3.3.46 \(\int (d+e x)^2 (b x+c x^2)^3 \, dx\) [246]

Optimal. Leaf size=127 \[ \frac {1}{4} b^3 d^2 x^4+\frac {1}{5} b^2 d (3 c d+2 b e) x^5+\frac {1}{6} b \left (3 c^2 d^2+6 b c d e+b^2 e^2\right ) x^6+\frac {1}{7} c \left (c^2 d^2+6 b c d e+3 b^2 e^2\right ) x^7+\frac {1}{8} c^2 e (2 c d+3 b e) x^8+\frac {1}{9} c^3 e^2 x^9 \]

[Out]

1/4*b^3*d^2*x^4+1/5*b^2*d*(2*b*e+3*c*d)*x^5+1/6*b*(b^2*e^2+6*b*c*d*e+3*c^2*d^2)*x^6+1/7*c*(3*b^2*e^2+6*b*c*d*e
+c^2*d^2)*x^7+1/8*c^2*e*(3*b*e+2*c*d)*x^8+1/9*c^3*e^2*x^9

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Rubi [A]
time = 0.08, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {712} \begin {gather*} \frac {1}{4} b^3 d^2 x^4+\frac {1}{7} c x^7 \left (3 b^2 e^2+6 b c d e+c^2 d^2\right )+\frac {1}{6} b x^6 \left (b^2 e^2+6 b c d e+3 c^2 d^2\right )+\frac {1}{5} b^2 d x^5 (2 b e+3 c d)+\frac {1}{8} c^2 e x^8 (3 b e+2 c d)+\frac {1}{9} c^3 e^2 x^9 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^2*(b*x + c*x^2)^3,x]

[Out]

(b^3*d^2*x^4)/4 + (b^2*d*(3*c*d + 2*b*e)*x^5)/5 + (b*(3*c^2*d^2 + 6*b*c*d*e + b^2*e^2)*x^6)/6 + (c*(c^2*d^2 +
6*b*c*d*e + 3*b^2*e^2)*x^7)/7 + (c^2*e*(2*c*d + 3*b*e)*x^8)/8 + (c^3*e^2*x^9)/9

Rule 712

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin {align*} \int (d+e x)^2 \left (b x+c x^2\right )^3 \, dx &=\int \left (b^3 d^2 x^3+b^2 d (3 c d+2 b e) x^4+b \left (3 c^2 d^2+6 b c d e+b^2 e^2\right ) x^5+c \left (c^2 d^2+6 b c d e+3 b^2 e^2\right ) x^6+c^2 e (2 c d+3 b e) x^7+c^3 e^2 x^8\right ) \, dx\\ &=\frac {1}{4} b^3 d^2 x^4+\frac {1}{5} b^2 d (3 c d+2 b e) x^5+\frac {1}{6} b \left (3 c^2 d^2+6 b c d e+b^2 e^2\right ) x^6+\frac {1}{7} c \left (c^2 d^2+6 b c d e+3 b^2 e^2\right ) x^7+\frac {1}{8} c^2 e (2 c d+3 b e) x^8+\frac {1}{9} c^3 e^2 x^9\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 127, normalized size = 1.00 \begin {gather*} \frac {1}{4} b^3 d^2 x^4+\frac {1}{5} b^2 d (3 c d+2 b e) x^5+\frac {1}{6} b \left (3 c^2 d^2+6 b c d e+b^2 e^2\right ) x^6+\frac {1}{7} c \left (c^2 d^2+6 b c d e+3 b^2 e^2\right ) x^7+\frac {1}{8} c^2 e (2 c d+3 b e) x^8+\frac {1}{9} c^3 e^2 x^9 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^2*(b*x + c*x^2)^3,x]

[Out]

(b^3*d^2*x^4)/4 + (b^2*d*(3*c*d + 2*b*e)*x^5)/5 + (b*(3*c^2*d^2 + 6*b*c*d*e + b^2*e^2)*x^6)/6 + (c*(c^2*d^2 +
6*b*c*d*e + 3*b^2*e^2)*x^7)/7 + (c^2*e*(2*c*d + 3*b*e)*x^8)/8 + (c^3*e^2*x^9)/9

