3.214 \(\int \left (a+b x^n\right )^p \left (c+d x^n\right )^3 \, dx\)

Optimal. Leaf size=402 \[ \frac{d x \left (a+b x^n\right )^{p+1} \left (a^2 d^2 \left (2 n^2+3 n+1\right )-a b c d \left (n^2 (p+7)+n (2 p+9)+2\right )+b^2 c^2 \left (n^2 \left (p^2+6 p+11\right )+2 n (p+3)+1\right )\right )}{b^3 (n p+n+1) (n (p+2)+1) (n (p+3)+1)}-\frac{x \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \left (a^3 d^3 \left (2 n^2+3 n+1\right )-3 a^2 b c d^2 (n+1) (n (p+3)+1)+3 a b^2 c^2 d \left (n^2 \left (p^2+5 p+6\right )+n (2 p+5)+1\right )-b^3 c^3 \left (n^3 \left (p^3+6 p^2+11 p+6\right )+n^2 \left (3 p^2+12 p+11\right )+3 n (p+2)+1\right )\right ) \, _2F_1\left (\frac{1}{n},-p;1+\frac{1}{n};-\frac{b x^n}{a}\right )}{b^3 (n p+n+1) (n (p+2)+1) (n (p+3)+1)}-\frac{d x \left (c+d x^n\right ) \left (a+b x^n\right )^{p+1} (a d (2 n+1)-b c (n (p+5)+1))}{b^2 (n (p+2)+1) (n (p+3)+1)}+\frac{d x \left (c+d x^n\right )^2 \left (a+b x^n\right )^{p+1}}{b (n p+3 n+1)} \]

[Out]

(d*(a^2*d^2*(1 + 3*n + 2*n^2) - a*b*c*d*(2 + n^2*(7 + p) + n*(9 + 2*p)) + b^2*c^
2*(1 + 2*n*(3 + p) + n^2*(11 + 6*p + p^2)))*x*(a + b*x^n)^(1 + p))/(b^3*(1 + n +
n*p)*(1 + n*(2 + p))*(1 + n*(3 + p))) - (d*(a*d*(1 + 2*n) - b*c*(1 + n*(5 + p))
)*x*(a + b*x^n)^(1 + p)*(c + d*x^n))/(b^2*(1 + n*(2 + p))*(1 + n*(3 + p))) + (d*
x*(a + b*x^n)^(1 + p)*(c + d*x^n)^2)/(b*(1 + 3*n + n*p)) - ((a^3*d^3*(1 + 3*n +
2*n^2) - 3*a^2*b*c*d^2*(1 + n)*(1 + n*(3 + p)) + 3*a*b^2*c^2*d*(1 + n*(5 + 2*p)
+ n^2*(6 + 5*p + p^2)) - b^3*c^3*(1 + 3*n*(2 + p) + n^2*(11 + 12*p + 3*p^2) + n^
3*(6 + 11*p + 6*p^2 + p^3)))*x*(a + b*x^n)^p*Hypergeometric2F1[n^(-1), -p, 1 + n
^(-1), -((b*x^n)/a)])/(b^3*(1 + n + n*p)*(1 + n*(2 + p))*(1 + n*(3 + p))*(1 + (b
*x^n)/a)^p)

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Rubi [A]  time = 1.28482, antiderivative size = 401, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ \frac{d x \left (a+b x^n\right )^{p+1} \left (a^2 d^2 \left (2 n^2+3 n+1\right )-a b c d \left (n^2 (p+7)+n (2 p+9)+2\right )+b^2 c^2 \left (n^2 \left (p^2+6 p+11\right )+2 n (p+3)+1\right )\right )}{b^3 (n p+n+1) (n (p+2)+1) (n (p+3)+1)}-\frac{x \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \left (a^3 d^3 \left (2 n^2+3 n+1\right )-3 a^2 b c d^2 (n+1) (n (p+3)+1)+3 a b^2 c^2 d \left (n^2 \left (p^2+5 p+6\right )+n (2 p+5)+1\right )-b^3 c^3 \left (n^3 \left (p^3+6 p^2+11 p+6\right )+n^2 \left (3 p^2+12 p+11\right )+3 n (p+2)+1\right )\right ) \, _2F_1\left (\frac{1}{n},-p;1+\frac{1}{n};-\frac{b x^n}{a}\right )}{b^3 (n p+n+1) (n (p+2)+1) (n (p+3)+1)}-\frac{d x \left (c+d x^n\right ) \left (a+b x^n\right )^{p+1} (a d (2 n+1)-b (c n (p+5)+c))}{b^2 (n (p+2)+1) (n (p+3)+1)}+\frac{d x \left (c+d x^n\right )^2 \left (a+b x^n\right )^{p+1}}{b (n (p+3)+1)} \]

Antiderivative was successfully verified.

