Optimal. Leaf size=141 \[ -\frac{64 \left (a+b x^2\right )^{7/4} (12 b c-11 a d)}{231 a^4 e^3 (e x)^{7/2}}+\frac{16 \left (a+b x^2\right )^{3/4} (12 b c-11 a d)}{33 a^3 e^3 (e x)^{7/2}}-\frac{2 (12 b c-11 a d)}{11 a^2 e^3 (e x)^{7/2} \sqrt [4]{a+b x^2}}-\frac{2 c}{11 a e (e x)^{11/2} \sqrt [4]{a+b x^2}} \]
[Out]
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Rubi [A] time = 0.219422, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ -\frac{64 \left (a+b x^2\right )^{7/4} (12 b c-11 a d)}{231 a^4 e^3 (e x)^{7/2}}+\frac{16 \left (a+b x^2\right )^{3/4} (12 b c-11 a d)}{33 a^3 e^3 (e x)^{7/2}}-\frac{2 (12 b c-11 a d)}{11 a^2 e^3 (e x)^{7/2} \sqrt [4]{a+b x^2}}-\frac{2 c}{11 a e (e x)^{11/2} \sqrt [4]{a+b x^2}} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x^2)/((e*x)^(13/2)*(a + b*x^2)^(5/4)),x]
[Out]
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Rubi in Sympy [A] time = 23.334, size = 134, normalized size = 0.95 \[ - \frac{2 c}{11 a e \left (e x\right )^{\frac{11}{2}} \sqrt [4]{a + b x^{2}}} + \frac{2 \left (11 a d - 12 b c\right )}{11 a^{2} e^{3} \left (e x\right )^{\frac{7}{2}} \sqrt [4]{a + b x^{2}}} - \frac{16 \left (a + b x^{2}\right )^{\frac{3}{4}} \left (11 a d - 12 b c\right )}{33 a^{3} e^{3} \left (e x\right )^{\frac{7}{2}}} + \frac{64 \left (a + b x^{2}\right )^{\frac{7}{4}} \left (11 a d - 12 b c\right )}{231 a^{4} e^{3} \left (e x\right )^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x**2+c)/(e*x)**(13/2)/(b*x**2+a)**(5/4),x)
[Out]
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Mathematica [A] time = 0.159672, size = 94, normalized size = 0.67 \[ \frac{\sqrt{e x} \left (-6 a^3 \left (7 c+11 d x^2\right )+8 a^2 b x^2 \left (9 c+22 d x^2\right )+64 a b^2 x^4 \left (11 d x^2-3 c\right )-768 b^3 c x^6\right )}{231 a^4 e^7 x^6 \sqrt [4]{a+b x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x^2)/((e*x)^(13/2)*(a + b*x^2)^(5/4)),x]
[Out]
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Maple [A] time = 0.009, size = 86, normalized size = 0.6 \[ -{\frac{2\,x \left ( -352\,a{b}^{2}d{x}^{6}+384\,{b}^{3}c{x}^{6}-88\,{a}^{2}bd{x}^{4}+96\,a{b}^{2}c{x}^{4}+33\,{a}^{3}d{x}^{2}-36\,{a}^{2}bc{x}^{2}+21\,c{a}^{3} \right ) }{231\,{a}^{4}}{\frac{1}{\sqrt [4]{b{x}^{2}+a}}} \left ( ex \right ) ^{-{\frac{13}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x^2+c)/(e*x)^(13/2)/(b*x^2+a)^(5/4),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac{5}{4}} \left (e x\right )^{\frac{13}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)/((b*x^2 + a)^(5/4)*(e*x)^(13/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.218447, size = 122, normalized size = 0.87 \[ -\frac{2 \,{\left (32 \,{\left (12 \, b^{3} c - 11 \, a b^{2} d\right )} x^{6} + 8 \,{\left (12 \, a b^{2} c - 11 \, a^{2} b d\right )} x^{4} + 21 \, a^{3} c - 3 \,{\left (12 \, a^{2} b c - 11 \, a^{3} d\right )} x^{2}\right )}}{231 \,{\left (b x^{2} + a\right )}^{\frac{1}{4}} \sqrt{e x} a^{4} e^{6} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)/((b*x^2 + a)^(5/4)*(e*x)^(13/2)),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x**2+c)/(e*x)**(13/2)/(b*x**2+a)**(5/4),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac{5}{4}} \left (e x\right )^{\frac{13}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)/((b*x^2 + a)^(5/4)*(e*x)^(13/2)),x, algorithm="giac")
[Out]