3.2835 \(\int (c+d x)^3 \left (a+b (c+d x)^2\right )^2 \, dx\)

Optimal. Leaf size=51 \[ \frac{a^2 (c+d x)^4}{4 d}+\frac{a b (c+d x)^6}{3 d}+\frac{b^2 (c+d x)^8}{8 d} \]

[Out]

(a^2*(c + d*x)^4)/(4*d) + (a*b*(c + d*x)^6)/(3*d) + (b^2*(c + d*x)^8)/(8*d)

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Rubi [A]  time = 0.169338, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{a^2 (c+d x)^4}{4 d}+\frac{a b (c+d x)^6}{3 d}+\frac{b^2 (c+d x)^8}{8 d} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^3*(a + b*(c + d*x)^2)^2,x]

[Out]

(a^2*(c + d*x)^4)/(4*d) + (a*b*(c + d*x)^6)/(3*d) + (b^2*(c + d*x)^8)/(8*d)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{a^{2} \int ^{\left (c + d x\right )^{2}} x\, dx}{2 d} + \frac{a b \left (c + d x\right )^{6}}{3 d} + \frac{b^{2} \left (c + d x\right )^{8}}{8 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**3*(a+b*(d*x+c)**2)**2,x)

[Out]

a**2*Integral(x, (x, (c + d*x)**2))/(2*d) + a*b*(c + d*x)**6/(3*d) + b**2*(c + d
*x)**8/(8*d)

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Mathematica [B]  time = 0.0422352, size = 172, normalized size = 3.37 \[ \frac{1}{4} d^3 x^4 \left (a^2+20 a b c^2+35 b^2 c^4\right )+\frac{1}{3} c d^2 x^3 \left (3 a^2+20 a b c^2+21 b^2 c^4\right )+\frac{1}{2} c^2 d x^2 \left (3 a^2+10 a b c^2+7 b^2 c^4\right )+\frac{1}{6} b d^5 x^6 \left (2 a+21 b c^2\right )+b c d^4 x^5 \left (2 a+7 b c^2\right )+c^3 x \left (a+b c^2\right )^2+b^2 c d^6 x^7+\frac{1}{8} b^2 d^7 x^8 \]

Antiderivative was successfully verified.

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

[Out]

c^3*(a + b*c^2)^2*x + (c^2*(3*a^2 + 10*a*b*c^2 + 7*b^2*c^4)*d*x^2)/2 + (c*(3*a^2
+ 20*a*b*c^2 + 21*b^2*c^4)*d^2*x^3)/3 + ((a^2 + 20*a*b*c^2 + 35*b^2*c^4)*d^3*x^
4)/4 + b*c*(2*a + 7*b*c^2)*d^4*x^5 + (b*(2*a + 21*b*c^2)*d^5*x^6)/6 + b^2*c*d^6*
x^7 + (b^2*d^7*x^8)/8

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Maple [B]  time = 0.001, size = 324, normalized size = 6.4 \[{\frac{{d}^{7}{b}^{2}{x}^{8}}{8}}+c{d}^{6}{b}^{2}{x}^{7}+{\frac{ \left ( 15\,{c}^{2}{d}^{5}{b}^{2}+{d}^{3} \left ( 2\, \left ( b{c}^{2}+a \right ) b{d}^{2}+4\,{b}^{2}{c}^{2}{d}^{2} \right ) \right ){x}^{6}}{6}}+{\frac{ \left ( 13\,{c}^{3}{b}^{2}{d}^{4}+3\,c{d}^{2} \left ( 2\, \left ( b{c}^{2}+a \right ) b{d}^{2}+4\,{b}^{2}{c}^{2}{d}^{2} \right ) +4\,{d}^{4} \left ( b{c}^{2}+a \right ) bc \right ){x}^{5}}{5}}+{\frac{ \left ( 4\,{c}^{4}{b}^{2}{d}^{3}+3\,{c}^{2}d \left ( 2\, \left ( b{c}^{2}+a \right ) b{d}^{2}+4\,{b}^{2}{c}^{2}{d}^{2} \right ) +12\,{c}^{2}{d}^{3} \left ( b{c}^{2}+a \right ) b+{d}^{3} \left ( b{c}^{2}+a \right ) ^{2} \right ){x}^{4}}{4}}+{\frac{ \left ({c}^{3} \left ( 2\, \left ( b{c}^{2}+a \right ) b{d}^{2}+4\,{b}^{2}{c}^{2}{d}^{2} \right ) +12\,{c}^{3}{d}^{2} \left ( b{c}^{2}+a \right ) b+3\,c{d}^{2} \left ( b{c}^{2}+a \right ) ^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 4\,{c}^{4} \left ( b{c}^{2}+a \right ) bd+3\,{c}^{2}d \left ( b{c}^{2}+a \right ) ^{2} \right ){x}^{2}}{2}}+{c}^{3} \left ( b{c}^{2}+a \right ) ^{2}x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^3*(a+b*(d*x+c)^2)^2,x)