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

[Out]

(d*(a^2*d^2*(1 + 3*n + 2*n^2) - a*b*c*d*(2 + n^2*(7 + p) + n*(9 + 2*p)) + b^2*c^
2*(1 + 2*n*(3 + p) + n^2*(11 + 6*p + p^2)))*x*(a + b*x^n)^(1 + p))/(b^3*(1 + n +
n*p)*(1 + n*(2 + p))*(1 + n*(3 + p))) - (d*(a*d*(1 + 2*n) - b*(c + c*n*(5 + p))
)*x*(a + b*x^n)^(1 + p)*(c + d*x^n))/(b^2*(1 + n*(2 + p))*(1 + n*(3 + p))) + (d*
x*(a + b*x^n)^(1 + p)*(c + d*x^n)^2)/(b*(1 + n*(3 + p))) - ((a^3*d^3*(1 + 3*n +
2*n^2) - 3*a^2*b*c*d^2*(1 + n)*(1 + n*(3 + p)) + 3*a*b^2*c^2*d*(1 + n*(5 + 2*p)
+ n^2*(6 + 5*p + p^2)) - b^3*c^3*(1 + 3*n*(2 + p) + n^2*(11 + 12*p + 3*p^2) + n^
3*(6 + 11*p + 6*p^2 + p^3)))*x*(a + b*x^n)^p*Hypergeometric2F1[n^(-1), -p, 1 + n
^(-1), -((b*x^n)/a)])/(b^3*(1 + n + n*p)*(1 + n*(2 + p))*(1 + n*(3 + p))*(1 + (b
*x^n)/a)^p)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 0.444014, size = 168, normalized size = 0.42 \[ x \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \left (c^3 \, _2F_1\left (\frac{1}{n},-p;1+\frac{1}{n};-\frac{b x^n}{a}\right )+\frac{3 c^2 d x^n \, _2F_1\left (1+\frac{1}{n},-p;2+\frac{1}{n};-\frac{b x^n}{a}\right )}{n+1}+\frac{3 c d^2 x^{2 n} \, _2F_1\left (2+\frac{1}{n},-p;3+\frac{1}{n};-\frac{b x^n}{a}\right )}{2 n+1}+\frac{d^3 x^{3 n} \, _2F_1\left (3+\frac{1}{n},-p;4+\frac{1}{n};-\frac{b x^n}{a}\right )}{3 n+1}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(x*(a + b*x^n)^p*((3*c^2*d*x^n*Hypergeometric2F1[1 + n^(-1), -p, 2 + n^(-1), -((
b*x^n)/a)])/(1 + n) + (3*c*d^2*x^(2*n)*Hypergeometric2F1[2 + n^(-1), -p, 3 + n^(
-1), -((b*x^n)/a)])/(1 + 2*n) + (d^3*x^(3*n)*Hypergeometric2F1[3 + n^(-1), -p, 4
+ n^(-1), -((b*x^n)/a)])/(1 + 3*n) + c^3*Hypergeometric2F1[n^(-1), -p, 1 + n^(-
1), -((b*x^n)/a)]))/(1 + (b*x^n)/a)^p

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Maple [F]  time = 0.159, size = 0, normalized size = 0. \[ \int \left ( a+b{x}^{n} \right ) ^{p} \left ( c+d{x}^{n} \right ) ^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^n)^p*(c+d*x^n)^3,x)

[Out]

int((a+b*x^n)^p*(c+d*x^n)^3,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (d x^{n} + c\right )}^{3}{\left (b x^{n} + a\right )}^{p}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((d*x^n + c)^3*(b*x^n + a)^p, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (d^{3} x^{3 \, n} + 3 \, c d^{2} x^{2 \, n} + 3 \, c^{2} d x^{n} + c^{3}\right )}{\left (b x^{n} + a\right )}^{p}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((d^3*x^(3*n) + 3*c*d^2*x^(2*n) + 3*c^2*d*x^n + c^3)*(b*x^n + a)^p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError