3.4 \(\int (d+e x^2) (a+c x^4)^5 \, dx\)

Optimal. Leaf size=141 \[ a^5 d x+\frac {1}{3} a^5 e x^3+a^4 c d x^5+\frac {5}{7} a^4 c e x^7+\frac {10}{9} a^3 c^2 d x^9+\frac {10}{11} a^3 c^2 e x^{11}+\frac {10}{13} a^2 c^3 d x^{13}+\frac {2}{3} a^2 c^3 e x^{15}+\frac {5}{17} a c^4 d x^{17}+\frac {5}{19} a c^4 e x^{19}+\frac {1}{21} c^5 d x^{21}+\frac {1}{23} c^5 e x^{23} \]

[Out]

a^5*d*x+1/3*a^5*e*x^3+a^4*c*d*x^5+5/7*a^4*c*e*x^7+10/9*a^3*c^2*d*x^9+10/11*a^3*c^2*e*x^11+10/13*a^2*c^3*d*x^13
+2/3*a^2*c^3*e*x^15+5/17*a*c^4*d*x^17+5/19*a*c^4*e*x^19+1/21*c^5*d*x^21+1/23*c^5*e*x^23

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Rubi [A]  time = 0.08, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {1154} \[ \frac {10}{13} a^2 c^3 d x^{13}+\frac {10}{9} a^3 c^2 d x^9+\frac {2}{3} a^2 c^3 e x^{15}+\frac {10}{11} a^3 c^2 e x^{11}+a^4 c d x^5+\frac {5}{7} a^4 c e x^7+a^5 d x+\frac {1}{3} a^5 e x^3+\frac {5}{17} a c^4 d x^{17}+\frac {5}{19} a c^4 e x^{19}+\frac {1}{21} c^5 d x^{21}+\frac {1}{23} c^5 e x^{23} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x^2)*(a + c*x^4)^5,x]

[Out]

a^5*d*x + (a^5*e*x^3)/3 + a^4*c*d*x^5 + (5*a^4*c*e*x^7)/7 + (10*a^3*c^2*d*x^9)/9 + (10*a^3*c^2*e*x^11)/11 + (1
0*a^2*c^3*d*x^13)/13 + (2*a^2*c^3*e*x^15)/3 + (5*a*c^4*d*x^17)/17 + (5*a*c^4*e*x^19)/19 + (c^5*d*x^21)/21 + (c
^5*e*x^23)/23

Rule 1154

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

Rubi steps

\begin {align*} \int \left (d+e x^2\right ) \left (a+c x^4\right )^5 \, dx &=\int \left (a^5 d+a^5 e x^2+5 a^4 c d x^4+5 a^4 c e x^6+10 a^3 c^2 d x^8+10 a^3 c^2 e x^{10}+10 a^2 c^3 d x^{12}+10 a^2 c^3 e x^{14}+5 a c^4 d x^{16}+5 a c^4 e x^{18}+c^5 d x^{20}+c^5 e x^{22}\right ) \, dx\\ &=a^5 d x+\frac {1}{3} a^5 e x^3+a^4 c d x^5+\frac {5}{7} a^4 c e x^7+\frac {10}{9} a^3 c^2 d x^9+\frac {10}{11} a^3 c^2 e x^{11}+\frac {10}{13} a^2 c^3 d x^{13}+\frac {2}{3} a^2 c^3 e x^{15}+\frac {5}{17} a c^4 d x^{17}+\frac {5}{19} a c^4 e x^{19}+\frac {1}{21} c^5 d x^{21}+\frac {1}{23} c^5 e x^{23}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 141, normalized size = 1.00 \[ a^5 d x+\frac {1}{3} a^5 e x^3+a^4 c d x^5+\frac {5}{7} a^4 c e x^7+\frac {10}{9} a^3 c^2 d x^9+\frac {10}{11} a^3 c^2 e x^{11}+\frac {10}{13} a^2 c^3 d x^{13}+\frac {2}{3} a^2 c^3 e x^{15}+\frac {5}{17} a c^4 d x^{17}+\frac {5}{19} a c^4 e x^{19}+\frac {1}{21} c^5 d x^{21}+\frac {1}{23} c^5 e x^{23} \]

Antiderivative was successfully verified.

