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

Optimal. Leaf size=139 \[ -\frac {a^5 d}{x}+a^5 e x+\frac {5}{3} a^4 c d x^3+a^4 c e x^5+\frac {10}{7} a^3 c^2 d x^7+\frac {10}{9} a^3 c^2 e x^9+\frac {10}{11} a^2 c^3 d x^{11}+\frac {10}{13} a^2 c^3 e x^{13}+\frac {1}{3} a c^4 d x^{15}+\frac {5}{17} a c^4 e x^{17}+\frac {1}{19} c^5 d x^{19}+\frac {1}{21} c^5 e x^{21} \]

[Out]

-a^5*d/x+a^5*e*x+5/3*a^4*c*d*x^3+a^4*c*e*x^5+10/7*a^3*c^2*d*x^7+10/9*a^3*c^2*e*x^9+10/11*a^2*c^3*d*x^11+10/13*
a^2*c^3*e*x^13+1/3*a*c^4*d*x^15+5/17*a*c^4*e*x^17+1/19*c^5*d*x^19+1/21*c^5*e*x^21

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Rubi [A]  time = 0.08, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {1262} \[ \frac {10}{11} a^2 c^3 d x^{11}+\frac {10}{7} a^3 c^2 d x^7+\frac {10}{13} a^2 c^3 e x^{13}+\frac {10}{9} a^3 c^2 e x^9+\frac {5}{3} a^4 c d x^3+a^4 c e x^5-\frac {a^5 d}{x}+a^5 e x+\frac {1}{3} a c^4 d x^{15}+\frac {5}{17} a c^4 e x^{17}+\frac {1}{19} c^5 d x^{19}+\frac {1}{21} c^5 e x^{21} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a^5*d)/x) + a^5*e*x + (5*a^4*c*d*x^3)/3 + a^4*c*e*x^5 + (10*a^3*c^2*d*x^7)/7 + (10*a^3*c^2*e*x^9)/9 + (10*a
^2*c^3*d*x^11)/11 + (10*a^2*c^3*e*x^13)/13 + (a*c^4*d*x^15)/3 + (5*a*c^4*e*x^17)/17 + (c^5*d*x^19)/19 + (c^5*e
*x^21)/21

Rule 1262

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(f*x)^m*(d + e*x^2)^q*(a + c*x^4)^p, x], x] /; FreeQ[{a, c, d, e, f, m, q}, x] && IGtQ[p, 0] && IGtQ[q, -2]

Rubi steps

\begin {align*} \int \frac {\left (d+e x^2\right ) \left (a+c x^4\right )^5}{x^2} \, dx &=\int \left (a^5 e+\frac {a^5 d}{x^2}+5 a^4 c d x^2+5 a^4 c e x^4+10 a^3 c^2 d x^6+10 a^3 c^2 e x^8+10 a^2 c^3 d x^{10}+10 a^2 c^3 e x^{12}+5 a c^4 d x^{14}+5 a c^4 e x^{16}+c^5 d x^{18}+c^5 e x^{20}\right ) \, dx\\ &=-\frac {a^5 d}{x}+a^5 e x+\frac {5}{3} a^4 c d x^3+a^4 c e x^5+\frac {10}{7} a^3 c^2 d x^7+\frac {10}{9} a^3 c^2 e x^9+\frac {10}{11} a^2 c^3 d x^{11}+\frac {10}{13} a^2 c^3 e x^{13}+\frac {1}{3} a c^4 d x^{15}+\frac {5}{17} a c^4 e x^{17}+\frac {1}{19} c^5 d x^{19}+\frac {1}{21} c^5 e x^{21}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 139, normalized size = 1.00 \[ -\frac {a^5 d}{x}+a^5 e x+\frac {5}{3} a^4 c d x^3+a^4 c e x^5+\frac {10}{7} a^3 c^2 d x^7+\frac {10}{9} a^3 c^2 e x^9+\frac {10}{11} a^2 c^3 d x^{11}+\frac {10}{13} a^2 c^3 e x^{13}+\frac {1}{3} a c^4 d x^{15}+\frac {5}{17} a c^4 e x^{17}+\frac {1}{19} c^5 d x^{19}+\frac {1}{21} c^5 e x^{21} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a^5*d)/x) + a^5*e*x + (5*a^4*c*d*x^3)/3 + a^4*c*e*x^5 + (10*a^3*c^2*d*x^7)/7 + (10*a^3*c^2*e*x^9)/9 + (10*a
^2*c^3*d*x^11)/11 + (10*a^2*c^3*e*x^13)/13 + (a*c^4*d*x^15)/3 + (5*a*c^4*e*x^17)/17 + (c^5*d*x^19)/19 + (c^5*e
*x^21)/21

