3.292 \(\int (a+b x^n)^2 (d+e x^n)^3 \, dx\)

Optimal. Leaf size=158 \[ \frac{d x^{2 n+1} \left (3 a^2 e^2+6 a b d e+b^2 d^2\right )}{2 n+1}+\frac{e x^{3 n+1} \left (a^2 e^2+6 a b d e+3 b^2 d^2\right )}{3 n+1}+a^2 d^3 x+\frac{a d^2 x^{n+1} (3 a e+2 b d)}{n+1}+\frac{b e^2 x^{4 n+1} (2 a e+3 b d)}{4 n+1}+\frac{b^2 e^3 x^{5 n+1}}{5 n+1} \]

[Out]

a^2*d^3*x + (a*d^2*(2*b*d + 3*a*e)*x^(1 + n))/(1 + n) + (d*(b^2*d^2 + 6*a*b*d*e + 3*a^2*e^2)*x^(1 + 2*n))/(1 +
2*n) + (e*(3*b^2*d^2 + 6*a*b*d*e + a^2*e^2)*x^(1 + 3*n))/(1 + 3*n) + (b*e^2*(3*b*d + 2*a*e)*x^(1 + 4*n))/(1 +
4*n) + (b^2*e^3*x^(1 + 5*n))/(1 + 5*n)

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Rubi [A]  time = 0.12993, antiderivative size = 158, normalized size of antiderivative = 1., 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 = {373} \[ \frac{d x^{2 n+1} \left (3 a^2 e^2+6 a b d e+b^2 d^2\right )}{2 n+1}+\frac{e x^{3 n+1} \left (a^2 e^2+6 a b d e+3 b^2 d^2\right )}{3 n+1}+a^2 d^3 x+\frac{a d^2 x^{n+1} (3 a e+2 b d)}{n+1}+\frac{b e^2 x^{4 n+1} (2 a e+3 b d)}{4 n+1}+\frac{b^2 e^3 x^{5 n+1}}{5 n+1} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^n)^2*(d + e*x^n)^3,x]

[Out]

a^2*d^3*x + (a*d^2*(2*b*d + 3*a*e)*x^(1 + n))/(1 + n) + (d*(b^2*d^2 + 6*a*b*d*e + 3*a^2*e^2)*x^(1 + 2*n))/(1 +
2*n) + (e*(3*b^2*d^2 + 6*a*b*d*e + a^2*e^2)*x^(1 + 3*n))/(1 + 3*n) + (b*e^2*(3*b*d + 2*a*e)*x^(1 + 4*n))/(1 +
4*n) + (b^2*e^3*x^(1 + 5*n))/(1 + 5*n)

Rule 373

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n
)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && IGtQ[q, 0]

Rubi steps

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

Mathematica [A]  time = 0.158896, size = 149, normalized size = 0.94 \[ x \left (\frac{d x^{2 n} \left (3 a^2 e^2+6 a b d e+b^2 d^2\right )}{2 n+1}+\frac{e x^{3 n} \left (a^2 e^2+6 a b d e+3 b^2 d^2\right )}{3 n+1}+a^2 d^3+\frac{a d^2 x^n (3 a e+2 b d)}{n+1}+\frac{b e^2 x^{4 n} (2 a e+3 b d)}{4 n+1}+\frac{b^2 e^3 x^{5 n}}{5 n+1}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

x*(a^2*d^3 + (a*d^2*(2*b*d + 3*a*e)*x^n)/(1 + n) + (d*(b^2*d^2 + 6*a*b*d*e + 3*a^2*e^2)*x^(2*n))/(1 + 2*n) + (
e*(3*b^2*d^2 + 6*a*b*d*e + a^2*e^2)*x^(3*n))/(1 + 3*n) + (b*e^2*(3*b*d + 2*a*e)*x^(4*n))/(1 + 4*n) + (b^2*e^3*
x^(5*n))/(1 + 5*n))

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Maple [A]  time = 0.055, size = 164, normalized size = 1. \begin{align*}{a}^{2}{d}^{3}x+{\frac{{b}^{2}{e}^{3}x \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{5}}{1+5\,n}}+{\frac{d \left ( 3\,{a}^{2}{e}^{2}+6\,abde+{b}^{2}{d}^{2} \right ) x \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{1+2\,n}}+{\frac{e \left ({a}^{2}{e}^{2}+6\,abde+3\,{b}^{2}{d}^{2} \right ) x \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{1+3\,n}}+{\frac{a{d}^{2} \left ( 3\,ae+2\,bd \right ) x{{\rm e}^{n\ln \left ( x \right ) }}}{1+n}}+{\frac{b{e}^{2} \left ( 2\,ae+3\,bd \right ) x \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{4}}{1+4\,n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*x^n)^2*(d+e*x^n)^3,x)

