Optimal. Leaf size=294 \[ \frac {a d x \sqrt {a+b x^n+c x^{2 n}} F_1\left (\frac {1}{n};-\frac {3}{2},-\frac {3}{2};1+\frac {1}{n};-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{\sqrt {\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^n}{\sqrt {b^2-4 a c}+b}+1}}+\frac {a e x^{n+1} \sqrt {a+b x^n+c x^{2 n}} F_1\left (1+\frac {1}{n};-\frac {3}{2},-\frac {3}{2};2+\frac {1}{n};-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{(n+1) \sqrt {\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^n}{\sqrt {b^2-4 a c}+b}+1}} \]
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Rubi [A] time = 0.35, antiderivative size = 294, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {1432, 1348, 429, 1385, 510} \[ \frac {a d x \sqrt {a+b x^n+c x^{2 n}} F_1\left (\frac {1}{n};-\frac {3}{2},-\frac {3}{2};1+\frac {1}{n};-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{\sqrt {\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^n}{\sqrt {b^2-4 a c}+b}+1}}+\frac {a e x^{n+1} \sqrt {a+b x^n+c x^{2 n}} F_1\left (1+\frac {1}{n};-\frac {3}{2},-\frac {3}{2};2+\frac {1}{n};-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{(n+1) \sqrt {\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^n}{\sqrt {b^2-4 a c}+b}+1}} \]
Antiderivative was successfully verified.
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Rule 429
Rule 510
Rule 1348
Rule 1385
Rule 1432
Rubi steps
\begin {align*} \int \left (d+e x^n\right ) \left (a+b x^n+c x^{2 n}\right )^{3/2} \, dx &=\int \left (d \left (a+b x^n+c x^{2 n}\right )^{3/2}+e x^n \left (a+b x^n+c x^{2 n}\right )^{3/2}\right ) \, dx\\ &=d \int \left (a+b x^n+c x^{2 n}\right )^{3/2} \, dx+e \int x^n \left (a+b x^n+c x^{2 n}\right )^{3/2} \, dx\\ &=\frac {\left (a d \sqrt {a+b x^n+c x^{2 n}}\right ) \int \left (1+\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )^{3/2} \left (1+\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )^{3/2} \, dx}{\sqrt {1+\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}}}+\frac {\left (a e \sqrt {a+b x^n+c x^{2 n}}\right ) \int x^n \left (1+\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )^{3/2} \left (1+\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )^{3/2} \, dx}{\sqrt {1+\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}}}\\ &=\frac {a e x^{1+n} \sqrt {a+b x^n+c x^{2 n}} F_1\left (1+\frac {1}{n};-\frac {3}{2},-\frac {3}{2};2+\frac {1}{n};-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{(1+n) \sqrt {1+\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}}}+\frac {a d x \sqrt {a+b x^n+c x^{2 n}} F_1\left (\frac {1}{n};-\frac {3}{2},-\frac {3}{2};1+\frac {1}{n};-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{\sqrt {1+\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}}}\\ \end {align*}
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Mathematica [B] time = 4.53, size = 690, normalized size = 2.35 \[ \frac {x \left (3 n^2 x^n \sqrt {\frac {-\sqrt {b^2-4 a c}+b+2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {\sqrt {b^2-4 a c}+b+2 c x^n}{\sqrt {b^2-4 a c}+b}} F_1\left (1+\frac {1}{n};\frac {1}{2},\frac {1}{2};2+\frac {1}{n};-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}},\frac {2 c x^n}{\sqrt {b^2-4 a c}-b}\right ) \left (16 a^2 c^2 e \left (3 n^2+4 n+1\right )-4 a b^2 c e \left (6 n^2+14 n+5\right )+8 a b c^2 d \left (12 n^2+11 n+2\right )+b^4 e \left (3 n^2+8 n+4\right )-2 b^3 c d \left (4 n^2+9 n+2\right )\right )+2 (n+1) \left (3 a n^2 \sqrt {\frac {-\sqrt {b^2-4 a c}+b+2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {\sqrt {b^2-4 a c}+b+2 c x^n}{\sqrt {b^2-4 a c}+b}} F_1\left (\frac {1}{n};\frac {1}{2},\frac {1}{2};1+\frac {1}{n};-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}},\frac {2 c x^n}{\sqrt {b^2-4 a c}-b}\right ) \left (-4 a b c e (5 n+2)+8 a c^2 d \left (8 n^2+6 n+1\right )+b^3 e (3 n+2)-2 b^2 c d (4 n+1)\right )+\left (a+x^n \left (b+c x^n\right )\right ) \left (4 a c \left (3 b e (5 n+2) n^2+2 c \left (d (2 n+1) (4 n+1)^2+e \left (15 n^3+23 n^2+9 n+1\right ) x^n\right )\right )-3 b^3 e n^2 (3 n+2)+6 b^2 c n^2 \left (4 d n+d+e (n+1) x^n\right )+4 b c^2 (n+1) x^n \left (d \left (28 n^2+15 n+2\right )+e \left (18 n^2+13 n+2\right ) x^n\right )+8 c^3 \left (2 n^2+3 n+1\right ) x^{2 n} \left (4 d n+d+e (3 n+1) x^n\right )\right )\right )\right )}{16 c^2 (n+1)^2 (2 n+1) (3 n+1) (4 n+1) \sqrt {a+x^n \left (b+c x^n\right )}} \]
Warning: Unable to verify antiderivative.
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fricas [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.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (c x^{2 \, n} + b x^{n} + a\right )}^{\frac {3}{2}} {\left (e x^{n} + d\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.07, size = 0, normalized size = 0.00 \[ \int \left (e \,x^{n}+d \right ) \left (b \,x^{n}+c \,x^{2 n}+a \right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (c x^{2 \, n} + b x^{n} + a\right )}^{\frac {3}{2}} {\left (e x^{n} + d\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \left (d+e\,x^n\right )\,{\left (a+b\,x^n+c\,x^{2\,n}\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d + e x^{n}\right ) \left (a + b x^{n} + c x^{2 n}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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