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fricas [A]  time = 0.67, size = 127, normalized size = 0.91 \[ \frac {138567 \, c^{5} e x^{22} + 153153 \, c^{5} d x^{20} + 855855 \, a c^{4} e x^{18} + 969969 \, a c^{4} d x^{16} + 2238390 \, a^{2} c^{3} e x^{14} + 2645370 \, a^{2} c^{3} d x^{12} + 3233230 \, a^{3} c^{2} e x^{10} + 4157010 \, a^{3} c^{2} d x^{8} + 2909907 \, a^{4} c e x^{6} + 4849845 \, a^{4} c d x^{4} + 2909907 \, a^{5} e x^{2} - 2909907 \, a^{5} d}{2909907 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2909907*(138567*c^5*e*x^22 + 153153*c^5*d*x^20 + 855855*a*c^4*e*x^18 + 969969*a*c^4*d*x^16 + 2238390*a^2*c^3
*e*x^14 + 2645370*a^2*c^3*d*x^12 + 3233230*a^3*c^2*e*x^10 + 4157010*a^3*c^2*d*x^8 + 2909907*a^4*c*e*x^6 + 4849
845*a^4*c*d*x^4 + 2909907*a^5*e*x^2 - 2909907*a^5*d)/x

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giac [A]  time = 0.20, size = 127, normalized size = 0.91 \[ \frac {1}{21} \, c^{5} x^{21} e + \frac {1}{19} \, c^{5} d x^{19} + \frac {5}{17} \, a c^{4} x^{17} e + \frac {1}{3} \, a c^{4} d x^{15} + \frac {10}{13} \, a^{2} c^{3} x^{13} e + \frac {10}{11} \, a^{2} c^{3} d x^{11} + \frac {10}{9} \, a^{3} c^{2} x^{9} e + \frac {10}{7} \, a^{3} c^{2} d x^{7} + a^{4} c x^{5} e + \frac {5}{3} \, a^{4} c d x^{3} + a^{5} x e - \frac {a^{5} d}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/21*c^5*x^21*e + 1/19*c^5*d*x^19 + 5/17*a*c^4*x^17*e + 1/3*a*c^4*d*x^15 + 10/13*a^2*c^3*x^13*e + 10/11*a^2*c^
3*d*x^11 + 10/9*a^3*c^2*x^9*e + 10/7*a^3*c^2*d*x^7 + a^4*c*x^5*e + 5/3*a^4*c*d*x^3 + a^5*x*e - a^5*d/x

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maple [A]  time = 0.02, size = 122, normalized size = 0.88 \[ \frac {c^{5} e \,x^{21}}{21}+\frac {c^{5} d \,x^{19}}{19}+\frac {5 a \,c^{4} e \,x^{17}}{17}+\frac {a \,c^{4} d \,x^{15}}{3}+\frac {10 a^{2} c^{3} e \,x^{13}}{13}+\frac {10 a^{2} c^{3} d \,x^{11}}{11}+\frac {10 a^{3} c^{2} e \,x^{9}}{9}+\frac {10 a^{3} c^{2} d \,x^{7}}{7}+a^{4} c e \,x^{5}+\frac {5 a^{4} c d \,x^{3}}{3}+a^{5} e x -\frac {a^{5} d}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a^5*d/x+a^5*e*x+5/3*a^4*c*d*x^3+a^4*c*e*x^5+10/7*a^3*c^2*d*x^7+10/9*a^3*c^2*e*x^9+10/11*a^2*c^3*d*x^11+10/13*
a^2*c^3*e*x^13+1/3*a*c^4*d*x^15+5/17*a*c^4*e*x^17+1/19*c^5*d*x^19+1/21*c^5*e*x^21

