3.1.1 \(\int (d+e x^3)^5 (a+b x^3+c x^6) \, dx\) [1]

Optimal. Leaf size=163 \[ a d^5 x+\frac {1}{4} d^4 (b d+5 a e) x^4+\frac {1}{7} d^3 \left (c d^2+5 e (b d+2 a e)\right ) x^7+\frac {1}{2} d^2 e \left (c d^2+2 e (b d+a e)\right ) x^{10}+\frac {5}{13} d e^2 \left (2 c d^2+e (2 b d+a e)\right ) x^{13}+\frac {1}{16} e^3 \left (10 c d^2+e (5 b d+a e)\right ) x^{16}+\frac {1}{19} e^4 (5 c d+b e) x^{19}+\frac {1}{22} c e^5 x^{22} \]

[Out]

a*d^5*x+1/4*d^4*(5*a*e+b*d)*x^4+1/7*d^3*(c*d^2+5*e*(2*a*e+b*d))*x^7+1/2*d^2*e*(c*d^2+2*e*(a*e+b*d))*x^10+5/13*
d*e^2*(2*c*d^2+e*(a*e+2*b*d))*x^13+1/16*e^3*(10*c*d^2+e*(a*e+5*b*d))*x^16+1/19*e^4*(b*e+5*c*d)*x^19+1/22*c*e^5
*x^22

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Rubi [A]
time = 0.13, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {1421} \begin {gather*} \frac {1}{16} e^3 x^{16} \left (e (a e+5 b d)+10 c d^2\right )+\frac {5}{13} d e^2 x^{13} \left (e (a e+2 b d)+2 c d^2\right )+\frac {1}{2} d^2 e x^{10} \left (2 e (a e+b d)+c d^2\right )+\frac {1}{7} d^3 x^7 \left (5 e (2 a e+b d)+c d^2\right )+\frac {1}{4} d^4 x^4 (5 a e+b d)+a d^5 x+\frac {1}{19} e^4 x^{19} (b e+5 c d)+\frac {1}{22} c e^5 x^{22} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x^3)^5*(a + b*x^3 + c*x^6),x]

[Out]

a*d^5*x + (d^4*(b*d + 5*a*e)*x^4)/4 + (d^3*(c*d^2 + 5*e*(b*d + 2*a*e))*x^7)/7 + (d^2*e*(c*d^2 + 2*e*(b*d + a*e
))*x^10)/2 + (5*d*e^2*(2*c*d^2 + e*(2*b*d + a*e))*x^13)/13 + (e^3*(10*c*d^2 + e*(5*b*d + a*e))*x^16)/16 + (e^4
*(5*c*d + b*e)*x^19)/19 + (c*e^5*x^22)/22

Rule 1421

Int[((d_) + (e_.)*(x_)^(n_))^(q_.)*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x^n)^q*(a + b*x^n + c*x^(2*n)), x], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 -
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[q, 0]

Rubi steps

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

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Mathematica [A]
time = 0.04, size = 164, normalized size = 1.01 \begin {gather*} a d^5 x+\frac {1}{4} d^4 (b d+5 a e) x^4+\frac {1}{7} d^3 \left (c d^2+5 b d e+10 a e^2\right ) x^7+\frac {1}{2} d^2 e \left (c d^2+2 b d e+2 a e^2\right ) x^{10}+\frac {5}{13} d e^2 \left (2 c d^2+2 b d e+a e^2\right ) x^{13}+\frac {1}{16} e^3 \left (10 c d^2+5 b d e+a e^2\right ) x^{16}+\frac {1}{19} e^4 (5 c d+b e) x^{19}+\frac {1}{22} c e^5 x^{22} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x^3)^5*(a + b*x^3 + c*x^6),x]

[Out]

a*d^5*x + (d^4*(b*d + 5*a*e)*x^4)/4 + (d^3*(c*d^2 + 5*b*d*e + 10*a*e^2)*x^7)/7 + (d^2*e*(c*d^2 + 2*b*d*e + 2*a
*e^2)*x^10)/2 + (5*d*e^2*(2*c*d^2 + 2*b*d*e + a*e^2)*x^13)/13 + (e^3*(10*c*d^2 + 5*b*d*e + a*e^2)*x^16)/16 + (
e^4*(5*c*d + b*e)*x^19)/19 + (c*e^5*x^22)/22

