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

Optimal. Leaf size=142 \[ \frac {1}{2} a^5 e x^2+\frac {5}{4} a^4 c d x^4+\frac {5}{6} a^4 c e x^6+\frac {5}{4} a^3 c^2 d x^8+a^3 c^2 e x^{10}+\frac {5}{6} a^2 c^3 d x^{12}+\frac {5}{7} a^2 c^3 e x^{14}+\frac {5}{16} a c^4 d x^{16}+\frac {5}{18} a c^4 e x^{18}+\frac {1}{20} c^5 d x^{20}+\frac {1}{22} c^5 e x^{22}+a^5 d \log (x) \]

[Out]

1/2*a^5*e*x^2+5/4*a^4*c*d*x^4+5/6*a^4*c*e*x^6+5/4*a^3*c^2*d*x^8+a^3*c^2*e*x^10+5/6*a^2*c^3*d*x^12+5/7*a^2*c^3*
e*x^14+5/16*a*c^4*d*x^16+5/18*a*c^4*e*x^18+1/20*c^5*d*x^20+1/22*c^5*e*x^22+a^5*d*ln(x)

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Rubi [A]
time = 0.07, 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 = {1266, 780} \begin {gather*} a^5 d \log (x)+\frac {1}{2} a^5 e x^2+\frac {5}{4} a^4 c d x^4+\frac {5}{6} a^4 c e x^6+\frac {5}{4} a^3 c^2 d x^8+a^3 c^2 e x^{10}+\frac {5}{6} a^2 c^3 d x^{12}+\frac {5}{7} a^2 c^3 e x^{14}+\frac {5}{16} a c^4 d x^{16}+\frac {5}{18} a c^4 e x^{18}+\frac {1}{20} c^5 d x^{20}+\frac {1}{22} c^5 e x^{22} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((d + e*x^2)*(a + c*x^4)^5)/x,x]

[Out]

(a^5*e*x^2)/2 + (5*a^4*c*d*x^4)/4 + (5*a^4*c*e*x^6)/6 + (5*a^3*c^2*d*x^8)/4 + a^3*c^2*e*x^10 + (5*a^2*c^3*d*x^
12)/6 + (5*a^2*c^3*e*x^14)/7 + (5*a*c^4*d*x^16)/16 + (5*a*c^4*e*x^18)/18 + (c^5*d*x^20)/20 + (c^5*e*x^22)/22 +
a^5*d*Log[x]

Rule 780

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 1266

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} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {(d+e x) \left (a+c x^2\right )^5}{x} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (a^5 e+\frac {a^5 d}{x}+5 a^4 c d x+5 a^4 c e x^2+10 a^3 c^2 d x^3+10 a^3 c^2 e x^4+10 a^2 c^3 d x^5+10 a^2 c^3 e x^6+5 a c^4 d x^7+5 a c^4 e x^8+c^5 d x^9+c^5 e x^{10}\right ) \, dx,x,x^2\right )\\ &=\frac {1}{2} a^5 e x^2+\frac {5}{4} a^4 c d x^4+\frac {5}{6} a^4 c e x^6+\frac {5}{4} a^3 c^2 d x^8+a^3 c^2 e x^{10}+\frac {5}{6} a^2 c^3 d x^{12}+\frac {5}{7} a^2 c^3 e x^{14}+\frac {5}{16} a c^4 d x^{16}+\frac {5}{18} a c^4 e x^{18}+\frac {1}{20} c^5 d x^{20}+\frac {1}{22} c^5 e x^{22}+a^5 d \log (x)\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 142, normalized size = 1.00 \begin {gather*} \frac {1}{2} a^5 e x^2+\frac {5}{4} a^4 c d x^4+\frac {5}{6} a^4 c e x^6+\frac {5}{4} a^3 c^2 d x^8+a^3 c^2 e x^{10}+\frac {5}{6} a^2 c^3 d x^{12}+\frac {5}{7} a^2 c^3 e x^{14}+\frac {5}{16} a c^4 d x^{16}+\frac {5}{18} a c^4 e x^{18}+\frac {1}{20} c^5 d x^{20}+\frac {1}{22} c^5 e x^{22}+a^5 d \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a^5*e*x^2)/2 + (5*a^4*c*d*x^4)/4 + (5*a^4*c*e*x^6)/6 + (5*a^3*c^2*d*x^8)/4 + a^3*c^2*e*x^10 + (5*a^2*c^3*d*x^
12)/6 + (5*a^2*c^3*e*x^14)/7 + (5*a*c^4*d*x^16)/16 + (5*a*c^4*e*x^18)/18 + (c^5*d*x^20)/20 + (c^5*e*x^22)/22 +
a^5*d*Log[x]

