Optimal. Leaf size=398 \[ -\frac{a^{3/4} e^6 \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (77 \sqrt{a} B+25 A \sqrt{c}\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{20 c^{15/4} \sqrt{e x} \sqrt{a+c x^2}}+\frac{77 a^{5/4} B e^6 \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{10 c^{15/4} \sqrt{e x} \sqrt{a+c x^2}}-\frac{e^3 (e x)^{5/2} (9 A+11 B x)}{6 c^2 \sqrt{a+c x^2}}-\frac{e (e x)^{9/2} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}+\frac{5 A e^5 \sqrt{e x} \sqrt{a+c x^2}}{2 c^3}-\frac{77 a B e^6 x \sqrt{a+c x^2}}{10 c^{7/2} \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}+\frac{77 B e^4 (e x)^{3/2} \sqrt{a+c x^2}}{30 c^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.485742, antiderivative size = 398, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {819, 833, 842, 840, 1198, 220, 1196} \[ -\frac{a^{3/4} e^6 \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (77 \sqrt{a} B+25 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{20 c^{15/4} \sqrt{e x} \sqrt{a+c x^2}}+\frac{77 a^{5/4} B e^6 \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{10 c^{15/4} \sqrt{e x} \sqrt{a+c x^2}}-\frac{e^3 (e x)^{5/2} (9 A+11 B x)}{6 c^2 \sqrt{a+c x^2}}-\frac{e (e x)^{9/2} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}+\frac{5 A e^5 \sqrt{e x} \sqrt{a+c x^2}}{2 c^3}-\frac{77 a B e^6 x \sqrt{a+c x^2}}{10 c^{7/2} \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}+\frac{77 B e^4 (e x)^{3/2} \sqrt{a+c x^2}}{30 c^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 819
Rule 833
Rule 842
Rule 840
Rule 1198
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \frac{(e x)^{11/2} (A+B x)}{\left (a+c x^2\right )^{5/2}} \, dx &=-\frac{e (e x)^{9/2} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}+\frac{\int \frac{(e x)^{7/2} \left (\frac{9}{2} a A e^2+\frac{11}{2} a B e^2 x\right )}{\left (a+c x^2\right )^{3/2}} \, dx}{3 a c}\\ &=-\frac{e (e x)^{9/2} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}-\frac{e^3 (e x)^{5/2} (9 A+11 B x)}{6 c^2 \sqrt{a+c x^2}}+\frac{\int \frac{(e x)^{3/2} \left (\frac{45}{4} a^2 A e^4+\frac{77}{4} a^2 B e^4 x\right )}{\sqrt{a+c x^2}} \, dx}{3 a^2 c^2}\\ &=-\frac{e (e x)^{9/2} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}-\frac{e^3 (e x)^{5/2} (9 A+11 B x)}{6 c^2 \sqrt{a+c x^2}}+\frac{77 B e^4 (e x)^{3/2} \sqrt{a+c x^2}}{30 c^3}+\frac{2 \int \frac{\sqrt{e x} \left (-\frac{231}{8} a^3 B e^5+\frac{225}{8} a^2 A c e^5 x\right )}{\sqrt{a+c x^2}} \, dx}{15 a^2 c^3}\\ &=-\frac{e (e x)^{9/2} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}-\frac{e^3 (e x)^{5/2} (9 A+11 B x)}{6 c^2 \sqrt{a+c x^2}}+\frac{5 A e^5 \sqrt{e x} \sqrt{a+c x^2}}{2 c^3}+\frac{77 B e^4 (e x)^{3/2} \sqrt{a+c x^2}}{30 c^3}+\frac{4 \int \frac{-\frac{225}{16} a^3 A c e^6-\frac{693}{16} a^3 B c e^6 x}{\sqrt{e x} \sqrt{a+c x^2}} \, dx}{45 a^2 c^4}\\ &=-\frac{e (e x)^{9/2} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}-\frac{e^3 (e x)^{5/2} (9 A+11 B x)}{6 c^2 \sqrt{a+c x^2}}+\frac{5 A e^5 \sqrt{e x} \sqrt{a+c x^2}}{2 c^3}+\frac{77 B e^4 (e x)^{3/2} \sqrt{a+c x^2}}{30 c^3}+\frac{\left (4 \sqrt{x}\right ) \int \frac{-\frac{225}{16} a^3 A c e^6-\frac{693}{16} a^3 B c e^6 x}{\sqrt{x} \sqrt{a+c x^2}} \, dx}{45 a^2 c^4 \sqrt{e x}}\\ &=-\frac{e (e