[In]

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

[Out]

a^5*d*x + (a^5*e*x^3)/3 + a^4*c*d*x^5 + (5*a^4*c*e*x^7)/7 + (10*a^3*c^2*d*x^9)/9 + (10*a^3*c^2*e*x^11)/11 + (1
0*a^2*c^3*d*x^13)/13 + (2*a^2*c^3*e*x^15)/3 + (5*a*c^4*d*x^17)/17 + (5*a*c^4*e*x^19)/19 + (c^5*d*x^21)/21 + (c
^5*e*x^23)/23

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fricas [A]  time = 0.62, size = 121, normalized size = 0.86 \[ \frac {1}{23} x^{23} e c^{5} + \frac {1}{21} x^{21} d c^{5} + \frac {5}{19} x^{19} e c^{4} a + \frac {5}{17} x^{17} d c^{4} a + \frac {2}{3} x^{15} e c^{3} a^{2} + \frac {10}{13} x^{13} d c^{3} a^{2} + \frac {10}{11} x^{11} e c^{2} a^{3} + \frac {10}{9} x^{9} d c^{2} a^{3} + \frac {5}{7} x^{7} e c a^{4} + x^{5} d c a^{4} + \frac {1}{3} x^{3} e a^{5} + x d a^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/23*x^23*e*c^5 + 1/21*x^21*d*c^5 + 5/19*x^19*e*c^4*a + 5/17*x^17*d*c^4*a + 2/3*x^15*e*c^3*a^2 + 10/13*x^13*d*
c^3*a^2 + 10/11*x^11*e*c^2*a^3 + 10/9*x^9*d*c^2*a^3 + 5/7*x^7*e*c*a^4 + x^5*d*c*a^4 + 1/3*x^3*e*a^5 + x*d*a^5

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giac [A]  time = 0.22, size = 127, normalized size = 0.90 \[ \frac {1}{23} \, c^{5} x^{23} e + \frac {1}{21} \, c^{5} d x^{21} + \frac {5}{19} \, a c^{4} x^{19} e + \frac {5}{17} \, a c^{4} d x^{17} + \frac {2}{3} \, a^{2} c^{3} x^{15} e + \frac {10}{13} \, a^{2} c^{3} d x^{13} + \frac {10}{11} \, a^{3} c^{2} x^{11} e + \frac {10}{9} \, a^{3} c^{2} d x^{9} + \frac {5}{7} \, a^{4} c x^{7} e + a^{4} c d x^{5} + \frac {1}{3} \, a^{5} x^{3} e + a^{5} d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/23*c^5*x^23*e + 1/21*c^5*d*x^21 + 5/19*a*c^4*x^19*e + 5/17*a*c^4*d*x^17 + 2/3*a^2*c^3*x^15*e + 10/13*a^2*c^3
*d*x^13 + 10/11*a^3*c^2*x^11*e + 10/9*a^3*c^2*d*x^9 + 5/7*a^4*c*x^7*e + a^4*c*d*x^5 + 1/3*a^5*x^3*e + a^5*d*x

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maple [A]  time = 0.00, size = 122, normalized size = 0.87 \[ \frac {1}{23} c^{5} e \,x^{23}+\frac {1}{21} c^{5} d \,x^{21}+\frac {5}{19} a \,c^{4} e \,x^{19}+\frac {5}{17} a \,c^{4} d \,x^{17}+\frac {2}{3} a^{2} c^{3} e \,x^{15}+\frac {10}{13} a^{2} c^{3} d \,x^{13}+\frac {10}{11} a^{3} c^{2} e \,x^{11}+\frac {10}{9} a^{3} c^{2} d \,x^{9}+\frac {5}{7} a^{4} c e \,x^{7}+a^{4} c d \,x^{5}+\frac {1}{3} a^{5} e \,x^{3}+a^{5} d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^5*d*x+1/3*a^5*e*x^3+a^4*c*d*x^5+5/7*a^4*c*e*x^7+10/9*a^3*c^2*d*x^9+10/11*a^3*c^2*e*x^11+10/13*a^2*c^3*d*x^13
+2/3*a^2*c^3*e*x^15+5/17*a*c^4*d*x^17+5/19*a*c^4*e*x^19+1/21*c^5*d*x^21+1/23*c^5*e*x^23