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maxima [A]  time = 0.52, size = 121, normalized size = 0.87 \[ \frac {1}{21} \, c^{5} e x^{21} + \frac {1}{19} \, c^{5} d x^{19} + \frac {5}{17} \, a c^{4} e x^{17} + \frac {1}{3} \, a c^{4} d x^{15} + \frac {10}{13} \, a^{2} c^{3} e x^{13} + \frac {10}{11} \, a^{2} c^{3} d x^{11} + \frac {10}{9} \, a^{3} c^{2} e x^{9} + \frac {10}{7} \, a^{3} c^{2} d x^{7} + a^{4} c e x^{5} + \frac {5}{3} \, a^{4} c d x^{3} + a^{5} e x - \frac {a^{5} d}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/21*c^5*e*x^21 + 1/19*c^5*d*x^19 + 5/17*a*c^4*e*x^17 + 1/3*a*c^4*d*x^15 + 10/13*a^2*c^3*e*x^13 + 10/11*a^2*c^
3*d*x^11 + 10/9*a^3*c^2*e*x^9 + 10/7*a^3*c^2*d*x^7 + a^4*c*e*x^5 + 5/3*a^4*c*d*x^3 + a^5*e*x - a^5*d/x

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mupad [B]  time = 0.09, size = 121, normalized size = 0.87 \[ \frac {c^5\,d\,x^{19}}{19}-\frac {a^5\,d}{x}+\frac {c^5\,e\,x^{21}}{21}+a^5\,e\,x+\frac {10\,a^3\,c^2\,d\,x^7}{7}+\frac {10\,a^2\,c^3\,d\,x^{11}}{11}+\frac {10\,a^3\,c^2\,e\,x^9}{9}+\frac {10\,a^2\,c^3\,e\,x^{13}}{13}+\frac {5\,a^4\,c\,d\,x^3}{3}+\frac {a\,c^4\,d\,x^{15}}{3}+a^4\,c\,e\,x^5+\frac {5\,a\,c^4\,e\,x^{17}}{17} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c^5*d*x^19)/19 - (a^5*d)/x + (c^5*e*x^21)/21 + a^5*e*x + (10*a^3*c^2*d*x^7)/7 + (10*a^2*c^3*d*x^11)/11 + (10*
a^3*c^2*e*x^9)/9 + (10*a^2*c^3*e*x^13)/13 + (5*a^4*c*d*x^3)/3 + (a*c^4*d*x^15)/3 + a^4*c*e*x^5 + (5*a*c^4*e*x^
17)/17

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sympy [A]  time = 0.24, size = 143, normalized size = 1.03 \[ - \frac {a^{5} d}{x} + a^{5} e x + \frac {5 a^{4} c d x^{3}}{3} + a^{4} c e x^{5} + \frac {10 a^{3} c^{2} d x^{7}}{7} + \frac {10 a^{3} c^{2} e x^{9}}{9} + \frac {10 a^{2} c^{3} d x^{11}}{11} + \frac {10 a^{2} c^{3} e x^{13}}{13} + \frac {a c^{4} d x^{15}}{3} + \frac {5 a c^{4} e x^{17}}{17} + \frac {c^{5} d x^{19}}{19} + \frac {c^{5} e x^{21}}{21} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a**5*d/x + a**5*e*x + 5*a**4*c*d*x**3/3 + a**4*c*e*x**5 + 10*a**3*c**2*d*x**7/7 + 10*a**3*c**2*e*x**9/9 + 10*
a**2*c**3*d*x**11/11 + 10*a**2*c**3*e*x**13/13 + a*c**4*d*x**15/3 + 5*a*c**4*e*x**17/17 + c**5*d*x**19/19 + c*
*5*e*x**21/21

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