d\,x^2}{2}+\frac {5\,a\,c^4\,d\,x^{14}}{14}+\frac {5\,a^4\,c\,e\,x^4}{4}+\frac {5\,a\,c^4\,e\,x^{16}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + c*x^4)^5*(d + e*x^2))/x^3,x)

[Out]

(c^5*d*x^18)/18 - (a^5*d)/(2*x^2) + (c^5*e*x^20)/20 + a^5*e*log(x) + (5*a^3*c^2*d*x^6)/3 + a^2*c^3*d*x^10 + (5
*a^3*c^2*e*x^8)/4 + (5*a^2*c^3*e*x^12)/6 + (5*a^4*c*d*x^2)/2 + (5*a*c^4*d*x^14)/14 + (5*a^4*c*e*x^4)/4 + (5*a*
c^4*e*x^16)/16

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sympy [A]  time = 0.28, size = 150, normalized size = 1.06 \[ - \frac {a^{5} d}{2 x^{2}} + a^{5} e \log {\relax (x )} + \frac {5 a^{4} c d x^{2}}{2} + \frac {5 a^{4} c e x^{4}}{4} + \frac {5 a^{3} c^{2} d x^{6}}{3} + \frac {5 a^{3} c^{2} e x^{8}}{4} + a^{2} c^{3} d x^{10} + \frac {5 a^{2} c^{3} e x^{12}}{6} + \frac {5 a c^{4} d x^{14}}{14} + \frac {5 a c^{4} e x^{16}}{16} + \frac {c^{5} d x^{18}}{18} + \frac {c^{5} e x^{20}}{20} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x**2+d)*(c*x**4+a)**5/x**3,x)

[Out]

-a**5*d/(2*x**2) + a**5*e*log(x) + 5*a**4*c*d*x**2/2 + 5*a**4*c*e*x**4/4 + 5*a**3*c**2*d*x**6/3 + 5*a**3*c**2*
e*x**8/4 + a**2*c**3*d*x**10 + 5*a**2*c**3*e*x**12/6 + 5*a*c**4*d*x**14/14 + 5*a*c**4*e*x**16/16 + c**5*d*x**1
8/18 + c**5*e*x**20/20

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