[Out]

1/8*d^7*b^2*x^8+c*d^6*b^2*x^7+1/6*(15*c^2*d^5*b^2+d^3*(2*(b*c^2+a)*b*d^2+4*b^2*c
^2*d^2))*x^6+1/5*(13*c^3*b^2*d^4+3*c*d^2*(2*(b*c^2+a)*b*d^2+4*b^2*c^2*d^2)+4*d^4
*(b*c^2+a)*b*c)*x^5+1/4*(4*c^4*b^2*d^3+3*c^2*d*(2*(b*c^2+a)*b*d^2+4*b^2*c^2*d^2)
+12*c^2*d^3*(b*c^2+a)*b+d^3*(b*c^2+a)^2)*x^4+1/3*(c^3*(2*(b*c^2+a)*b*d^2+4*b^2*c
^2*d^2)+12*c^3*d^2*(b*c^2+a)*b+3*c*d^2*(b*c^2+a)^2)*x^3+1/2*(4*c^4*(b*c^2+a)*b*d
+3*c^2*d*(b*c^2+a)^2)*x^2+c^3*(b*c^2+a)^2*x

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Maxima [A]  time = 1.41231, size = 238, normalized size = 4.67 \[ \frac{1}{8} \, b^{2} d^{7} x^{8} + b^{2} c d^{6} x^{7} + \frac{1}{6} \,{\left (21 \, b^{2} c^{2} + 2 \, a b\right )} d^{5} x^{6} +{\left (7 \, b^{2} c^{3} + 2 \, a b c\right )} d^{4} x^{5} + \frac{1}{4} \,{\left (35 \, b^{2} c^{4} + 20 \, a b c^{2} + a^{2}\right )} d^{3} x^{4} + \frac{1}{3} \,{\left (21 \, b^{2} c^{5} + 20 \, a b c^{3} + 3 \, a^{2} c\right )} d^{2} x^{3} + \frac{1}{2} \,{\left (7 \, b^{2} c^{6} + 10 \, a b c^{4} + 3 \, a^{2} c^{2}\right )} d x^{2} +{\left (b^{2} c^{7} + 2 \, a b c^{5} + a^{2} c^{3}\right )} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((d*x + c)^2*b + a)^2*(d*x + c)^3,x, algorithm="maxima")

[Out]

1/8*b^2*d^7*x^8 + b^2*c*d^6*x^7 + 1/6*(21*b^2*c^2 + 2*a*b)*d^5*x^6 + (7*b^2*c^3
+ 2*a*b*c)*d^4*x^5 + 1/4*(35*b^2*c^4 + 20*a*b*c^2 + a^2)*d^3*x^4 + 1/3*(21*b^2*c
^5 + 20*a*b*c^3 + 3*a^2*c)*d^2*x^3 + 1/2*(7*b^2*c^6 + 10*a*b*c^4 + 3*a^2*c^2)*d*
x^2 + (b^2*c^7 + 2*a*b*c^5 + a^2*c^3)*x

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Fricas [A]  time = 0.186226, size = 1, normalized size = 0.02 \[ \frac{1}{8} x^{8} d^{7} b^{2} + x^{7} d^{6} c b^{2} + \frac{7}{2} x^{6} d^{5} c^{2} b^{2} + 7 x^{5} d^{4} c^{3} b^{2} + \frac{35}{4} x^{4} d^{3} c^{4} b^{2} + \frac{1}{3} x^{6} d^{5} b a + 7 x^{3} d^{2} c^{5} b^{2} + 2 x^{5} d^{4} c b a + \frac{7}{2} x^{2} d c^{6} b^{2} + 5 x^{4} d^{3} c^{2} b a + x c^{7} b^{2} + \frac{20}{3} x^{3} d^{2} c^{3} b a + 5 x^{2} d c^{4} b a + \frac{1}{4} x^{4} d^{3} a^{2} + 2 x c^{5} b a + x^{3} d^{2} c a^{2} + \frac{3}{2} x^{2} d c^{2} a^{2} + x c^{3} a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((d*x + c)^2*b + a)^2*(d*x + c)^3,x, algorithm="fricas")

[Out]