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maxima [A]  time = 0.46, size = 121, normalized size = 0.86 \[ \frac {1}{23} \, c^{5} e x^{23} + \frac {1}{21} \, c^{5} d x^{21} + \frac {5}{19} \, a c^{4} e x^{19} + \frac {5}{17} \, a c^{4} d x^{17} + \frac {2}{3} \, a^{2} c^{3} e x^{15} + \frac {10}{13} \, a^{2} c^{3} d x^{13} + \frac {10}{11} \, a^{3} c^{2} e x^{11} + \frac {10}{9} \, a^{3} c^{2} d x^{9} + \frac {5}{7} \, a^{4} c e x^{7} + a^{4} c d x^{5} + \frac {1}{3} \, a^{5} e x^{3} + a^{5} d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/23*c^5*e*x^23 + 1/21*c^5*d*x^21 + 5/19*a*c^4*e*x^19 + 5/17*a*c^4*d*x^17 + 2/3*a^2*c^3*e*x^15 + 10/13*a^2*c^3
*d*x^13 + 10/11*a^3*c^2*e*x^11 + 10/9*a^3*c^2*d*x^9 + 5/7*a^4*c*e*x^7 + a^4*c*d*x^5 + 1/3*a^5*e*x^3 + a^5*d*x

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mupad [B]  time = 0.07, size = 121, normalized size = 0.86 \[ \frac {e\,a^5\,x^3}{3}+d\,a^5\,x+\frac {5\,e\,a^4\,c\,x^7}{7}+d\,a^4\,c\,x^5+\frac {10\,e\,a^3\,c^2\,x^{11}}{11}+\frac {10\,d\,a^3\,c^2\,x^9}{9}+\frac {2\,e\,a^2\,c^3\,x^{15}}{3}+\frac {10\,d\,a^2\,c^3\,x^{13}}{13}+\frac {5\,e\,a\,c^4\,x^{19}}{19}+\frac {5\,d\,a\,c^4\,x^{17}}{17}+\frac {e\,c^5\,x^{23}}{23}+\frac {d\,c^5\,x^{21}}{21} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^4)^5*(d + e*x^2),x)

[Out]

(a^5*e*x^3)/3 + (c^5*d*x^21)/21 + (c^5*e*x^23)/23 + a^5*d*x + (10*a^3*c^2*d*x^9)/9 + (10*a^2*c^3*d*x^13)/13 +
(10*a^3*c^2*e*x^11)/11 + (2*a^2*c^3*e*x^15)/3 + a^4*c*d*x^5 + (5*a*c^4*d*x^17)/17 + (5*a^4*c*e*x^7)/7 + (5*a*c
^4*e*x^19)/19

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sympy [A]  time = 0.09, size = 148, normalized size = 1.05 \[ a^{5} d x + \frac {a^{5} e x^{3}}{3} + a^{4} c d x^{5} + \frac {5 a^{4} c e x^{7}}{7} + \frac {10 a^{3} c^{2} d x^{9}}{9} + \frac {10 a^{3} c^{2} e x^{11}}{11} + \frac {10 a^{2} c^{3} d x^{13}}{13} + \frac {2 a^{2} c^{3} e x^{15}}{3} + \frac {5 a c^{4} d x^{17}}{17} + \frac {5 a c^{4} e x^{19}}{19} + \frac {c^{5} d x^{21}}{21} + \frac {c^{5} e x^{23}}{23} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**5*d*x + a**5*e*x**3/3 + a**4*c*d*x**5 + 5*a**4*c*e*x**7/7 + 10*a**3*c**2*d*x**9/9 + 10*a**3*c**2*e*x**11/11
+ 10*a**2*c**3*d*x**13/13 + 2*a**2*c**3*e*x**15/3 + 5*a*c**4*d*x**17/17 + 5*a*c**4*e*x**19/19 + c**5*d*x**21/
21 + c**5*e*x**23/23

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