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Maple [A]
time = 0.11, size = 123, normalized size = 0.87

method	result	size
default	\(\frac {a^{5} e \,x^{2}}{2}+\frac {5 a^{4} c d \,x^{4}}{4}+\frac {5 a^{4} c e \,x^{6}}{6}+\frac {5 a^{3} c^{2} d \,x^{8}}{4}+a^{3} c^{2} e \,x^{10}+\frac {5 a^{2} c^{3} d \,x^{12}}{6}+\frac {5 a^{2} c^{3} e \,x^{14}}{7}+\frac {5 a \,c^{4} d \,x^{16}}{16}+\frac {5 a \,c^{4} e \,x^{18}}{18}+\frac {c^{5} d \,x^{20}}{20}+\frac {c^{5} e \,x^{22}}{22}+a^{5} d \ln \left (x \right )\)	\(123\)
norman	\(\frac {a^{5} e \,x^{2}}{2}+\frac {5 a^{4} c d \,x^{4}}{4}+\frac {5 a^{4} c e \,x^{6}}{6}+\frac {5 a^{3} c^{2} d \,x^{8}}{4}+a^{3} c^{2} e \,x^{10}+\frac {5 a^{2} c^{3} d \,x^{12}}{6}+\frac {5 a^{2} c^{3} e \,x^{14}}{7}+\frac {5 a \,c^{4} d \,x^{16}}{16}+\frac {5 a \,c^{4} e \,x^{18}}{18}+\frac {c^{5} d \,x^{20}}{20}+\frac {c^{5} e \,x^{22}}{22}+a^{5} d \ln \left (x \right )\)	\(123\)
risch	\(\frac {a^{5} e \,x^{2}}{2}+\frac {5 a^{4} c d \,x^{4}}{4}+\frac {5 a^{4} c e \,x^{6}}{6}+\frac {5 a^{3} c^{2} d \,x^{8}}{4}+a^{3} c^{2} e \,x^{10}+\frac {5 a^{2} c^{3} d \,x^{12}}{6}+\frac {5 a^{2} c^{3} e \,x^{14}}{7}+\frac {5 a \,c^{4} d \,x^{16}}{16}+\frac {5 a \,c^{4} e \,x^{18}}{18}+\frac {c^{5} d \,x^{20}}{20}+\frac {c^{5} e \,x^{22}}{22}+a^{5} d \ln \left (x \right )\)	\(123\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a^5*e*x^2+5/4*a^4*c*d*x^4+5/6*a^4*c*e*x^6+5/4*a^3*c^2*d*x^8+a^3*c^2*e*x^10+5/6*a^2*c^3*d*x^12+5/7*a^2*c^3*
e*x^14+5/16*a*c^4*d*x^16+5/18*a*c^4*e*x^18+1/20*c^5*d*x^20+1/22*c^5*e*x^22+a^5*d*ln(x)

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Maxima [A]
time = 0.27, size = 131, normalized size = 0.92 \begin {gather*} \frac {1}{22} \, c^{5} x^{22} e + \frac {1}{20} \, c^{5} d x^{20} + \frac {5}{18} \, a c^{4} x^{18} e + \frac {5}{16} \, a c^{4} d x^{16} + \frac {5}{7} \, a^{2} c^{3} x^{14} e + \frac {5}{6} \, a^{2} c^{3} d x^{12} + a^{3} c^{2} x^{10} e + \frac {5}{4} \, a^{3} c^{2} d x^{8} + \frac {5}{6} \, a^{4} c x^{6} e + \frac {5}{4} \, a^{4} c d x^{4} + \frac {1}{2} \, a^{5} x^{2} e + \frac {1}{2} \, a^{5} d \log \left (x^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/22*c^5*x^22*e + 1/20*c^5*d*x^20 + 5/18*a*c^4*x^18*e + 5/16*a*c^4*d*x^16 + 5/7*a^2*c^3*x^14*e + 5/6*a^2*c^3*d
*x^12 + a^3*c^2*x^10*e + 5/4*a^3*c^2*d*x^8 + 5/6*a^4*c*x^6*e + 5/4*a^4*c*d*x^4 + 1/2*a^5*x^2*e + 1/2*a^5*d*log
(x^2)

