3.2840 \(\int (c+d x)^5 \left (a+b (c+d x)^2\right )^p \, dx\)

Optimal. Leaf size=93 \[ \frac{a^2 \left (a+b (c+d x)^2\right )^{p+1}}{2 b^3 d (p+1)}-\frac{a \left (a+b (c+d x)^2\right )^{p+2}}{b^3 d (p+2)}+\frac{\left (a+b (c+d x)^2\right )^{p+3}}{2 b^3 d (p+3)} \]

[Out]

(a^2*(a + b*(c + d*x)^2)^(1 + p))/(2*b^3*d*(1 + p)) - (a*(a + b*(c + d*x)^2)^(2
+ p))/(b^3*d*(2 + p)) + (a + b*(c + d*x)^2)^(3 + p)/(2*b^3*d*(3 + p))

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Rubi [A]  time = 0.212051, antiderivative size = 93, 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 \[ \frac{a^2 \left (a+b (c+d x)^2\right )^{p+1}}{2 b^3 d (p+1)}-\frac{a \left (a+b (c+d x)^2\right )^{p+2}}{b^3 d (p+2)}+\frac{\left (a+b (c+d x)^2\right )^{p+3}}{2 b^3 d (p+3)} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^5*(a + b*(c + d*x)^2)^p,x]

[Out]

(a^2*(a + b*(c + d*x)^2)^(1 + p))/(2*b^3*d*(1 + p)) - (a*(a + b*(c + d*x)^2)^(2
+ p))/(b^3*d*(2 + p)) + (a + b*(c + d*x)^2)^(3 + p)/(2*b^3*d*(3 + p))

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Rubi in Sympy [A]  time = 25.7369, size = 73, normalized size = 0.78 \[ \frac{a^{2} \left (a + b \left (c + d x\right )^{2}\right )^{p + 1}}{2 b^{3} d \left (p + 1\right )} - \frac{a \left (a + b \left (c + d x\right )^{2}\right )^{p + 2}}{b^{3} d \left (p + 2\right )} + \frac{\left (a + b \left (c + d x\right )^{2}\right )^{p + 3}}{2 b^{3} d \left (p + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**5*(a+b*(d*x+c)**2)**p,x)

[Out]

a**2*(a + b*(c + d*x)**2)**(p + 1)/(2*b**3*d*(p + 1)) - a*(a + b*(c + d*x)**2)**
(p + 2)/(b**3*d*(p + 2)) + (a + b*(c + d*x)**2)**(p + 3)/(2*b**3*d*(p + 3))

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Mathematica [A]  time = 0.132039, size = 91, normalized size = 0.98 \[ \frac{\left (a+b (c+d x)^2\right )^p \left (2 a^3-2 a^2 b p (c+d x)^2+a b^2 p (p+1) (c+d x)^4+b^3 (p+1) (p+2) (c+d x)^6\right )}{2 b^3 d (p+1) (p+2) (p+3)} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^5*(a + b*(c + d*x)^2)^p,x]

[Out]

((a + b*(c + d*x)^2)^p*(2*a^3 - 2*a^2*b*p*(c + d*x)^2 + a*b^2*p*(1 + p)*(c + d*x
)^4 + b^3*(1 + p)*(2 + p)*(c + d*x)^6))/(2*b^3*d*(1 + p)*(2 + p)*(3 + p))

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Maple [B]  time = 0.022, size = 289, normalized size = 3.1 \[{\frac{ \left ( b{d}^{2}{x}^{2}+2\,bcdx+b{c}^{2}+a \right ) ^{1+p} \left ({b}^{2}{d}^{4}{p}^{2}{x}^{4}+4\,{b}^{2}c{d}^{3}{p}^{2}{x}^{3}+3\,{b}^{2}{d}^{4}p{x}^{4}+6\,{b}^{2}{c}^{2}{d}^{2}{p}^{2}{x}^{2}+12\,{b}^{2}c{d}^{3}p{x}^{3}+2\,{d}^{4}{x}^{4}{b}^{2}+4\,{b}^{2}{c}^{3}d{p}^{2}x+18\,{b}^{2}{c}^{2}{d}^{2}p{x}^{2}+8\,c{d}^{3}{x}^{3}{b}^{2}+{b}^{2}{c}^{4}{p}^{2}+12\,{b}^{2}{c}^{3}dpx+12\,{b}^{2}{c}^{2}{d}^{2}{x}^{2}-2\,ab{d}^{2}p{x}^{2}+3\,{b}^{2}{c}^{4}p+8\,{b}^{2}{c}^{3}dx-4\,abcdpx-2\,ab{d}^{2}{x}^{2}+2\,{b}^{2}{c}^{4}-2\,ab{c}^{2}p-4\,abcdx-2\,ab{c}^{2}+2\,{a}^{2} \right ) }{2\,{b}^{3}d \left ({p}^{3}+6\,{p}^{2}+11\,p+6 \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^5*(a+b*(d*x+c)^2)^p,x)