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Maple [A]
time = 0.44, size = 128, normalized size = 1.01

method	result	size
norman	\(\frac {c^{3} e^{2} x^{9}}{9}+\left (\frac {3}{8} e^{2} b \,c^{2}+\frac {1}{4} d e \,c^{3}\right ) x^{8}+\left (\frac {3}{7} b^{2} e^{2} c +\frac {6}{7} d e b \,c^{2}+\frac {1}{7} c^{3} d^{2}\right ) x^{7}+\left (\frac {1}{6} b^{3} e^{2}+b^{2} d c e +\frac {1}{2} d^{2} b \,c^{2}\right ) x^{6}+\left (\frac {2}{5} b^{3} d e +\frac {3}{5} b^{2} c \,d^{2}\right ) x^{5}+\frac {b^{3} d^{2} x^{4}}{4}\)	\(125\)
default	\(\frac {c^{3} e^{2} x^{9}}{9}+\frac {\left (3 e^{2} b \,c^{2}+2 d e \,c^{3}\right ) x^{8}}{8}+\frac {\left (3 b^{2} e^{2} c +6 d e b \,c^{2}+c^{3} d^{2}\right ) x^{7}}{7}+\frac {\left (b^{3} e^{2}+6 b^{2} d c e +3 d^{2} b \,c^{2}\right ) x^{6}}{6}+\frac {\left (2 b^{3} d e +3 b^{2} c \,d^{2}\right ) x^{5}}{5}+\frac {b^{3} d^{2} x^{4}}{4}\)	\(128\)
gosper	\(\frac {x^{4} \left (280 c^{3} e^{2} x^{5}+945 x^{4} e^{2} b \,c^{2}+630 x^{4} d e \,c^{3}+1080 x^{3} b^{2} e^{2} c +2160 x^{3} d e b \,c^{2}+360 x^{3} c^{3} d^{2}+420 x^{2} b^{3} e^{2}+2520 x^{2} b^{2} d c e +1260 x^{2} d^{2} b \,c^{2}+1008 x \,b^{3} d e +1512 x \,b^{2} c \,d^{2}+630 d^{2} b^{3}\right )}{2520}\)	\(134\)
risch	\(\frac {1}{9} c^{3} e^{2} x^{9}+\frac {3}{8} x^{8} e^{2} b \,c^{2}+\frac {1}{4} x^{8} d e \,c^{3}+\frac {3}{7} x^{7} b^{2} e^{2} c +\frac {6}{7} x^{7} d e b \,c^{2}+\frac {1}{7} x^{7} c^{3} d^{2}+\frac {1}{6} x^{6} b^{3} e^{2}+x^{6} b^{2} d c e +\frac {1}{2} x^{6} d^{2} b \,c^{2}+\frac {2}{5} x^{5} b^{3} d e +\frac {3}{5} b^{2} c \,d^{2} x^{5}+\frac {1}{4} b^{3} d^{2} x^{4}\)	\(135\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^2*(c*x^2+b*x)^3,x,method=_RETURNVERBOSE)

[Out]

1/9*c^3*e^2*x^9+1/8*(3*b*c^2*e^2+2*c^3*d*e)*x^8+1/7*(3*b^2*c*e^2+6*b*c^2*d*e+c^3*d^2)*x^7+1/6*(b^3*e^2+6*b^2*c
*d*e+3*b*c^2*d^2)*x^6+1/5*(2*b^3*d*e+3*b^2*c*d^2)*x^5+1/4*b^3*d^2*x^4

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Maxima [A]
time = 0.29, size = 127, normalized size = 1.00 \begin {gather*} \frac {1}{9} \, c^{3} x^{9} e^{2} + \frac {1}{4} \, b^{3} d^{2} x^{4} + \frac {1}{8} \, {\left (2 \, c^{3} d e + 3 \, b c^{2} e^{2}\right )} x^{8} + \frac {1}{7} \, {\left (c^{3} d^{2} + 6 \, b c^{2} d e + 3 \, b^{2} c e^{2}\right )} x^{7} + \frac {1}{6} \, {\left (3 \, b c^{2} d^{2} + 6 \, b^{2} c d e + b^{3} e^{2}\right )} x^{6} + \frac {1}{5} \, {\left (3 \, b^{2} c d^{2} + 2 \, b^{3} d e\right )} x^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*(c*x^2+b*x)^3,x, algorithm="maxima")

[Out]

1/9*c^3*x^9*e^2 + 1/4*b^3*d^2*x^4 + 1/8*(2*c^3*d*e + 3*b*c^2*e^2)*x^8 + 1/7*(c^3*d^2 + 6*b*c^2*d*e + 3*b^2*c*e
^2)*x^7 + 1/6*(3*b*c^2*d^2 + 6*b^2*c*d*e + b^3*e^2)*x^6 + 1/5*(3*b^2*c*d^2 + 2*b^3*d*e)*x^5

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Fricas [A]
time = 1.55, size = 129, normalized size = 1.02 \begin {gather*} \frac {1}{7} \, c^{3} d^{2} x^{7} + \frac {1}{2} \, b c^{2} d^{2} x^{6} + \frac {3}{5} \, b^{2} c d^{2} x^{5} + \frac {1}{4} \, b^{3} d^{2} x^{4} + \frac {1}{504} \, {\left (56 \, c^{3} x^{9} + 189 \, b c^{2} x^{8} + 216 \, b^{2} c x^{7} + 84 \, b^{3} x^{6}\right )} e^{2} + \frac {1}{140} \, {\left (35 \, c^{3} d x^{8} + 120 \, b c^{2} d x^{7} + 140 \, b^{2} c d x^{6} + 56 \, b^{3} d x^{5}\right )} e \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*(c*x^2+b*x)^3,x, algorithm="fricas")

