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 = 0.347993, antiderivative size = 294, normalized size of antiderivative = 1., 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 1432
Rule 1348
Rule 429
Rule 1385
Rule 510
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*}
Mathematica [B] time = 4.54331, size = 690, normalized size = 2.35 \[ \frac{x \left (3 n^2 x^n \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}} F_1\left (1+\frac{1}{n};\frac{1}{2},\frac{1}{2};2+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right ) \left (16 a^2 c^2 e \left (3 n^2+4 n+1\right )-4 a b^2 c e \left (6 n^2+14 n+5\right )+8 a b c^2 d \left (12 n^2+11 n+2\right )-2 b^3 c d \left (4 n^2+9 n+2\right )+b^4 e \left (3 n^2+8 n+4\right )\right )+2 (n+1) \left (3 a n^2 \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}} F_1\left (\frac{1}{n};\frac{1}{2},\frac{1}{2};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right ) \left (-4 a b c e (5 n+2)+8 a c^2 d \left (8 n^2+6 n+1\right )-2 b^2 c d (4 n+1)+b^3 e (3 n+2)\right )+\left (a+x^n \left (b+c x^n\right )\right ) \left (4 a c \left (3 b e (5 n+2) n^2+2 c \left (d (2 n+1) (4 n+1)^2+e \left (15 n^3+23 n^2+9 n+1\right ) x^n\right )\right )+6 b^2 c n^2 \left (4 d n+d+e (n+1) x^n\right )-3 b^3 e n^2 (3 n+2)+4 b c^2 (n+1) x^n \left (d \left (28 n^2+15 n+2\right )+e \left (18 n^2+13 n+2\right ) x^n\right )+8 c^3 \left (2 n^2+3 n+1\right ) x^{2 n} \left (4 d n+d+e (3 n+1) x^n\right )\right )\right )\right )}{16 c^2 (n+1)^2 (2 n+1) (3 n+1) (4 n+1) \sqrt{a+x^n \left (b+c x^n\right )}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.059, size = 0, normalized size = 0. \begin{align*} \int \left ( d+e{x}^{n} \right ) \left ( a+b{x}^{n}+c{x}^{2\,n} \right ) ^{{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c x^{2 \, n} + b x^{n} + a\right )}^{\frac{3}{2}}{\left (e x^{n} + d\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c x^{2 \, n} + b x^{n} + a\right )}^{\frac{3}{2}}{\left (e x^{n} + d\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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