Optimal. Leaf size=141 \[ \frac{64 \left (a+b x^2\right )^{9/4} (12 b c-13 a d)}{585 a^4 e^3 (e x)^{9/2}}-\frac{16 \left (a+b x^2\right )^{5/4} (12 b c-13 a d)}{65 a^3 e^3 (e x)^{9/2}}+\frac{2 \sqrt [4]{a+b x^2} (12 b c-13 a d)}{13 a^2 e^3 (e x)^{9/2}}-\frac{2 c \sqrt [4]{a+b x^2}}{13 a e (e x)^{13/2}} \]
[Out]
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Rubi [A] time = 0.222218, 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 \[ \frac{64 \left (a+b x^2\right )^{9/4} (12 b c-13 a d)}{585 a^4 e^3 (e x)^{9/2}}-\frac{16 \left (a+b x^2\right )^{5/4} (12 b c-13 a d)}{65 a^3 e^3 (e x)^{9/2}}+\frac{2 \sqrt [4]{a+b x^2} (12 b c-13 a d)}{13 a^2 e^3 (e x)^{9/2}}-\frac{2 c \sqrt [4]{a+b x^2}}{13 a e (e x)^{13/2}} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x^2)/((e*x)^(15/2)*(a + b*x^2)^(3/4)),x]
[Out]
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Rubi in Sympy [A] time = 23.4163, size = 136, normalized size = 0.96 \[ - \frac{2 c \sqrt [4]{a + b x^{2}}}{13 a e \left (e x\right )^{\frac{13}{2}}} - \frac{2 \sqrt [4]{a + b x^{2}} \left (13 a d - 12 b c\right )}{13 a^{2} e^{3} \left (e x\right )^{\frac{9}{2}}} + \frac{16 \left (a + b x^{2}\right )^{\frac{5}{4}} \left (13 a d - 12 b c\right )}{65 a^{3} e^{3} \left (e x\right )^{\frac{9}{2}}} - \frac{64 \left (a + b x^{2}\right )^{\frac{9}{4}} \left (13 a d - 12 b c\right )}{585 a^{4} e^{3} \left (e x\right )^{\frac{9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x**2+c)/(e*x)**(15/2)/(b*x**2+a)**(3/4),x)
[Out]
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Mathematica [A] time = 0.119092, size = 94, normalized size = 0.67 \[ -\frac{2 \sqrt{e x} \sqrt [4]{a+b x^2} \left (5 a^3 \left (9 c+13 d x^2\right )-4 a^2 b x^2 \left (15 c+26 d x^2\right )+32 a b^2 x^4 \left (3 c+13 d x^2\right )-384 b^3 c x^6\right )}{585 a^4 e^8 x^7} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x^2)/((e*x)^(15/2)*(a + b*x^2)^(3/4)),x]
[Out]
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Maple [A] time = 0.009, size = 86, normalized size = 0.6 \[ -{\frac{2\,x \left ( 416\,a{b}^{2}d{x}^{6}-384\,{b}^{3}c{x}^{6}-104\,{a}^{2}bd{x}^{4}+96\,a{b}^{2}c{x}^{4}+65\,{a}^{3}d{x}^{2}-60\,{a}^{2}bc{x}^{2}+45\,c{a}^{3} \right ) }{585\,{a}^{4}}\sqrt [4]{b{x}^{2}+a} \left ( ex \right ) ^{-{\frac{15}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x^2+c)/(e*x)^(15/2)/(b*x^2+a)^(3/4),x)
[Out]
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Maxima [A] time = 1.43236, size = 176, normalized size = 1.25 \[ -\frac{2 \,{\left (\frac{45 \,{\left (b x^{2} + a\right )}^{\frac{1}{4}} b^{2}}{\sqrt{x}} - \frac{18 \,{\left (b x^{2} + a\right )}^{\frac{5}{4}} b}{x^{\frac{5}{2}}} + \frac{5 \,{\left (b x^{2} + a\right )}^{\frac{9}{4}}}{x^{\frac{9}{2}}}\right )} d}{45 \, a^{3} e^{\frac{15}{2}}} + \frac{2 \,{\left (\frac{195 \,{\left (b x^{2} + a\right )}^{\frac{1}{4}} b^{3}}{\sqrt{x}} - \frac{117 \,{\left (b x^{2} + a\right )}^{\frac{5}{4}} b^{2}}{x^{\frac{5}{2}}} + \frac{65 \,{\left (b x^{2} + a\right )}^{\frac{9}{4}} b}{x^{\frac{9}{2}}} - \frac{15 \,{\left (b x^{2} + a\right )}^{\frac{13}{4}}}{x^{\frac{13}{2}}}\right )} c}{195 \, a^{4} e^{\frac{15}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)/((b*x^2 + a)^(3/4)*(e*x)^(15/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.222813, size = 122, normalized size = 0.87 \[ \frac{2 \,{\left (32 \,{\left (12 \, b^{3} c - 13 \, a b^{2} d\right )} x^{6} - 8 \,{\left (12 \, a b^{2} c - 13 \, a^{2} b d\right )} x^{4} - 45 \, a^{3} c + 5 \,{\left (12 \, a^{2} b c - 13 \, a^{3} d\right )} x^{2}\right )}{\left (b x^{2} + a\right )}^{\frac{1}{4}} \sqrt{e x}}{585 \, a^{4} e^{8} x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)/((b*x^2 + a)^(3/4)*(e*x)^(15/2)),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)**(15/2)/(b*x**2+a)**(3/4),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac{3}{4}} \left (e x\right )^{\frac{15}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)/((b*x^2 + a)^(3/4)*(e*x)^(15/2)),x, algorithm="giac")
[Out]