x)^{9/2} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}-\frac{e^3 (e x)^{5/2} (9 A+11 B x)}{6 c^2 \sqrt{a+c x^2}}+\frac{5 A e^5 \sqrt{e x} \sqrt{a+c x^2}}{2 c^3}+\frac{77 B e^4 (e x)^{3/2} \sqrt{a+c x^2}}{30 c^3}+\frac{\left (8 \sqrt{x}\right ) \operatorname{Subst}\left (\int \frac{-\frac{225}{16} a^3 A c e^6-\frac{693}{16} a^3 B c e^6 x^2}{\sqrt{a+c x^4}} \, dx,x,\sqrt{x}\right )}{45 a^2 c^4 \sqrt{e x}}\\ &=-\frac{e (e x)^{9/2} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}-\frac{e^3 (e x)^{5/2} (9 A+11 B x)}{6 c^2 \sqrt{a+c x^2}}+\frac{5 A e^5 \sqrt{e x} \sqrt{a+c x^2}}{2 c^3}+\frac{77 B e^4 (e x)^{3/2} \sqrt{a+c x^2}}{30 c^3}+\frac{\left (77 a^{3/2} B e^6 \sqrt{x}\right ) \operatorname{Subst}\left (\int \frac{1-\frac{\sqrt{c} x^2}{\sqrt{a}}}{\sqrt{a+c x^4}} \, dx,x,\sqrt{x}\right )}{10 c^{7/2} \sqrt{e x}}-\frac{\left (a \left (77 \sqrt{a} B+25 A \sqrt{c}\right ) e^6 \sqrt{x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+c x^4}} \, dx,x,\sqrt{x}\right )}{10 c^{7/2} \sqrt{e x}}\\ &=-\frac{e (e x)^{9/2} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}-\frac{e^3 (e x)^{5/2} (9 A+11 B x)}{6 c^2 \sqrt{a+c x^2}}+\frac{5 A e^5 \sqrt{e x} \sqrt{a+c x^2}}{2 c^3}+\frac{77 B e^4 (e x)^{3/2} \sqrt{a+c x^2}}{30 c^3}-\frac{77 a B e^6 x \sqrt{a+c x^2}}{10 c^{7/2} \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}+\frac{77 a^{5/4} B e^6 \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{10 c^{15/4} \sqrt{e x} \sqrt{a+c x^2}}-\frac{a^{3/4} \left (77 \sqrt{a} B+25 A \sqrt{c}\right ) e^6 \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{20 c^{15/4} \sqrt{e x} \sqrt{a+c x^2}}\\ \end{align*}
Mathematica [C] time = 0.130346, size = 165, normalized size = 0.41 \[ \frac{e^5 \sqrt{e x} \left (75 a^2 A+77 a^2 B x-75 a A \left (a+c x^2\right ) \sqrt{\frac{c x^2}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{c x^2}{a}\right )+105 a A c x^2-77 a B x \left (a+c x^2\right ) \sqrt{\frac{c x^2}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{c x^2}{a}\right )+99 a B c x^3+20 A c^2 x^4+12 B c^2 x^5\right )}{30 c^3 \left (a+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.058, size = 615, normalized size = 1.6 \begin{align*} -{\frac{{e}^{5}}{60\,x{c}^{4}} \left ( 75\,A\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ) \sqrt{-ac}{x}^{2}ac+462\,B\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){x}^{2}{a}^{2}c-231\,B\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){x}^{2}{a}^{2}c-24\,B{c}^{3}{x}^{6}+75\,A\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ) \sqrt{-ac}{a}^{2}-40\,A{c}^{3}{x}^{5}+462\,B\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){a}^{3}-231\,B\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){a}^{3}-198\,aB{c}^{2}{x}^{4}-210\,aA{c}^{2}{x}^{3}-154\,{a}^{2}Bc{x}^{2}-150\,{a}^{2}Acx \right ) \sqrt{ex} \left ( c{x}^{2}+a \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B x + A\right )} \left (e x\right )^{\frac{11}{2}}}{{\left (c x^{2} + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (B e^{5} x^{6} + A e^{5} x^{5}\right )} \sqrt{c x^{2} + a} \sqrt{e x}}{c^{3} x^{6} + 3 \, a c^{2} x^{4} + 3 \, a^{2} c x^{2} + a^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B x + A\right )} \left (e x\right )^{\frac{11}{2}}}{{\left (c x^{2} + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]