3.1425 \(\int (b d+2 c d x)^5 \left (a+b x+c x^2\right )^p \, dx\)

Optimal. Leaf size=121 \[ \frac{2 d^5 \left (b^2-4 a c\right ) (b+2 c x)^2 \left (a+b x+c x^2\right )^{p+1}}{(p+2) (p+3)}+\frac{2 d^5 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{p+1}}{(p+1) (p+2) (p+3)}+\frac{d^5 (b+2 c x)^4 \left (a+b x+c x^2\right )^{p+1}}{p+3} \]

[Out]

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

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Rubi [A]  time = 0.182108, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{2 d^5 \left (b^2-4 a c\right ) (b+2 c x)^2 \left (a+b x+c x^2\right )^{p+1}}{(p+2) (p+3)}+\frac{2 d^5 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{p+1}}{(p+1) (p+2) (p+3)}+\frac{d^5 (b+2 c x)^4 \left (a+b x+c x^2\right )^{p+1}}{p+3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 41.9559, size = 109, normalized size = 0.9 \[ \frac{d^{5} \left (b + 2 c x\right )^{4} \left (a + b x + c x^{2}\right )^{p + 1}}{p + 3} + \frac{2 d^{5} \left (b + 2 c x\right )^{2} \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{p + 1}}{\left (p + 2\right ) \left (p + 3\right )} + \frac{2 d^{5} \left (- 4 a c + b^{2}\right )^{2} \left (a + b x + c x^{2}\right )^{p + 1}}{\left (p + 1\right ) \left (p + 2\right ) \left (p + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.118788, size = 132, normalized size = 1.09 \[ \frac{d^5 (a+x (b+c x))^p \left (-\frac{p \left (b^2-4 a c\right ) (b+2 c x)^4}{2 (p+2) (p+3)}-\frac{p \left (b^2-4 a c\right )^2 (b+2 c x)^2}{(p+1) (p+2) (p+3)}-\frac{\left (b^2-4 a c\right )^3}{(p+1) (p+2) (p+3)}+\frac{(b+2 c x)^6}{2 p+6}\right )}{2 c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.019, size = 233, normalized size = 1.9 \[{\frac{ \left ( c{x}^{2}+bx+a \right ) ^{1+p}{d}^{5} \left ( 16\,{c}^{4}{p}^{2}{x}^{4}+32\,b{c}^{3}{p}^{2}{x}^{3}+48\,{c}^{4}p{x}^{4}+24\,{b}^{2}{c}^{2}{p}^{2}{x}^{2}+96\,b{c}^{3}p{x}^{3}+32\,{c}^{4}{x}^{4}-32\,a{c}^{3}p{x}^{2}+8\,{b}^{3}c{p}^{2}x+80\,{b}^{2}{c}^{2}p{x}^{2}+64\,b{c}^{3}{x}^{3}-32\,ab{c}^{2}px-32\,{x}^{2}a{c}^{3}+{b}^{4}{p}^{2}+32\,{b}^{3}cpx+56\,{x}^{2}{b}^{2}{c}^{2}-8\,a{b}^{2}cp-32\,xab{c}^{2}+5\,{b}^{4}p+24\,{b}^{3}cx+32\,{a}^{2}{c}^{2}-24\,ac{b}^{2}+6\,{b}^{4} \right ) }{{p}^{3}+6\,{p}^{2}+11\,p+6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(c*x^2+b*x+a)^(1+p)*d^5*(16*c^4*p^2*x^4+32*b*c^3*p^2*x^3+48*c^4*p*x^4+24*b^2*c^2
*p^2*x^2+96*b*c^3*p*x^3+32*c^4*x^4-32*a*c^3*p*x^2+8*b^3*c*p^2*x+80*b^2*c^2*p*x^2
+64*b*c^3*x^3-32*a*b*c^2*p*x-32*a*c^3*x^2+b^4*p^2+32*b^3*c*p*x+56*b^2*c^2*x^2-8*
a*b^2*c*p-32*a*b*c^2*x+5*b^4*p+24*b^3*c*x+32*a^2*c^2-24*a*b^2*c+6*b^4)/(p^3+6*p^
2+11*p+6)

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Maxima [A]  time = 0.751076, size = 398, normalized size = 3.29 \[ \frac{{\left (16 \,{\left (p^{2} + 3 \, p + 2\right )} c^{5} d^{5} x^{6} + 48 \,{\left (p^{2} + 3 \, p + 2\right )} b c^{4} d^{5} x^{5} +{\left (p^{2} + 5 \, p + 6\right )} a b^{4} d^{5} - 8 \, a^{2} b^{2} c d^{5}{\left (p + 3\right )} + 32 \, a^{3} c^{2} d^{5} + 8 \,{\left ({\left (7 \, p^{2} + 22 \, p + 15\right )} b^{2} c^{3} d^{5} + 2 \,{\left (p^{2} + p\right )} a c^{4} d^{5}\right )} x^{4} + 16 \,{\left ({\left (2 \, p^{2} + 7 \, p + 5\right )} b^{3} c^{2} d^{5} + 2 \,{\left (p^{2} + p\right )} a b c^{3} d^{5}\right )} x^{3} +{\left ({\left (9 \, p^{2} + 37 \, p + 30\right )} b^{4} c d^{5} + 8 \,{\left (3 \, p^{2} + 5 \, p\right )} a b^{2} c^{2} d^{5} - 32 \, a^{2} c^{3} d^{5} p\right )} x^{2} +{\left ({\left (p^{2} + 5 \, p + 6\right )} b^{5} d^{5} + 8 \,{\left (p^{2} + 3 \, p\right )} a b^{3} c d^{5} - 32 \, a^{2} b c^{2} d^{5} p\right )} x\right )}{\left (c x^{2} + b x + a\right )}^{p}}{p^{3} + 6 \, p^{2} + 11 \, p + 6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(16*(p^2 + 3*p + 2)*c^5*d^5*x^6 + 48*(p^2 + 3*p + 2)*b*c^4*d^5*x^5 + (p^2 + 5*p
+ 6)*a*b^4*d^5 - 8*a^2*b^2*c*d^5*(p + 3) + 32*a^3*c^2*d^5 + 8*((7*p^2 + 22*p + 1
5)*b^2*c^3*d^5 + 2*(p^2 + p)*a*c^4*d^5)*x^4 + 16*((2*p^2 + 7*p + 5)*b^3*c^2*d^5
+ 2*(p^2 + p)*a*b*c^3*d^5)*x^3 + ((9*p^2 + 37*p + 30)*b^4*c*d^5 + 8*(3*p^2 + 5*p
)*a*b^2*c^2*d^5 - 32*a^2*c^3*d^5*p)*x^2 + ((p^2 + 5*p + 6)*b^5*d^5 + 8*(p^2 + 3*
p)*a*b^3*c*d^5 - 32*a^2*b*c^2*d^5*p)*x)*(c*x^2 + b*x + a)^p/(p^3 + 6*p^2 + 11*p
+ 6)

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Fricas [A]  time = 0.23213, size = 541, normalized size = 4.47 \[ \frac{{\left (a b^{4} d^{5} p^{2} +{\left (5 \, a b^{4} - 8 \, a^{2} b^{2} c\right )} d^{5} p + 16 \,{\left (c^{5} d^{5} p^{2} + 3 \, c^{5} d^{5} p + 2 \, c^{5} d^{5}\right )} x^{6} + 2 \,{\left (3 \, a b^{4} - 12 \, a^{2} b^{2} c + 16 \, a^{3} c^{2}\right )} d^{5} + 48 \,{\left (b c^{4} d^{5} p^{2} + 3 \, b c^{4} d^{5} p + 2 \, b c^{4} d^{5}\right )} x^{5} + 8 \,{\left (15 \, b^{2} c^{3} d^{5} +{\left (7 \, b^{2} c^{3} + 2 \, a c^{4}\right )} d^{5} p^{2} + 2 \,{\left (11 \, b^{2} c^{3} + a c^{4}\right )} d^{5} p\right )} x^{4} + 16 \,{\left (5 \, b^{3} c^{2} d^{5} + 2 \,{\left (b^{3} c^{2} + a b c^{3}\right )} d^{5} p^{2} +{\left (7 \, b^{3} c^{2} + 2 \, a b c^{3}\right )} d^{5} p\right )} x^{3} +{\left (30 \, b^{4} c d^{5} + 3 \,{\left (3 \, b^{4} c + 8 \, a b^{2} c^{2}\right )} d^{5} p^{2} +{\left (37 \, b^{4} c + 40 \, a b^{2} c^{2} - 32 \, a^{2} c^{3}\right )} d^{5} p\right )} x^{2} +{\left (6 \, b^{5} d^{5} +{\left (b^{5} + 8 \, a b^{3} c\right )} d^{5} p^{2} +{\left (5 \, b^{5} + 24 \, a b^{3} c - 32 \, a^{2} b c^{2}\right )} d^{5} p\right )} x\right )}{\left (c x^{2} + b x + a\right )}^{p}}{p^{3} + 6 \, p^{2} + 11 \, p + 6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a*b^4*d^5*p^2 + (5*a*b^4 - 8*a^2*b^2*c)*d^5*p + 16*(c^5*d^5*p^2 + 3*c^5*d^5*p +
2*c^5*d^5)*x^6 + 2*(3*a*b^4 - 12*a^2*b^2*c + 16*a^3*c^2)*d^5 + 48*(b*c^4*d^5*p^
2 + 3*b*c^4*d^5*p + 2*b*c^4*d^5)*x^5 + 8*(15*b^2*c^3*d^5 + (7*b^2*c^3 + 2*a*c^4)
*d^5*p^2 + 2*(11*b^2*c^3 + a*c^4)*d^5*p)*x^4 + 16*(5*b^3*c^2*d^5 + 2*(b^3*c^2 +
a*b*c^3)*d^5*p^2 + (7*b^3*c^2 + 2*a*b*c^3)*d^5*p)*x^3 + (30*b^4*c*d^5 + 3*(3*b^4
*c + 8*a*b^2*c^2)*d^5*p^2 + (37*b^4*c + 40*a*b^2*c^2 - 32*a^2*c^3)*d^5*p)*x^2 +
(6*b^5*d^5 + (b^5 + 8*a*b^3*c)*d^5*p^2 + (5*b^5 + 24*a*b^3*c - 32*a^2*b*c^2)*d^5
*p)*x)*(c*x^2 + b*x + a)^p/(p^3 + 6*p^2 + 11*p + 6)

