Optimal. Leaf size=149 \[ \frac {a x^2 \sqrt {a+b x^n+c x^{2 n}} F_1\left (\frac {2}{n};-\frac {3}{2},-\frac {3}{2};\frac {n+2}{n};-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{2 \sqrt {\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^n}{\sqrt {b^2-4 a c}+b}+1}} \]
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Rubi [A] time = 0.11, antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1385, 510} \[ \frac {a x^2 \sqrt {a+b x^n+c x^{2 n}} F_1\left (\frac {2}{n};-\frac {3}{2},-\frac {3}{2};\frac {n+2}{n};-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{2 \sqrt {\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^n}{\sqrt {b^2-4 a c}+b}+1}} \]
Antiderivative was successfully verified.
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Rule 510
Rule 1385
Rubi steps
\begin {align*} \int x \left (a+b x^n+c x^{2 n}\right )^{3/2} \, dx &=\frac {\left (a \sqrt {a+b x^n+c x^{2 n}}\right ) \int x \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 x^2 \sqrt {a+b x^n+c x^{2 n}} F_1\left (\frac {2}{n};-\frac {3}{2},-\frac {3}{2};\frac {2+n}{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 )}{2 \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 = 1.49, size = 520, normalized size = 3.49 \[ \frac {x^2 \left (2 (n+2) \left (16 a^2 c \left (2 n^2+3 n+1\right )+a \left (3 b^2 n^2+2 b c \left (23 n^2+42 n+16\right ) x^n+8 c^2 \left (5 n^2+9 n+4\right ) x^{2 n}\right )+x^n \left (b+c x^n\right ) \left (3 b^2 n^2+2 b c \left (7 n^2+18 n+8\right ) x^n+8 c^2 \left (n^2+3 n+2\right ) x^{2 n}\right )\right )-3 b n^2 x^n \left (b^2 (n+4)-4 a c (3 n+4)\right ) \sqrt {\frac {-\sqrt {b^2-4 a c}+b+2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {\sqrt {b^2-4 a c}+b+2 c x^n}{\sqrt {b^2-4 a c}+b}} F_1\left (\frac {n+2}{n};\frac {1}{2},\frac {1}{2};2+\frac {2}{n};-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}},\frac {2 c x^n}{\sqrt {b^2-4 a c}-b}\right )-6 a n^2 (n+2) \left (b^2-4 a c (n+1)\right ) \sqrt {\frac {-\sqrt {b^2-4 a c}+b+2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {\sqrt {b^2-4 a c}+b+2 c x^n}{\sqrt {b^2-4 a c}+b}} F_1\left (\frac {2}{n};\frac {1}{2},\frac {1}{2};\frac {n+2}{n};-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}},\frac {2 c x^n}{\sqrt {b^2-4 a c}-b}\right )\right )}{16 c (n+1) (n+2)^2 (3 n+2) \sqrt {a+x^n \left (b+c x^n\right )}} \]
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 \[ \int {\left (c x^{2 \, n} + b x^{n} + a\right )}^{\frac {3}{2}} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int \left (b \,x^{n}+c \,x^{2 n}+a \right )^{\frac {3}{2}} x\, dx \]
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 \[ \int {\left (c x^{2 \, n} + b x^{n} + a\right )}^{\frac {3}{2}} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,{\left (a+b\,x^n+c\,x^{2\,n}\right )}^{3/2} \,d x \]
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 \[ \int x \left (a + b x^{n} + c x^{2 n}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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