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]
________________________________________________________________________________________
Rubi [A] time = 0.58, antiderivative size = 401, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {416, 528, 388, 246, 245} \[ -\frac {x \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \left (-3 a^2 b c d^2 (n+1) (n (p+3)+1)+a^3 d^3 \left (2 n^2+3 n+1\right )+3 a b^2 c^2 d \left (n^2 \left (p^2+5 p+6\right )+n (2 p+5)+1\right )-b^3 c^3 \left (n^3 \left (p^3+6 p^2+11 p+6\right )+n^2 \left (3 p^2+12 p+11\right )+3 n (p+2)+1\right )\right ) \, _2F_1\left (\frac {1}{n},-p;1+\frac {1}{n};-\frac {b x^n}{a}\right )}{b^3 (n p+n+1) (n (p+2)+1) (n (p+3)+1)}+\frac {d x \left (a+b x^n\right )^{p+1} \left (a^2 d^2 \left (2 n^2+3 n+1\right )-a b c d \left (n^2 (p+7)+n (2 p+9)+2\right )+b^2 c^2 \left (n^2 \left (p^2+6 p+11\right )+2 n (p+3)+1\right )\right )}{b^3 (n p+n+1) (n (p+2)+1) (n (p+3)+1)}-\frac {d x \left (c+d x^n\right ) \left (a+b x^n\right )^{p+1} (a d (2 n+1)-b (c n (p+5)+c))}{b^2 (n (p+2)+1) (n (p+3)+1)}+\frac {d x \left (c+d x^n\right )^2 \left (a+b x^n\right )^{p+1}}{b (n (p+3)+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 245
Rule 246
Rule 388
Rule 416
Rule 528
Rubi steps
\begin {align*} \int \left (a+b x^n\right )^p \left (c+d x^n\right )^3 \, dx &=\frac {d x \left (a+b x^n\right )^{1+p} \left (c+d x^n\right )^2}{b (1+n (3+p))}+\frac {\int \left (a+b x^n\right )^p \left (c+d x^n\right ) \left (-c (a d-b (c+c n (3+p)))-d (a d (1+2 n)-b (c+c n (5+p))) x^n\right ) \, dx}{b (1+n (3+p))}\\ &=-\frac {d (a d (1+2 n)-b (c+c n (5+p))) x \left (a+b x^n\right )^{1+p} \left (c+d x^n\right )}{b^2 (1+n (2+p)) (1+n (3+p))}+\frac {d x \left (a+b x^n\right )^{1+p} \left (c+d x^n\right )^2}{b (1+n (3+p))}+\frac {\int \left (a+b x^n\right )^p \left (c \left (a^2 d^2 (1+2 n)-a b c d (2+n (7+2 p))+b^2 c^2 \left (1+n (5+2 p)+n^2 \left (6+5 p+p^2\right )\right )\right )+d \left (a^2 d^2 \left (1+3 n+2 n^2\right )-a b c d \left (2+n^2 (7+p)+n (9+2 p)\right )+b^2 c^2 \left (1+2 n (3+p)+n^2 \left (11+6 p+p^2\right )\right )\right ) x^n\right ) \, dx}{b^2 (1+n (2+p)) (1+n (3+p))}\\ &=\frac {d \left (a^2 d^2 \left (1+3 n+2 n^2\right )-a b c d \left (2+n^2 (7+p)+n (9+2 p)\right )+b^2 c^2 \left (1+2 n (3+p)+n^2 \left (11+6 p+p^2\right )\right )\right ) x \left (a+b x^n\right )^{1+p}}{b^3 (1+n+n p) (1+n (2+p)) (1+n (3+p))}-\frac {d (a d (1+2 n)-b (c+c n (5+p))) x \left (a+b x^n\right )^{1+p} \left (c+d x^n\right )}{b^2 (1+n (2+p)) (1+n (3+p))}+\frac {d x \left (a+b x^n\right )^{1+p} \left (c+d x^n\right )^2}{b (1+n (3+p))}-\frac {\left (a^3 d^3 \left (1+3 n+2 n^2\right )-3 a^2 b c d^2 (1+n) (1+n (3+p))+3 a b^2 c^2 d \left (1+n (5+2 p)+n^2 \left (6+5 p+p^2\right )\right )-b^3 c^3 \left (1+3 n (2+p)+n^2 \left (11+12 p+3 p^2\right )+n^3 \left (6+11 p+6 p^2+p^3\right )\right )\right ) \int \left (a+b x^n\right )^p \, dx}{b^3 (1+n+n p) (1+n (2+p)) (1+n (3+p))}\\ &=\frac {d \left (a^2 d^2 \left (1+3 n+2 n^2\right )-a b c d \left (2+n^2 (7+p)+n (9+2 p)\right )+b^2 c^2 \left (1+2 n (3+p)+n^2 \left (11+6 p+p^2\right )\right )\right ) x \left (a+b x^n\right )^{1+p}}{b^3 (1+n+n p) (1+n (2+p)) (1+n (3+p))}-\frac {d (a d (1+2 n)-b (c+c n (5+p))) x \left (a+b x^n\right )^{1+p} \left (c+d x^n\right )}{b^2 (1+n (2+p)) (1+n (3+p))}+\frac {d x \left (a+b x^n\right )^{1+p} \left (c+d x^n\right )^2}{b (1+n (3+p))}-\frac {\left (\left (a^3 d^3 \left (1+3 n+2 n^2\right )-3 a^2 b c d^2 (1+n) (1+n (3+p))+3 a b^2 c^2 d \left (1+n (5+2 p)+n^2 \left (6+5 p+p^2\right )\right )-b^3 c^3 \left (1+3 n (2+p)+n^2 \left (11+12 p+3 p^2\right )+n^3 \left (6+11 p+6 p^2+p^3\right )\right )\right ) \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p}\right ) \int \left (1+\frac {b x^n}{a}\right )^p \, dx}{b^3 (1+n+n p) (1+n (2+p)) (1+n (3+p))}\\ &=\frac {d \left (a^2 d^2 \left (1+3 n+2 n^2\right )-a b c d \left (2+n^2 (7+p)+n (9+2 p)\right )+b^2 c^2 \left (1+2 n (3+p)+n^2 \left (11+6 p+p^2\right )\right )\right ) x \left (a+b x^n\right )^{1+p}}{b^3 (1+n+n p) (1+n (2+p)) (1+n (3+p))}-\frac {d (a d (1+2 n)-b (c+c n (5+p))) x \left (a+b x^n\right )^{1+p} \left (c+d x^n\right )}{b^2 (1+n (2+p)) (1+n (3+p))}+\frac {d x \left (a+b x^n\right )^{1+p} \left (c+d x^n\right )^2}{b (1+n (3+p))}-\frac {\left (a^3 d^3 \left (1+3 n+2 n^2\right )-3 a^2 b c d^2 (1+n) (1+n (3+p))+3 a b^2 c^2 d \left (1+n (5+2 p)+n^2 \left (6+5 p+p^2\right )\right )-b^3 c^3 \left (1+3 n (2+p)+n^2 \left (11+12 p+3 p^2\right )+n^3 \left (6+11 p+6 p^2+p^3\right )\right )\right ) x \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \, _2F_1\left (\frac {1}{n},-p;1+\frac {1}{n};-\frac {b x^n}{a}\right )}{b^3 (1+n+n p) (1+n (2+p)) (1+n (3+p))}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 5.32, 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]
[Out]
________________________________________________________________________________________
fricas [F] time = 1.11, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (d^{3} x^{3 \, n} + 3 \, c d^{2} x^{2 \, n} + 3 \, c^{2} d x^{n} + c^{3}\right )} {\left (b x^{n} + a\right )}^{p}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.74, size = 0, normalized size = 0.00 \[ \int \left (d \,x^{n}+c \right )^{3} \left (b \,x^{n}+a \right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x^{n} + c\right )}^{3} {\left (b x^{n} + a\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (a+b\,x^n\right )}^p\,{\left (c+d\,x^n\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 93.23, size = 199, normalized size = 0.50 \[ \frac {a^{p} c^{3} x \Gamma \left (\frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{n}, - p \\ 1 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (1 + \frac {1}{n}\right )} + \frac {3 a^{p} c^{2} d x x^{n} \Gamma \left (1 + \frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, 1 + \frac {1}{n} \\ 2 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (2 + \frac {1}{n}\right )} + \frac {3 a^{p} c d^{2} x x^{2 n} \Gamma \left (2 + \frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, 2 + \frac {1}{n} \\ 3 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (3 + \frac {1}{n}\right )} + \frac {a^{p} d^{3} x x^{3 n} \Gamma \left (3 + \frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, 3 + \frac {1}{n} \\ 4 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (4 + \frac {1}{n}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________