[Out]

1/2*(b*d^2*x^2+2*b*c*d*x+b*c^2+a)^(1+p)*(b^2*d^4*p^2*x^4+4*b^2*c*d^3*p^2*x^3+3*b
^2*d^4*p*x^4+6*b^2*c^2*d^2*p^2*x^2+12*b^2*c*d^3*p*x^3+2*b^2*d^4*x^4+4*b^2*c^3*d*
p^2*x+18*b^2*c^2*d^2*p*x^2+8*b^2*c*d^3*x^3+b^2*c^4*p^2+12*b^2*c^3*d*p*x+12*b^2*c
^2*d^2*x^2-2*a*b*d^2*p*x^2+3*b^2*c^4*p+8*b^2*c^3*d*x-4*a*b*c*d*p*x-2*a*b*d^2*x^2
+2*b^2*c^4-2*a*b*c^2*p-4*a*b*c*d*x-2*a*b*c^2+2*a^2)/b^3/d/(p^3+6*p^2+11*p+6)

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Maxima [A]  time = 1.5068, size = 405, normalized size = 4.35 \[ \frac{{\left ({\left (p^{2} + 3 \, p + 2\right )} b^{3} d^{6} x^{6} + 6 \,{\left (p^{2} + 3 \, p + 2\right )} b^{3} c d^{5} x^{5} +{\left (p^{2} + 3 \, p + 2\right )} b^{3} c^{6} +{\left (p^{2} + p\right )} a b^{2} c^{4} - 2 \, a^{2} b c^{2} p +{\left (15 \,{\left (p^{2} + 3 \, p + 2\right )} b^{3} c^{2} d^{4} +{\left (p^{2} + p\right )} a b^{2} d^{4}\right )} x^{4} + 4 \,{\left (5 \,{\left (p^{2} + 3 \, p + 2\right )} b^{3} c^{3} d^{3} +{\left (p^{2} + p\right )} a b^{2} c d^{3}\right )} x^{3} + 2 \, a^{3} +{\left (15 \,{\left (p^{2} + 3 \, p + 2\right )} b^{3} c^{4} d^{2} + 6 \,{\left (p^{2} + p\right )} a b^{2} c^{2} d^{2} - 2 \, a^{2} b d^{2} p\right )} x^{2} + 2 \,{\left (3 \,{\left (p^{2} + 3 \, p + 2\right )} b^{3} c^{5} d + 2 \,{\left (p^{2} + p\right )} a b^{2} c^{3} d - 2 \, a^{2} b c d p\right )} x\right )}{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )}^{p}}{2 \,{\left (p^{3} + 6 \, p^{2} + 11 \, p + 6\right )} b^{3} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^5*((d*x + c)^2*b + a)^p,x, algorithm="maxima")

[Out]

1/2*((p^2 + 3*p + 2)*b^3*d^6*x^6 + 6*(p^2 + 3*p + 2)*b^3*c*d^5*x^5 + (p^2 + 3*p
+ 2)*b^3*c^6 + (p^2 + p)*a*b^2*c^4 - 2*a^2*b*c^2*p + (15*(p^2 + 3*p + 2)*b^3*c^2
*d^4 + (p^2 + p)*a*b^2*d^4)*x^4 + 4*(5*(p^2 + 3*p + 2)*b^3*c^3*d^3 + (p^2 + p)*a
*b^2*c*d^3)*x^3 + 2*a^3 + (15*(p^2 + 3*p + 2)*b^3*c^4*d^2 + 6*(p^2 + p)*a*b^2*c^
2*d^2 - 2*a^2*b*d^2*p)*x^2 + 2*(3*(p^2 + 3*p + 2)*b^3*c^5*d + 2*(p^2 + p)*a*b^2*
c^3*d - 2*a^2*b*c*d*p)*x)*(b*d^2*x^2 + 2*b*c*d*x + b*c^2 + a)^p/((p^3 + 6*p^2 +
11*p + 6)*b^3*d)