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Maple [A]
time = 0.22, size = 169, normalized size = 1.04

method	result	size
norman	\(a \,d^{5} x +\left (\frac {5}{4} d^{4} e a +\frac {1}{4} d^{5} b \right ) x^{4}+\left (\frac {10}{7} a \,d^{3} e^{2}+\frac {5}{7} d^{4} e b +\frac {1}{7} d^{5} c \right ) x^{7}+\left (a \,d^{2} e^{3}+d^{3} e^{2} b +\frac {1}{2} c \,d^{4} e \right ) x^{10}+\left (\frac {5}{13} d \,e^{4} a +\frac {10}{13} d^{2} e^{3} b +\frac {10}{13} d^{3} e^{2} c \right ) x^{13}+\left (\frac {1}{16} e^{5} a +\frac {5}{16} b d \,e^{4}+\frac {5}{8} d^{2} e^{3} c \right ) x^{16}+\left (\frac {1}{19} e^{5} b +\frac {5}{19} d \,e^{4} c \right ) x^{19}+\frac {c \,e^{5} x^{22}}{22}\)	\(165\)
default	\(\frac {c \,e^{5} x^{22}}{22}+\frac {\left (e^{5} b +5 d \,e^{4} c \right ) x^{19}}{19}+\frac {\left (e^{5} a +5 b d \,e^{4}+10 d^{2} e^{3} c \right ) x^{16}}{16}+\frac {\left (5 d \,e^{4} a +10 d^{2} e^{3} b +10 d^{3} e^{2} c \right ) x^{13}}{13}+\frac {\left (10 a \,d^{2} e^{3}+10 d^{3} e^{2} b +5 c \,d^{4} e \right ) x^{10}}{10}+\frac {\left (10 a \,d^{3} e^{2}+5 d^{4} e b +d^{5} c \right ) x^{7}}{7}+\frac {\left (5 d^{4} e a +d^{5} b \right ) x^{4}}{4}+a \,d^{5} x\)	\(169\)
gosper	\(a \,d^{5} x +\frac {5}{4} x^{4} d^{4} e a +\frac {1}{4} x^{4} d^{5} b +\frac {10}{7} x^{7} a \,d^{3} e^{2}+\frac {5}{7} x^{7} d^{4} e b +\frac {1}{7} x^{7} d^{5} c +x^{10} a \,d^{2} e^{3}+x^{10} d^{3} e^{2} b +\frac {1}{2} x^{10} c \,d^{4} e +\frac {5}{13} x^{13} d \,e^{4} a +\frac {10}{13} x^{13} d^{2} e^{3} b +\frac {10}{13} x^{13} d^{3} e^{2} c +\frac {1}{16} x^{16} e^{5} a +\frac {5}{16} x^{16} b d \,e^{4}+\frac {5}{8} x^{16} d^{2} e^{3} c +\frac {1}{19} x^{19} e^{5} b +\frac {5}{19} x^{19} d \,e^{4} c +\frac {1}{22} c \,e^{5} x^{22}\)	\(183\)
risch	\(a \,d^{5} x +\frac {5}{4} x^{4} d^{4} e a +\frac {1}{4} x^{4} d^{5} b +\frac {10}{7} x^{7} a \,d^{3} e^{2}+\frac {5}{7} x^{7} d^{4} e b +\frac {1}{7} x^{7} d^{5} c +x^{10} a \,d^{2} e^{3}+x^{10} d^{3} e^{2} b +\frac {1}{2} x^{10} c \,d^{4} e +\frac {5}{13} x^{13} d \,e^{4} a +\frac {10}{13} x^{13} d^{2} e^{3} b +\frac {10}{13} x^{13} d^{3} e^{2} c +\frac {1}{16} x^{16} e^{5} a +\frac {5}{16} x^{16} b d \,e^{4}+\frac {5}{8} x^{16} d^{2} e^{3} c +\frac {1}{19} x^{19} e^{5} b +\frac {5}{19} x^{19} d \,e^{4} c +\frac {1}{22} c \,e^{5} x^{22}\)	\(183\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^3+d)^5*(c*x^6+b*x^3+a),x,method=_RETURNVERBOSE)

[Out]

1/22*c*e^5*x^22+1/19*(b*e^5+5*c*d*e^4)*x^19+1/16*(a*e^5+5*b*d*e^4+10*c*d^2*e^3)*x^16+1/13*(5*a*d*e^4+10*b*d^2*
e^3+10*c*d^3*e^2)*x^13+1/10*(10*a*d^2*e^3+10*b*d^3*e^2+5*c*d^4*e)*x^10+1/7*(10*a*d^3*e^2+5*b*d^4*e+c*d^5)*x^7+
1/4*(5*a*d^4*e+b*d^5)*x^4+a*d^5*x

