Optimal. Leaf size=104 \[ \frac{8 \sqrt [4]{a+b x^2} (8 b c-5 a d)}{15 a^3 e^3 \sqrt{e x}}-\frac{2 (8 b c-5 a d)}{15 a^2 e^3 \sqrt{e x} \left (a+b x^2\right )^{3/4}}-\frac{2 c}{5 a e (e x)^{5/2} \left (a+b x^2\right )^{3/4}} \]
[Out]
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Rubi [A] time = 0.170531, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{8 \sqrt [4]{a+b x^2} (8 b c-5 a d)}{15 a^3 e^3 \sqrt{e x}}-\frac{2 (8 b c-5 a d)}{15 a^2 e^3 \sqrt{e x} \left (a+b x^2\right )^{3/4}}-\frac{2 c}{5 a e (e x)^{5/2} \left (a+b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x^2)/((e*x)^(7/2)*(a + b*x^2)^(7/4)),x]
[Out]
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Rubi in Sympy [A] time = 17.1046, size = 97, normalized size = 0.93 \[ - \frac{2 c}{5 a e \left (e x\right )^{\frac{5}{2}} \left (a + b x^{2}\right )^{\frac{3}{4}}} + \frac{2 \left (5 a d - 8 b c\right )}{15 a^{2} e^{3} \sqrt{e x} \left (a + b x^{2}\right )^{\frac{3}{4}}} - \frac{8 \sqrt [4]{a + b x^{2}} \left (5 a d - 8 b c\right )}{15 a^{3} e^{3} \sqrt{e x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x**2+c)/(e*x)**(7/2)/(b*x**2+a)**(7/4),x)
[Out]
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Mathematica [A] time = 0.100183, size = 66, normalized size = 0.63 \[ \frac{x \left (-6 a^2 \left (c+5 d x^2\right )+8 a b x^2 \left (6 c-5 d x^2\right )+64 b^2 c x^4\right )}{15 a^3 (e x)^{7/2} \left (a+b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x^2)/((e*x)^(7/2)*(a + b*x^2)^(7/4)),x]
[Out]
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Maple [A] time = 0.009, size = 62, normalized size = 0.6 \[ -{\frac{2\,x \left ( 20\,{x}^{4}abd-32\,{b}^{2}c{x}^{4}+15\,{x}^{2}{a}^{2}d-24\,abc{x}^{2}+3\,{a}^{2}c \right ) }{15\,{a}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{4}}} \left ( ex \right ) ^{-{\frac{7}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x^2+c)/(e*x)^(7/2)/(b*x^2+a)^(7/4),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac{7}{4}} \left (e x\right )^{\frac{7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)/((b*x^2 + a)^(7/4)*(e*x)^(7/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.237579, size = 109, normalized size = 1.05 \[ \frac{2 \,{\left (4 \,{\left (8 \, b^{2} c - 5 \, a b d\right )} x^{4} - 3 \, a^{2} c + 3 \,{\left (8 \, a b c - 5 \, a^{2} d\right )} x^{2}\right )}{\left (b x^{2} + a\right )}^{\frac{1}{4}} \sqrt{e x}}{15 \,{\left (a^{3} b e^{4} x^{5} + a^{4} e^{4} x^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)/((b*x^2 + a)^(7/4)*(e*x)^(7/2)),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)**(7/2)/(b*x**2+a)**(7/4),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac{7}{4}} \left (e x\right )^{\frac{7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)/((b*x^2 + a)^(7/4)*(e*x)^(7/2)),x, algorithm="giac")
[Out]