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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((2*c*d*x+b*d)**5*(c*x**2+b*x+a)**p,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.235653, size = 1276, normalized size = 10.55 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(16*c^5*d^5*p^2*x^6*e^(p*ln(c*x^2 + b*x + a)) + 48*b*c^4*d^5*p^2*x^5*e^(p*ln(c*x
^2 + b*x + a)) + 48*c^5*d^5*p*x^6*e^(p*ln(c*x^2 + b*x + a)) + 56*b^2*c^3*d^5*p^2
*x^4*e^(p*ln(c*x^2 + b*x + a)) + 16*a*c^4*d^5*p^2*x^4*e^(p*ln(c*x^2 + b*x + a))
+ 144*b*c^4*d^5*p*x^5*e^(p*ln(c*x^2 + b*x + a)) + 32*c^5*d^5*x^6*e^(p*ln(c*x^2 +
b*x + a)) + 32*b^3*c^2*d^5*p^2*x^3*e^(p*ln(c*x^2 + b*x + a)) + 32*a*b*c^3*d^5*p
^2*x^3*e^(p*ln(c*x^2 + b*x + a)) + 176*b^2*c^3*d^5*p*x^4*e^(p*ln(c*x^2 + b*x + a
)) + 16*a*c^4*d^5*p*x^4*e^(p*ln(c*x^2 + b*x + a)) + 96*b*c^4*d^5*x^5*e^(p*ln(c*x
^2 + b*x + a)) + 9*b^4*c*d^5*p^2*x^2*e^(p*ln(c*x^2 + b*x + a)) + 24*a*b^2*c^2*d^
5*p^2*x^2*e^(p*ln(c*x^2 + b*x + a)) + 112*b^3*c^2*d^5*p*x^3*e^(p*ln(c*x^2 + b*x
+ a)) + 32*a*b*c^3*d^5*p*x^3*e^(p*ln(c*x^2 + b*x + a)) + 120*b^2*c^3*d^5*x^4*e^(
p*ln(c*x^2 + b*x + a)) + b^5*d^5*p^2*x*e^(p*ln(c*x^2 + b*x + a)) + 8*a*b^3*c*d^5
*p^2*x*e^(p*ln(c*x^2 + b*x + a)) + 37*b^4*c*d^5*p*x^2*e^(p*ln(c*x^2 + b*x + a))
+ 40*a*b^2*c^2*d^5*p*x^2*e^(p*ln(c*x^2 + b*x + a)) - 32*a^2*c^3*d^5*p*x^2*e^(p*l
n(c*x^2 + b*x + a)) + 80*b^3*c^2*d^5*x^3*e^(p*ln(c*x^2 + b*x + a)) + a*b^4*d^5*p
^2*e^(p*ln(c*x^2 + b*x + a)) + 5*b^5*d^5*p*x*e^(p*ln(c*x^2 + b*x + a)) + 24*a*b^
3*c*d^5*p*x*e^(p*ln(c*x^2 + b*x + a)) - 32*a^2*b*c^2*d^5*p*x*e^(p*ln(c*x^2 + b*x
+ a)) + 30*b^4*c*d^5*x^2*e^(p*ln(c*x^2 + b*x + a)) + 5*a*b^4*d^5*p*e^(p*ln(c*x^
2 + b*x + a)) - 8*a^2*b^2*c*d^5*p*e^(p*ln(c*x^2 + b*x + a)) + 6*b^5*d^5*x*e^(p*l
n(c*x^2 + b*x + a)) + 6*a*b^4*d^5*e^(p*ln(c*x^2 + b*x + a)) - 24*a^2*b^2*c*d^5*e
^(p*ln(c*x^2 + b*x + a)) + 32*a^3*c^2*d^5*e^(p*ln(c*x^2 + b*x + a)))/(p^3 + 6*p^
2 + 11*p + 6)