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Maxima [A]
time = 0.27, size = 157, normalized size = 0.96 \begin {gather*} \frac {1}{22} \, c x^{22} e^{5} + \frac {1}{19} \, {\left (5 \, c d e^{4} + b e^{5}\right )} x^{19} + \frac {1}{16} \, {\left (10 \, c d^{2} e^{3} + 5 \, b d e^{4} + a e^{5}\right )} x^{16} + \frac {5}{13} \, {\left (2 \, c d^{3} e^{2} + 2 \, b d^{2} e^{3} + a d e^{4}\right )} x^{13} + \frac {1}{2} \, {\left (c d^{4} e + 2 \, b d^{3} e^{2} + 2 \, a d^{2} e^{3}\right )} x^{10} + \frac {1}{7} \, {\left (c d^{5} + 5 \, b d^{4} e + 10 \, a d^{3} e^{2}\right )} x^{7} + a d^{5} x + \frac {1}{4} \, {\left (b d^{5} + 5 \, a d^{4} e\right )} x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^3+d)^5*(c*x^6+b*x^3+a),x, algorithm="maxima")

[Out]

1/22*c*x^22*e^5 + 1/19*(5*c*d*e^4 + b*e^5)*x^19 + 1/16*(10*c*d^2*e^3 + 5*b*d*e^4 + a*e^5)*x^16 + 5/13*(2*c*d^3
*e^2 + 2*b*d^2*e^3 + a*d*e^4)*x^13 + 1/2*(c*d^4*e + 2*b*d^3*e^2 + 2*a*d^2*e^3)*x^10 + 1/7*(c*d^5 + 5*b*d^4*e +
10*a*d^3*e^2)*x^7 + a*d^5*x + 1/4*(b*d^5 + 5*a*d^4*e)*x^4

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Fricas [A]
time = 0.32, size = 170, normalized size = 1.04 \begin {gather*} \frac {1}{7} \, c d^{5} x^{7} + \frac {1}{4} \, b d^{5} x^{4} + a d^{5} x + \frac {1}{3344} \, {\left (152 \, c x^{22} + 176 \, b x^{19} + 209 \, a x^{16}\right )} e^{5} + \frac {5}{3952} \, {\left (208 \, c d x^{19} + 247 \, b d x^{16} + 304 \, a d x^{13}\right )} e^{4} + \frac {1}{104} \, {\left (65 \, c d^{2} x^{16} + 80 \, b d^{2} x^{13} + 104 \, a d^{2} x^{10}\right )} e^{3} + \frac {1}{91} \, {\left (70 \, c d^{3} x^{13} + 91 \, b d^{3} x^{10} + 130 \, a d^{3} x^{7}\right )} e^{2} + \frac {1}{28} \, {\left (14 \, c d^{4} x^{10} + 20 \, b d^{4} x^{7} + 35 \, a d^{4} x^{4}\right )} e \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^3+d)^5*(c*x^6+b*x^3+a),x, algorithm="fricas")

[Out]

1/7*c*d^5*x^7 + 1/4*b*d^5*x^4 + a*d^5*x + 1/3344*(152*c*x^22 + 176*b*x^19 + 209*a*x^16)*e^5 + 5/3952*(208*c*d*
x^19 + 247*b*d*x^16 + 304*a*d*x^13)*e^4 + 1/104*(65*c*d^2*x^16 + 80*b*d^2*x^13 + 104*a*d^2*x^10)*e^3 + 1/91*(7
0*c*d^3*x^13 + 91*b*d^3*x^10 + 130*a*d^3*x^7)*e^2 + 1/28*(14*c*d^4*x^10 + 20*b*d^4*x^7 + 35*a*d^4*x^4)*e

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Sympy [A]
time = 0.02, size = 187, normalized size = 1.15 \begin {gather*} a d^{5} x + \frac {c e^{5} x^{22}}{22} + x^{19} \left (\frac {b e^{5}}{19} + \frac {5 c d e^{4}}{19}\right ) + x^{16} \left (\frac {a e^{5}}{16} + \frac {5 b d e^{4}}{16} + \frac {5 c d^{2} e^{3}}{8}\right ) + x^{13} \cdot \left (\frac {5 a d e^{4}}{13} + \frac {10 b d^{2} e^{3}}{13} + \frac {10 c d^{3} e^{2}}{13}\right ) + x^{10} \left (a d^{2} e^{3} + b d^{3} e^{2} + \frac {c d^{4} e}{2}\right ) + x^{7} \cdot \left (\frac {10 a d^{3} e^{2}}{7} + \frac {5 b d^{4} e}{7} + \frac {c d^{5}}{7}\right ) + x^{4} \cdot \left (\frac {5 a d^{4} e}{4} + \frac {b d^{5}}{4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x**3+d)**5*(c*x**6+b*x**3+a),x)