[Out]

a^2*d^3*x+b^2*e^3/(1+5*n)*x*exp(n*ln(x))^5+d*(3*a^2*e^2+6*a*b*d*e+b^2*d^2)/(1+2*n)*x*exp(n*ln(x))^2+e*(a^2*e^2
+6*a*b*d*e+3*b^2*d^2)/(1+3*n)*x*exp(n*ln(x))^3+a*d^2*(3*a*e+2*b*d)/(1+n)*x*exp(n*ln(x))+b*e^2*(2*a*e+3*b*d)/(1
+4*n)*x*exp(n*ln(x))^4

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.70087, size = 1434, normalized size = 9.08 \begin{align*} \frac{{\left (24 \, b^{2} e^{3} n^{4} + 50 \, b^{2} e^{3} n^{3} + 35 \, b^{2} e^{3} n^{2} + 10 \, b^{2} e^{3} n + b^{2} e^{3}\right )} x x^{5 \, n} +{\left (3 \, b^{2} d e^{2} + 2 \, a b e^{3} + 30 \,{\left (3 \, b^{2} d e^{2} + 2 \, a b e^{3}\right )} n^{4} + 61 \,{\left (3 \, b^{2} d e^{2} + 2 \, a b e^{3}\right )} n^{3} + 41 \,{\left (3 \, b^{2} d e^{2} + 2 \, a b e^{3}\right )} n^{2} + 11 \,{\left (3 \, b^{2} d e^{2} + 2 \, a b e^{3}\right )} n\right )} x x^{4 \, n} +{\left (3 \, b^{2} d^{2} e + 6 \, a b d e^{2} + a^{2} e^{3} + 40 \,{\left (3 \, b^{2} d^{2} e + 6 \, a b d e^{2} + a^{2} e^{3}\right )} n^{4} + 78 \,{\left (3 \, b^{2} d^{2} e + 6 \, a b d e^{2} + a^{2} e^{3}\right )} n^{3} + 49 \,{\left (3 \, b^{2} d^{2} e + 6 \, a b d e^{2} + a^{2} e^{3}\right )} n^{2} + 12 \,{\left (3 \, b^{2} d^{2} e + 6 \, a b d e^{2} + a^{2} e^{3}\right )} n\right )} x x^{3 \, n} +{\left (b^{2} d^{3} + 6 \, a b d^{2} e + 3 \, a^{2} d e^{2} + 60 \,{\left (b^{2} d^{3} + 6 \, a b d^{2} e + 3 \, a^{2} d e^{2}\right )} n^{4} + 107 \,{\left (b^{2} d^{3} + 6 \, a b d^{2} e + 3 \, a^{2} d e^{2}\right )} n^{3} + 59 \,{\left (b^{2} d^{3} + 6 \, a b d^{2} e + 3 \, a^{2} d e^{2}\right )} n^{2} + 13 \,{\left (b^{2} d^{3} + 6 \, a b d^{2} e + 3 \, a^{2} d e^{2}\right )} n\right )} x x^{2 \, n} +{\left (2 \, a b d^{3} + 3 \, a^{2} d^{2} e + 120 \,{\left (2 \, a b d^{3} + 3 \, a^{2} d^{2} e\right )} n^{4} + 154 \,{\left (2 \, a b d^{3} + 3 \, a^{2} d^{2} e\right )} n^{3} + 71 \,{\left (2 \, a b d^{3} + 3 \, a^{2} d^{2} e\right )} n^{2} + 14 \,{\left (2 \, a b d^{3} + 3 \, a^{2} d^{2} e\right )} n\right )} x x^{n} +{\left (120 \, a^{2} d^{3} n^{5} + 274 \, a^{2} d^{3} n^{4} + 225 \, a^{2} d^{3} n^{3} + 85 \, a^{2} d^{3} n^{2} + 15 \, a^{2} d^{3} n + a^{2} d^{3}\right )} x}{120 \, n^{5} + 274 \, n^{4} + 225 \, n^{3} + 85 \, n^{2} + 15 \, n + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((24*b^2*e^3*n^4 + 50*b^2*e^3*n^3 + 35*b^2*e^3*n^2 + 10*b^2*e^3*n + b^2*e^3)*x*x^(5*n) + (3*b^2*d*e^2 + 2*a*b*
e^3 + 30*(3*b^2*d*e^2 + 2*a*b*e^3)*n^4 + 61*(3*b^2*d*e^2 + 2*a*b*e^3)*n^3 + 41*(3*b^2*d*e^2 + 2*a*b*e^3)*n^2 +
11*(3*b^2*d*e^2 + 2*a*b*e^3)*n)*x*x^(4*n) + (3*b^2*d^2*e + 6*a*b*d*e^2 + a^2*e^3 + 40*(3*b^2*d^2*e + 6*a*b*d*
e^2 + a^2*e^3)*n^4 + 78*(3*b^2*d^2*e + 6*a*b*d*e^2 + a^2*e^3)*n^3 + 49*(3*b^2*d^2*e + 6*a*b*d*e^2 + a^2*e^3)*n
^2 + 12*(3*b^2*d^2*e + 6*a*b*d*e^2 + a^2*e^3)*n)*x*x^(3*n) + (b^2*d^3 + 6*a*b*d^2*e + 3*a^2*d*e^2 + 60*(b^2*d^
3 + 6*a*b*d^2*e + 3*a^2*d*e^2)*n^4 + 107*(b^2*d^3 + 6*a*b*d^2*e + 3*a^2*d*e^2)*n^3 + 59*(b^2*d^3 + 6*a*b*d^2*e
+ 3*a^2*d*e^2)*n^2 + 13*(b^2*d^3 + 6*a*b*d^2*e + 3*a^2*d*e^2)*n)*x*x^(2*n) + (2*a*b*d^3 + 3*a^2*d^2*e + 120*(
2*a*b*d^3 + 3*a^2*d^2*e)*n^4 + 154*(2*a*b*d^3 + 3*a^2*d^2*e)*n^3 + 71*(2*a*b*d^3 + 3*a^2*d^2*e)*n^2 + 14*(2*a*
b*d^3 + 3*a^2*d^2*e)*n)*x*x^n + (120*a^2*d^3*n^5 + 274*a^2*d^3*n^4 + 225*a^2*d^3*n^3 + 85*a^2*d^3*n^2 + 15*a^2
*d^3*n + a^2*d^3)*x)/(120*n^5 + 274*n^4 + 225*n^3 + 85*n^2 + 15*n + 1)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError

