3.1777 \(\int \frac{\left (a c+(b c+a d) x+b d x^2\right )^3}{(a+b x)^2} \, dx\)

Optimal. Leaf size=38 \[ \frac{b (c+d x)^5}{5 d^2}-\frac{(c+d x)^4 (b c-a d)}{4 d^2} \]

[Out]

-((b*c - a*d)*(c + d*x)^4)/(4*d^2) + (b*(c + d*x)^5)/(5*d^2)

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Rubi [A]  time = 0.0651665, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ \frac{b (c+d x)^5}{5 d^2}-\frac{(c+d x)^4 (b c-a d)}{4 d^2} \]

Antiderivative was successfully verified.

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

[Out]

-((b*c - a*d)*(c + d*x)^4)/(4*d^2) + (b*(c + d*x)^5)/(5*d^2)

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Rubi in Sympy [A]  time = 17.7, size = 31, normalized size = 0.82 \[ \frac{b \left (c + d x\right )^{5}}{5 d^{2}} + \frac{\left (c + d x\right )^{4} \left (a d - b c\right )}{4 d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*(c + d*x)**5/(5*d**2) + (c + d*x)**4*(a*d - b*c)/(4*d**2)

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Mathematica [A]  time = 0.0143925, size = 67, normalized size = 1.76 \[ \frac{1}{2} c^2 x^2 (3 a d+b c)+\frac{1}{4} d^2 x^4 (a d+3 b c)+c d x^3 (a d+b c)+a c^3 x+\frac{1}{5} b d^3 x^5 \]

Antiderivative was successfully verified.

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

[Out]

a*c^3*x + (c^2*(b*c + 3*a*d)*x^2)/2 + c*d*(b*c + a*d)*x^3 + (d^2*(3*b*c + a*d)*x
^4)/4 + (b*d^3*x^5)/5

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Maple [B]  time = 0.002, size = 94, normalized size = 2.5 \[{\frac{b{d}^{3}{x}^{5}}{5}}+{\frac{ \left ( 2\,c{d}^{2}b+{d}^{2} \left ( ad+bc \right ) \right ){x}^{4}}{4}}+{\frac{ \left ({c}^{2}bd+2\,cd \left ( ad+bc \right ) +ac{d}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ({c}^{2} \left ( ad+bc \right ) +2\,a{c}^{2}d \right ){x}^{2}}{2}}+xa{c}^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*b*d^3*x^5+1/4*(2*c*d^2*b+d^2*(a*d+b*c))*x^4+1/3*(c^2*b*d+2*c*d*(a*d+b*c)+a*c
*d^2)*x^3+1/2*(c^2*(a*d+b*c)+2*a*c^2*d)*x^2+x*a*c^3

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Maxima [A]  time = 0.712436, size = 93, normalized size = 2.45 \[ \frac{1}{5} \, b d^{3} x^{5} + a c^{3} x + \frac{1}{4} \,{\left (3 \, b c d^{2} + a d^{3}\right )} x^{4} +{\left (b c^{2} d + a c d^{2}\right )} x^{3} + \frac{1}{2} \,{\left (b c^{3} + 3 \, a c^{2} d\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*b*d^3*x^5 + a*c^3*x + 1/4*(3*b*c*d^2 + a*d^3)*x^4 + (b*c^2*d + a*c*d^2)*x^3
+ 1/2*(b*c^3 + 3*a*c^2*d)*x^2

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Fricas [A]  time = 0.200183, size = 93, normalized size = 2.45 \[ \frac{1}{5} \, b d^{3} x^{5} + a c^{3} x + \frac{1}{4} \,{\left (3 \, b c d^{2} + a d^{3}\right )} x^{4} +{\left (b c^{2} d + a c d^{2}\right )} x^{3} + \frac{1}{2} \,{\left (b c^{3} + 3 \, a c^{2} d\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*b*d^3*x^5 + a*c^3*x + 1/4*(3*b*c*d^2 + a*d^3)*x^4 + (b*c^2*d + a*c*d^2)*x^3
+ 1/2*(b*c^3 + 3*a*c^2*d)*x^2

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Sympy [A]  time = 0.343412, size = 73, normalized size = 1.92 \[ a c^{3} x + \frac{b d^{3} x^{5}}{5} + x^{4} \left (\frac{a d^{3}}{4} + \frac{3 b c d^{2}}{4}\right ) + x^{3} \left (a c d^{2} + b c^{2} d\right ) + x^{2} \left (\frac{3 a c^{2} d}{2} + \frac{b c^{3}}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*c**3*x + b*d**3*x**5/5 + x**4*(a*d**3/4 + 3*b*c*d**2/4) + x**3*(a*c*d**2 + b*c
**2*d) + x**2*(3*a*c**2*d/2 + b*c**3/2)

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GIAC/XCAS [A]  time = 0.210334, size = 209, normalized size = 5.5 \[ \frac{{\left (\frac{10 \, b^{3} c^{3}}{{\left (b x + a\right )}^{3}} + \frac{20 \, b^{2} c^{2} d}{{\left (b x + a\right )}^{2}} - \frac{30 \, a b^{2} c^{2} d}{{\left (b x + a\right )}^{3}} + \frac{15 \, b c d^{2}}{b x + a} - \frac{40 \, a b c d^{2}}{{\left (b x + a\right )}^{2}} + \frac{30 \, a^{2} b c d^{2}}{{\left (b x + a\right )}^{3}} - \frac{15 \, a d^{3}}{b x + a} + \frac{20 \, a^{2} d^{3}}{{\left (b x + a\right )}^{2}} - \frac{10 \, a^{3} d^{3}}{{\left (b x + a\right )}^{3}} + 4 \, d^{3}\right )}{\left (b x + a\right )}^{5}}{20 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/20*(10*b^3*c^3/(b*x + a)^3 + 20*b^2*c^2*d/(b*x + a)^2 - 30*a*b^2*c^2*d/(b*x +
a)^3 + 15*b*c*d^2/(b*x + a) - 40*a*b*c*d^2/(b*x + a)^2 + 30*a^2*b*c*d^2/(b*x + a
)^3 - 15*a*d^3/(b*x + a) + 20*a^2*d^3/(b*x + a)^2 - 10*a^3*d^3/(b*x + a)^3 + 4*d
^3)*(b*x + a)^5/b^4