Optimal. Leaf size=182 \[ \frac{8 b^{5/2} (e x)^{3/2} \left (\frac{a}{b x^2}+1\right )^{3/4} (10 b c-11 a d) F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{77 a^{7/2} e^8 \left (a+b x^2\right )^{3/4}}-\frac{4 b \sqrt [4]{a+b x^2} (10 b c-11 a d)}{77 a^3 e^5 (e x)^{3/2}}+\frac{2 \sqrt [4]{a+b x^2} (10 b c-11 a d)}{77 a^2 e^3 (e x)^{7/2}}-\frac{2 c \sqrt [4]{a+b x^2}}{11 a e (e x)^{11/2}} \]
[Out]
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Rubi [A] time = 0.402303, antiderivative size = 182, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ \frac{8 b^{5/2} (e x)^{3/2} \left (\frac{a}{b x^2}+1\right )^{3/4} (10 b c-11 a d) F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{77 a^{7/2} e^8 \left (a+b x^2\right )^{3/4}}-\frac{4 b \sqrt [4]{a+b x^2} (10 b c-11 a d)}{77 a^3 e^5 (e x)^{3/2}}+\frac{2 \sqrt [4]{a+b x^2} (10 b c-11 a d)}{77 a^2 e^3 (e x)^{7/2}}-\frac{2 c \sqrt [4]{a+b x^2}}{11 a e (e x)^{11/2}} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x^2)/((e*x)^(13/2)*(a + b*x^2)^(3/4)),x]
[Out]
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Rubi in Sympy [A] time = 43.2482, size = 173, normalized size = 0.95 \[ - \frac{2 c \sqrt [4]{a + b x^{2}}}{11 a e \left (e x\right )^{\frac{11}{2}}} - \frac{2 \sqrt [4]{a + b x^{2}} \left (11 a d - 10 b c\right )}{77 a^{2} e^{3} \left (e x\right )^{\frac{7}{2}}} + \frac{4 b \sqrt [4]{a + b x^{2}} \left (11 a d - 10 b c\right )}{77 a^{3} e^{5} \left (e x\right )^{\frac{3}{2}}} - \frac{8 b^{\frac{5}{2}} \left (e x\right )^{\frac{3}{2}} \left (11 a d - 10 b c\right ) \left (\frac{a}{b x^{2}} + 1\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{2}\middle | 2\right )}{77 a^{\frac{7}{2}} e^{8} \left (a + b x^{2}\right )^{\frac{3}{4}}} \]
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)**(3/4),x)
[Out]
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Mathematica [C] time = 0.218361, size = 132, normalized size = 0.73 \[ \frac{\sqrt{e x} \left (8 b^2 x^6 \left (\frac{b x^2}{a}+1\right )^{3/4} (11 a d-10 b c) \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};-\frac{b x^2}{a}\right )-2 \left (a+b x^2\right ) \left (a^2 \left (7 c+11 d x^2\right )-2 a b x^2 \left (5 c+11 d x^2\right )+20 b^2 c x^4\right )\right )}{77 a^3 e^7 x^6 \left (a+b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x^2)/((e*x)^(13/2)*(a + b*x^2)^(3/4)),x]
[Out]
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Maple [F] time = 0.062, size = 0, normalized size = 0. \[ \int{(d{x}^{2}+c) \left ( ex \right ) ^{-{\frac{13}{2}}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{4}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x^2+c)/(e*x)^(13/2)/(b*x^2+a)^(3/4),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac{3}{4}} \left (e x\right )^{\frac{13}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)/((b*x^2 + a)^(3/4)*(e*x)^(13/2)),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac{3}{4}} \sqrt{e x} e^{6} x^{6}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)/((b*x^2 + a)^(3/4)*(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)**(3/4),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac{3}{4}} \left (e x\right )^{\frac{13}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)/((b*x^2 + a)^(3/4)*(e*x)^(13/2)),x, algorithm="giac")
[Out]