[Out]

1/7*c^3*d^2*x^7 + 1/2*b*c^2*d^2*x^6 + 3/5*b^2*c*d^2*x^5 + 1/4*b^3*d^2*x^4 + 1/504*(56*c^3*x^9 + 189*b*c^2*x^8
+ 216*b^2*c*x^7 + 84*b^3*x^6)*e^2 + 1/140*(35*c^3*d*x^8 + 120*b*c^2*d*x^7 + 140*b^2*c*d*x^6 + 56*b^3*d*x^5)*e

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Sympy [A]
time = 0.02, size = 138, normalized size = 1.09 \begin {gather*} \frac {b^{3} d^{2} x^{4}}{4} + \frac {c^{3} e^{2} x^{9}}{9} + x^{8} \cdot \left (\frac {3 b c^{2} e^{2}}{8} + \frac {c^{3} d e}{4}\right ) + x^{7} \cdot \left (\frac {3 b^{2} c e^{2}}{7} + \frac {6 b c^{2} d e}{7} + \frac {c^{3} d^{2}}{7}\right ) + x^{6} \left (\frac {b^{3} e^{2}}{6} + b^{2} c d e + \frac {b c^{2} d^{2}}{2}\right ) + x^{5} \cdot \left (\frac {2 b^{3} d e}{5} + \frac {3 b^{2} c d^{2}}{5}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**2*(c*x**2+b*x)**3,x)

[Out]

b**3*d**2*x**4/4 + c**3*e**2*x**9/9 + x**8*(3*b*c**2*e**2/8 + c**3*d*e/4) + x**7*(3*b**2*c*e**2/7 + 6*b*c**2*d
*e/7 + c**3*d**2/7) + x**6*(b**3*e**2/6 + b**2*c*d*e + b*c**2*d**2/2) + x**5*(2*b**3*d*e/5 + 3*b**2*c*d**2/5)

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Giac [A]
time = 0.63, size = 134, normalized size = 1.06 \begin {gather*} \frac {1}{9} \, c^{3} x^{9} e^{2} + \frac {1}{4} \, c^{3} d x^{8} e + \frac {1}{7} \, c^{3} d^{2} x^{7} + \frac {3}{8} \, b c^{2} x^{8} e^{2} + \frac {6}{7} \, b c^{2} d x^{7} e + \frac {1}{2} \, b c^{2} d^{2} x^{6} + \frac {3}{7} \, b^{2} c x^{7} e^{2} + b^{2} c d x^{6} e + \frac {3}{5} \, b^{2} c d^{2} x^{5} + \frac {1}{6} \, b^{3} x^{6} e^{2} + \frac {2}{5} \, b^{3} d x^{5} e + \frac {1}{4} \, b^{3} d^{2} x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*(c*x^2+b*x)^3,x, algorithm="giac")

[Out]

1/9*c^3*x^9*e^2 + 1/4*c^3*d*x^8*e + 1/7*c^3*d^2*x^7 + 3/8*b*c^2*x^8*e^2 + 6/7*b*c^2*d*x^7*e + 1/2*b*c^2*d^2*x^
6 + 3/7*b^2*c*x^7*e^2 + b^2*c*d*x^6*e + 3/5*b^2*c*d^2*x^5 + 1/6*b^3*x^6*e^2 + 2/5*b^3*d*x^5*e + 1/4*b^3*d^2*x^
4

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Mupad [B]
time = 0.05, size = 118, normalized size = 0.93 \begin {gather*} x^6\,\left (\frac {b^3\,e^2}{6}+b^2\,c\,d\,e+\frac {b\,c^2\,d^2}{2}\right )+x^7\,\left (\frac {3\,b^2\,c\,e^2}{7}+\frac {6\,b\,c^2\,d\,e}{7}+\frac {c^3\,d^2}{7}\right )+\frac {b^3\,d^2\,x^4}{4}+\frac {c^3\,e^2\,x^9}{9}+\frac {b^2\,d\,x^5\,\left (2\,b\,e+3\,c\,d\right )}{5}+\frac {c^2\,e\,x^8\,\left (3\,b\,e+2\,c\,d\right )}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x + c*x^2)^3*(d + e*x)^2,x)

[Out]

x^6*((b^3*e^2)/6 + (b*c^2*d^2)/2 + b^2*c*d*e) + x^7*((c^3*d^2)/7 + (3*b^2*c*e^2)/7 + (6*b*c^2*d*e)/7) + (b^3*d
^2*x^4)/4 + (c^3*e^2*x^9)/9 + (b^2*d*x^5*(2*b*e + 3*c*d))/5 + (c^2*e*x^8*(3*b*e + 2*c*d))/8

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