Optimal. Leaf size=298 \[ \frac{d x \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1} F_1\left (\frac{1}{n};\frac{5}{2},\frac{5}{2};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{a^2 \sqrt{a+b x^n+c x^{2 n}}}+\frac{e x^{n+1} \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1} F_1\left (1+\frac{1}{n};\frac{5}{2},\frac{5}{2};2+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{a^2 (n+1) \sqrt{a+b x^n+c x^{2 n}}} \]
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Rubi [A] time = 0.347054, antiderivative size = 298, 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} \[ \frac{d x \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1} F_1\left (\frac{1}{n};\frac{5}{2},\frac{5}{2};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{a^2 \sqrt{a+b x^n+c x^{2 n}}}+\frac{e x^{n+1} \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1} F_1\left (1+\frac{1}{n};\frac{5}{2},\frac{5}{2};2+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{a^2 (n+1) \sqrt{a+b x^n+c x^{2 n}}} \]
Antiderivative was successfully verified.
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Rule 1432
Rule 1348
Rule 429
Rule 1385
Rule 510
Rubi steps
\begin{align*} \int \frac{d+e x^n}{\left (a+b x^n+c x^{2 n}\right )^{5/2}} \, dx &=\int \left (\frac{d}{\left (a+b x^n+c x^{2 n}\right )^{5/2}}+\frac{e x^n}{\left (a+b x^n+c x^{2 n}\right )^{5/2}}\right ) \, dx\\ &=d \int \frac{1}{\left (a+b x^n+c x^{2 n}\right )^{5/2}} \, dx+e \int \frac{x^n}{\left (a+b x^n+c x^{2 n}\right )^{5/2}} \, dx\\ &=\frac{\left (d \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}}}\right ) \int \frac{1}{\left (1+\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )^{5/2} \left (1+\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )^{5/2}} \, dx}{a^2 \sqrt{a+b x^n+c x^{2 n}}}+\frac{\left (e \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}}}\right ) \int \frac{x^n}{\left (1+\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )^{5/2} \left (1+\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )^{5/2}} \, dx}{a^2 \sqrt{a+b x^n+c x^{2 n}}}\\ &=\frac{e x^{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}}} F_1\left (1+\frac{1}{n};\frac{5}{2},\frac{5}{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 )}{a^2 (1+n) \sqrt{a+b x^n+c x^{2 n}}}+\frac{d x \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}}} F_1\left (\frac{1}{n};\frac{5}{2},\frac{5}{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 )}{a^2 \sqrt{a+b x^n+c x^{2 n}}}\\ \end{align*}
Mathematica [B] time = 6.55774, size = 6648, normalized size = 22.31 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.013, size = 0, normalized size = 0. \begin{align*} \int{(d+e{x}^{n}) \left ( a+b{x}^{n}+c{x}^{2\,n} \right ) ^{-{\frac{5}{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 \frac{e x^{n} + d}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{\frac{5}{2}}}\,{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 \frac{e x^{n} + d}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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