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Fricas [A]  time = 0.232473, size = 591, normalized size = 6.35 \[ \frac{{\left (2 \, b^{3} c^{6} +{\left (b^{3} d^{6} p^{2} + 3 \, b^{3} d^{6} p + 2 \, b^{3} d^{6}\right )} x^{6} + 6 \,{\left (b^{3} c d^{5} p^{2} + 3 \, b^{3} c d^{5} p + 2 \, b^{3} c d^{5}\right )} x^{5} +{\left (30 \, b^{3} c^{2} d^{4} +{\left (15 \, b^{3} c^{2} + a b^{2}\right )} d^{4} p^{2} +{\left (45 \, b^{3} c^{2} + a b^{2}\right )} d^{4} p\right )} x^{4} + 4 \,{\left (10 \, b^{3} c^{3} d^{3} +{\left (5 \, b^{3} c^{3} + a b^{2} c\right )} d^{3} p^{2} +{\left (15 \, b^{3} c^{3} + a b^{2} c\right )} d^{3} p\right )} x^{3} + 2 \, a^{3} +{\left (b^{3} c^{6} + a b^{2} c^{4}\right )} p^{2} +{\left (30 \, b^{3} c^{4} d^{2} + 3 \,{\left (5 \, b^{3} c^{4} + 2 \, a b^{2} c^{2}\right )} d^{2} p^{2} +{\left (45 \, b^{3} c^{4} + 6 \, a b^{2} c^{2} - 2 \, a^{2} b\right )} d^{2} p\right )} x^{2} +{\left (3 \, b^{3} c^{6} + a b^{2} c^{4} - 2 \, a^{2} b c^{2}\right )} p + 2 \,{\left (6 \, b^{3} c^{5} d +{\left (3 \, b^{3} c^{5} + 2 \, a b^{2} c^{3}\right )} d p^{2} +{\left (9 \, b^{3} c^{5} + 2 \, a b^{2} c^{3} - 2 \, a^{2} b c\right )} d p\right )} x\right )}{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )}^{p}}{2 \,{\left (b^{3} d p^{3} + 6 \, b^{3} d p^{2} + 11 \, b^{3} d p + 6 \, b^{3} d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^5*((d*x + c)^2*b + a)^p,x, algorithm="fricas")

[Out]

1/2*(2*b^3*c^6 + (b^3*d^6*p^2 + 3*b^3*d^6*p + 2*b^3*d^6)*x^6 + 6*(b^3*c*d^5*p^2
+ 3*b^3*c*d^5*p + 2*b^3*c*d^5)*x^5 + (30*b^3*c^2*d^4 + (15*b^3*c^2 + a*b^2)*d^4*
p^2 + (45*b^3*c^2 + a*b^2)*d^4*p)*x^4 + 4*(10*b^3*c^3*d^3 + (5*b^3*c^3 + a*b^2*c
)*d^3*p^2 + (15*b^3*c^3 + a*b^2*c)*d^3*p)*x^3 + 2*a^3 + (b^3*c^6 + a*b^2*c^4)*p^
2 + (30*b^3*c^4*d^2 + 3*(5*b^3*c^4 + 2*a*b^2*c^2)*d^2*p^2 + (45*b^3*c^4 + 6*a*b^
2*c^2 - 2*a^2*b)*d^2*p)*x^2 + (3*b^3*c^6 + a*b^2*c^4 - 2*a^2*b*c^2)*p + 2*(6*b^3
*c^5*d + (3*b^3*c^5 + 2*a*b^2*c^3)*d*p^2 + (9*b^3*c^5 + 2*a*b^2*c^3 - 2*a^2*b*c)
*d*p)*x)*(b*d^2*x^2 + 2*b*c*d*x + b*c^2 + a)^p/(b^3*d*p^3 + 6*b^3*d*p^2 + 11*b^3
*d*p + 6*b^3*d)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x+c)**5*(a+b*(d*x+c)**2)**p,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.220907, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^5*((d*x + c)^2*b + a)^p,x, algorithm="giac")

[Out]

Done