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Fricas [A]
time = 0.38, size = 122, normalized size = 0.86 \begin {gather*} \frac {1}{20} \, c^{5} d x^{20} + \frac {5}{16} \, a c^{4} d x^{16} + \frac {5}{6} \, a^{2} c^{3} d x^{12} + \frac {5}{4} \, a^{3} c^{2} d x^{8} + \frac {5}{4} \, a^{4} c d x^{4} + a^{5} d \log \left (x\right ) + \frac {1}{1386} \, {\left (63 \, c^{5} x^{22} + 385 \, a c^{4} x^{18} + 990 \, a^{2} c^{3} x^{14} + 1386 \, a^{3} c^{2} x^{10} + 1155 \, a^{4} c x^{6} + 693 \, a^{5} x^{2}\right )} e \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/20*c^5*d*x^20 + 5/16*a*c^4*d*x^16 + 5/6*a^2*c^3*d*x^12 + 5/4*a^3*c^2*d*x^8 + 5/4*a^4*c*d*x^4 + a^5*d*log(x)
+ 1/1386*(63*c^5*x^22 + 385*a*c^4*x^18 + 990*a^2*c^3*x^14 + 1386*a^3*c^2*x^10 + 1155*a^4*c*x^6 + 693*a^5*x^2)*
e

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Sympy [A]
time = 0.08, size = 150, normalized size = 1.06 \begin {gather*} a^{5} d \log {\left (x \right )} + \frac {a^{5} e x^{2}}{2} + \frac {5 a^{4} c d x^{4}}{4} + \frac {5 a^{4} c e x^{6}}{6} + \frac {5 a^{3} c^{2} d x^{8}}{4} + a^{3} c^{2} e x^{10} + \frac {5 a^{2} c^{3} d x^{12}}{6} + \frac {5 a^{2} c^{3} e x^{14}}{7} + \frac {5 a c^{4} d x^{16}}{16} + \frac {5 a c^{4} e x^{18}}{18} + \frac {c^{5} d x^{20}}{20} + \frac {c^{5} e x^{22}}{22} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**5*d*log(x) + a**5*e*x**2/2 + 5*a**4*c*d*x**4/4 + 5*a**4*c*e*x**6/6 + 5*a**3*c**2*d*x**8/4 + a**3*c**2*e*x**
10 + 5*a**2*c**3*d*x**12/6 + 5*a**2*c**3*e*x**14/7 + 5*a*c**4*d*x**16/16 + 5*a*c**4*e*x**18/18 + c**5*d*x**20/
20 + c**5*e*x**22/22

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Giac [A]
time = 4.15, size = 131, normalized size = 0.92 \begin {gather*} \frac {1}{22} \, c^{5} x^{22} e + \frac {1}{20} \, c^{5} d x^{20} + \frac {5}{18} \, a c^{4} x^{18} e + \frac {5}{16} \, a c^{4} d x^{16} + \frac {5}{7} \, a^{2} c^{3} x^{14} e + \frac {5}{6} \, a^{2} c^{3} d x^{12} + a^{3} c^{2} x^{10} e + \frac {5}{4} \, a^{3} c^{2} d x^{8} + \frac {5}{6} \, a^{4} c x^{6} e + \frac {5}{4} \, a^{4} c d x^{4} + \frac {1}{2} \, a^{5} x^{2} e + \frac {1}{2} \, a^{5} d \log \left (x^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/22*c^5*x^22*e + 1/20*c^5*d*x^20 + 5/18*a*c^4*x^18*e + 5/16*a*c^4*d*x^16 + 5/7*a^2*c^3*x^14*e + 5/6*a^2*c^3*d
*x^12 + a^3*c^2*x^10*e + 5/4*a^3*c^2*d*x^8 + 5/6*a^4*c*x^6*e + 5/4*a^4*c*d*x^4 + 1/2*a^5*x^2*e + 1/2*a^5*d*log
(x^2)

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Mupad [B]
time = 0.11, size = 122, normalized size = 0.86 \begin {gather*} \frac {a^5\,e\,x^2}{2}+\frac {c^5\,d\,x^{20}}{20}+\frac {c^5\,e\,x^{22}}{22}+a^5\,d\,\ln \left (x\right )+\frac {5\,a^3\,c^2\,d\,x^8}{4}+\frac {5\,a^2\,c^3\,d\,x^{12}}{6}+a^3\,c^2\,e\,x^{10}+\frac {5\,a^2\,c^3\,e\,x^{14}}{7}+\frac {5\,a^4\,c\,d\,x^4}{4}+\frac {5\,a\,c^4\,d\,x^{16}}{16}+\frac {5\,a^4\,c\,e\,x^6}{6}+\frac {5\,a\,c^4\,e\,x^{18}}{18} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + c*x^4)^5*(d + e*x^2))/x,x)

[Out]

(a^5*e*x^2)/2 + (c^5*d*x^20)/20 + (c^5*e*x^22)/22 + a^5*d*log(x) + (5*a^3*c^2*d*x^8)/4 + (5*a^2*c^3*d*x^12)/6
+ a^3*c^2*e*x^10 + (5*a^2*c^3*e*x^14)/7 + (5*a^4*c*d*x^4)/4 + (5*a*c^4*d*x^16)/16 + (5*a^4*c*e*x^6)/6 + (5*a*c
^4*e*x^18)/18

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