1/8*x^8*d^7*b^2 + x^7*d^6*c*b^2 + 7/2*x^6*d^5*c^2*b^2 + 7*x^5*d^4*c^3*b^2 + 35/4
*x^4*d^3*c^4*b^2 + 1/3*x^6*d^5*b*a + 7*x^3*d^2*c^5*b^2 + 2*x^5*d^4*c*b*a + 7/2*x
^2*d*c^6*b^2 + 5*x^4*d^3*c^2*b*a + x*c^7*b^2 + 20/3*x^3*d^2*c^3*b*a + 5*x^2*d*c^
4*b*a + 1/4*x^4*d^3*a^2 + 2*x*c^5*b*a + x^3*d^2*c*a^2 + 3/2*x^2*d*c^2*a^2 + x*c^
3*a^2

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Sympy [A]  time = 0.244625, size = 209, normalized size = 4.1 \[ b^{2} c d^{6} x^{7} + \frac{b^{2} d^{7} x^{8}}{8} + x^{6} \left (\frac{a b d^{5}}{3} + \frac{7 b^{2} c^{2} d^{5}}{2}\right ) + x^{5} \left (2 a b c d^{4} + 7 b^{2} c^{3} d^{4}\right ) + x^{4} \left (\frac{a^{2} d^{3}}{4} + 5 a b c^{2} d^{3} + \frac{35 b^{2} c^{4} d^{3}}{4}\right ) + x^{3} \left (a^{2} c d^{2} + \frac{20 a b c^{3} d^{2}}{3} + 7 b^{2} c^{5} d^{2}\right ) + x^{2} \left (\frac{3 a^{2} c^{2} d}{2} + 5 a b c^{4} d + \frac{7 b^{2} c^{6} d}{2}\right ) + x \left (a^{2} c^{3} + 2 a b c^{5} + b^{2} c^{7}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x+c)**3*(a+b*(d*x+c)**2)**2,x)

[Out]

b**2*c*d**6*x**7 + b**2*d**7*x**8/8 + x**6*(a*b*d**5/3 + 7*b**2*c**2*d**5/2) + x
**5*(2*a*b*c*d**4 + 7*b**2*c**3*d**4) + x**4*(a**2*d**3/4 + 5*a*b*c**2*d**3 + 35
*b**2*c**4*d**3/4) + x**3*(a**2*c*d**2 + 20*a*b*c**3*d**2/3 + 7*b**2*c**5*d**2)
+ x**2*(3*a**2*c**2*d/2 + 5*a*b*c**4*d + 7*b**2*c**6*d/2) + x*(a**2*c**3 + 2*a*b
*c**5 + b**2*c**7)

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GIAC/XCAS [A]  time = 0.213998, size = 279, normalized size = 5.47 \[ \frac{1}{8} \, b^{2} d^{7} x^{8} + b^{2} c d^{6} x^{7} + \frac{7}{2} \, b^{2} c^{2} d^{5} x^{6} + 7 \, b^{2} c^{3} d^{4} x^{5} + \frac{35}{4} \, b^{2} c^{4} d^{3} x^{4} + \frac{1}{3} \, a b d^{5} x^{6} + 7 \, b^{2} c^{5} d^{2} x^{3} + 2 \, a b c d^{4} x^{5} + \frac{7}{2} \, b^{2} c^{6} d x^{2} + 5 \, a b c^{2} d^{3} x^{4} + b^{2} c^{7} x + \frac{20}{3} \, a b c^{3} d^{2} x^{3} + 5 \, a b c^{4} d x^{2} + \frac{1}{4} \, a^{2} d^{3} x^{4} + 2 \, a b c^{5} x + a^{2} c d^{2} x^{3} + \frac{3}{2} \, a^{2} c^{2} d x^{2} + a^{2} c^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((d*x + c)^2*b + a)^2*(d*x + c)^3,x, algorithm="giac")

[Out]

1/8*b^2*d^7*x^8 + b^2*c*d^6*x^7 + 7/2*b^2*c^2*d^5*x^6 + 7*b^2*c^3*d^4*x^5 + 35/4
*b^2*c^4*d^3*x^4 + 1/3*a*b*d^5*x^6 + 7*b^2*c^5*d^2*x^3 + 2*a*b*c*d^4*x^5 + 7/2*b
^2*c^6*d*x^2 + 5*a*b*c^2*d^3*x^4 + b^2*c^7*x + 20/3*a*b*c^3*d^2*x^3 + 5*a*b*c^4*
d*x^2 + 1/4*a^2*d^3*x^4 + 2*a*b*c^5*x + a^2*c*d^2*x^3 + 3/2*a^2*c^2*d*x^2 + a^2*
c^3*x