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Giac [B]  time = 1.64841, size = 1278, normalized size = 8.09 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(120*a^2*d^3*n^5*x + 60*b^2*d^3*n^4*x*x^(2*n) + 240*a*b*d^3*n^4*x*x^n + 120*b^2*d^2*n^4*x*x^(3*n)*e + 360*a*b*
d^2*n^4*x*x^(2*n)*e + 360*a^2*d^2*n^4*x*x^n*e + 274*a^2*d^3*n^4*x + 107*b^2*d^3*n^3*x*x^(2*n) + 308*a*b*d^3*n^
3*x*x^n + 90*b^2*d*n^4*x*x^(4*n)*e^2 + 240*a*b*d*n^4*x*x^(3*n)*e^2 + 180*a^2*d*n^4*x*x^(2*n)*e^2 + 234*b^2*d^2
*n^3*x*x^(3*n)*e + 642*a*b*d^2*n^3*x*x^(2*n)*e + 462*a^2*d^2*n^3*x*x^n*e + 225*a^2*d^3*n^3*x + 59*b^2*d^3*n^2*
x*x^(2*n) + 142*a*b*d^3*n^2*x*x^n + 24*b^2*n^4*x*x^(5*n)*e^3 + 60*a*b*n^4*x*x^(4*n)*e^3 + 40*a^2*n^4*x*x^(3*n)
*e^3 + 183*b^2*d*n^3*x*x^(4*n)*e^2 + 468*a*b*d*n^3*x*x^(3*n)*e^2 + 321*a^2*d*n^3*x*x^(2*n)*e^2 + 147*b^2*d^2*n
^2*x*x^(3*n)*e + 354*a*b*d^2*n^2*x*x^(2*n)*e + 213*a^2*d^2*n^2*x*x^n*e + 85*a^2*d^3*n^2*x + 13*b^2*d^3*n*x*x^(
2*n) + 28*a*b*d^3*n*x*x^n + 50*b^2*n^3*x*x^(5*n)*e^3 + 122*a*b*n^3*x*x^(4*n)*e^3 + 78*a^2*n^3*x*x^(3*n)*e^3 +
123*b^2*d*n^2*x*x^(4*n)*e^2 + 294*a*b*d*n^2*x*x^(3*n)*e^2 + 177*a^2*d*n^2*x*x^(2*n)*e^2 + 36*b^2*d^2*n*x*x^(3*
n)*e + 78*a*b*d^2*n*x*x^(2*n)*e + 42*a^2*d^2*n*x*x^n*e + 15*a^2*d^3*n*x + b^2*d^3*x*x^(2*n) + 2*a*b*d^3*x*x^n
+ 35*b^2*n^2*x*x^(5*n)*e^3 + 82*a*b*n^2*x*x^(4*n)*e^3 + 49*a^2*n^2*x*x^(3*n)*e^3 + 33*b^2*d*n*x*x^(4*n)*e^2 +
72*a*b*d*n*x*x^(3*n)*e^2 + 39*a^2*d*n*x*x^(2*n)*e^2 + 3*b^2*d^2*x*x^(3*n)*e + 6*a*b*d^2*x*x^(2*n)*e + 3*a^2*d^
2*x*x^n*e + a^2*d^3*x + 10*b^2*n*x*x^(5*n)*e^3 + 22*a*b*n*x*x^(4*n)*e^3 + 12*a^2*n*x*x^(3*n)*e^3 + 3*b^2*d*x*x
^(4*n)*e^2 + 6*a*b*d*x*x^(3*n)*e^2 + 3*a^2*d*x*x^(2*n)*e^2 + b^2*x*x^(5*n)*e^3 + 2*a*b*x*x^(4*n)*e^3 + a^2*x*x
^(3*n)*e^3)/(120*n^5 + 274*n^4 + 225*n^3 + 85*n^2 + 15*n + 1)