[Out]

a*d**5*x + c*e**5*x**22/22 + x**19*(b*e**5/19 + 5*c*d*e**4/19) + x**16*(a*e**5/16 + 5*b*d*e**4/16 + 5*c*d**2*e
**3/8) + x**13*(5*a*d*e**4/13 + 10*b*d**2*e**3/13 + 10*c*d**3*e**2/13) + x**10*(a*d**2*e**3 + b*d**3*e**2 + c*
d**4*e/2) + x**7*(10*a*d**3*e**2/7 + 5*b*d**4*e/7 + c*d**5/7) + x**4*(5*a*d**4*e/4 + b*d**5/4)

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Giac [A]
time = 3.35, size = 173, normalized size = 1.06 \begin {gather*} \frac {1}{22} \, c x^{22} e^{5} + \frac {5}{19} \, c d x^{19} e^{4} + \frac {1}{19} \, b x^{19} e^{5} + \frac {5}{8} \, c d^{2} x^{16} e^{3} + \frac {5}{16} \, b d x^{16} e^{4} + \frac {1}{16} \, a x^{16} e^{5} + \frac {10}{13} \, c d^{3} x^{13} e^{2} + \frac {10}{13} \, b d^{2} x^{13} e^{3} + \frac {5}{13} \, a d x^{13} e^{4} + \frac {1}{2} \, c d^{4} x^{10} e + b d^{3} x^{10} e^{2} + a d^{2} x^{10} e^{3} + \frac {1}{7} \, c d^{5} x^{7} + \frac {5}{7} \, b d^{4} x^{7} e + \frac {10}{7} \, a d^{3} x^{7} e^{2} + \frac {1}{4} \, b d^{5} x^{4} + \frac {5}{4} \, a d^{4} x^{4} e + a d^{5} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^3+d)^5*(c*x^6+b*x^3+a),x, algorithm="giac")

[Out]

1/22*c*x^22*e^5 + 5/19*c*d*x^19*e^4 + 1/19*b*x^19*e^5 + 5/8*c*d^2*x^16*e^3 + 5/16*b*d*x^16*e^4 + 1/16*a*x^16*e
^5 + 10/13*c*d^3*x^13*e^2 + 10/13*b*d^2*x^13*e^3 + 5/13*a*d*x^13*e^4 + 1/2*c*d^4*x^10*e + b*d^3*x^10*e^2 + a*d
^2*x^10*e^3 + 1/7*c*d^5*x^7 + 5/7*b*d^4*x^7*e + 10/7*a*d^3*x^7*e^2 + 1/4*b*d^5*x^4 + 5/4*a*d^4*x^4*e + a*d^5*x

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Mupad [B]
time = 1.60, size = 158, normalized size = 0.97 \begin {gather*} x^4\,\left (\frac {b\,d^5}{4}+\frac {5\,a\,e\,d^4}{4}\right )+x^{19}\,\left (\frac {b\,e^5}{19}+\frac {5\,c\,d\,e^4}{19}\right )+x^7\,\left (\frac {c\,d^5}{7}+\frac {5\,b\,d^4\,e}{7}+\frac {10\,a\,d^3\,e^2}{7}\right )+x^{16}\,\left (\frac {5\,c\,d^2\,e^3}{8}+\frac {5\,b\,d\,e^4}{16}+\frac {a\,e^5}{16}\right )+\frac {c\,e^5\,x^{22}}{22}+a\,d^5\,x+\frac {d^2\,e\,x^{10}\,\left (c\,d^2+2\,b\,d\,e+2\,a\,e^2\right )}{2}+\frac {5\,d\,e^2\,x^{13}\,\left (2\,c\,d^2+2\,b\,d\,e+a\,e^2\right )}{13} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x^3)^5*(a + b*x^3 + c*x^6),x)

[Out]

x^4*((b*d^5)/4 + (5*a*d^4*e)/4) + x^19*((b*e^5)/19 + (5*c*d*e^4)/19) + x^7*((c*d^5)/7 + (10*a*d^3*e^2)/7 + (5*
b*d^4*e)/7) + x^16*((a*e^5)/16 + (5*c*d^2*e^3)/8 + (5*b*d*e^4)/16) + (c*e^5*x^22)/22 + a*d^5*x + (d^2*e*x^10*(
2*a*e^2 + c*d^2 + 2*b*d*e))/2 + (5*d*e^2*x^13*(a*e^2 + 2*c*d^2 + 2*b*d*e))/13

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