Optimal. Leaf size=152 \[ -\frac{a^{5/2} c^2 (c x)^{3/2} \left (\frac{a}{b x^2}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{12 b^{3/2} \left (a+b x^2\right )^{3/4}}-\frac{a^2 c^3 \sqrt{c x} \sqrt [4]{a+b x^2}}{12 b^2}+\frac{(c x)^{9/2} \sqrt [4]{a+b x^2}}{5 c}+\frac{a c (c x)^{5/2} \sqrt [4]{a+b x^2}}{30 b} \]
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Rubi [A] time = 0.311851, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368 \[ -\frac{a^{5/2} c^2 (c x)^{3/2} \left (\frac{a}{b x^2}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{12 b^{3/2} \left (a+b x^2\right )^{3/4}}-\frac{a^2 c^3 \sqrt{c x} \sqrt [4]{a+b x^2}}{12 b^2}+\frac{(c x)^{9/2} \sqrt [4]{a+b x^2}}{5 c}+\frac{a c (c x)^{5/2} \sqrt [4]{a+b x^2}}{30 b} \]
Antiderivative was successfully verified.
[In] Int[(c*x)^(7/2)*(a + b*x^2)^(1/4),x]
[Out]
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Rubi in Sympy [A] time = 33.2618, size = 133, normalized size = 0.88 \[ - \frac{a^{\frac{5}{2}} c^{2} \left (c x\right )^{\frac{3}{2}} \left (\frac{a}{b x^{2}} + 1\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{2}\middle | 2\right )}{12 b^{\frac{3}{2}} \left (a + b x^{2}\right )^{\frac{3}{4}}} - \frac{a^{2} c^{3} \sqrt{c x} \sqrt [4]{a + b x^{2}}}{12 b^{2}} + \frac{a c \left (c x\right )^{\frac{5}{2}} \sqrt [4]{a + b x^{2}}}{30 b} + \frac{\left (c x\right )^{\frac{9}{2}} \sqrt [4]{a + b x^{2}}}{5 c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x)**(7/2)*(b*x**2+a)**(1/4),x)
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Mathematica [C] time = 0.0680213, size = 98, normalized size = 0.64 \[ \frac{c^3 \sqrt{c x} \left (5 a^3 \left (\frac{b x^2}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};-\frac{b x^2}{a}\right )-5 a^3-3 a^2 b x^2+14 a b^2 x^4+12 b^3 x^6\right )}{60 b^2 \left (a+b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
[In] Integrate[(c*x)^(7/2)*(a + b*x^2)^(1/4),x]
[Out]
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Maple [F] time = 0.049, size = 0, normalized size = 0. \[ \int \left ( cx \right ) ^{{\frac{7}{2}}}\sqrt [4]{b{x}^{2}+a}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x)^(7/2)*(b*x^2+a)^(1/4),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{2} + a\right )}^{\frac{1}{4}} \left (c x\right )^{\frac{7}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^(1/4)*(c*x)^(7/2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x^{2} + a\right )}^{\frac{1}{4}} \sqrt{c x} c^{3} x^{3}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^(1/4)*(c*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((c*x)**(7/2)*(b*x**2+a)**(1/4),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{2} + a\right )}^{\frac{1}{4}} \left (c x\right )^{\frac{7}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^(1/4)*(c*x)^(7/2),x, algorithm="giac")
[Out]