Optimal. Leaf size=102 \[ \frac{(b c-a d)^{3/2} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{a^{5/2}}+\frac{\sqrt{c+d x^2} (3 b c-4 a d)}{3 a^2 x}-\frac{c \sqrt{c+d x^2}}{3 a x^3} \]
[Out]
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Rubi [A] time = 0.372934, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{(b c-a d)^{3/2} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{a^{5/2}}+\frac{\sqrt{c+d x^2} (3 b c-4 a d)}{3 a^2 x}-\frac{c \sqrt{c+d x^2}}{3 a x^3} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x^2)^(3/2)/(x^4*(a + b*x^2)),x]
[Out]
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Rubi in Sympy [A] time = 47.7446, size = 88, normalized size = 0.86 \[ - \frac{c \sqrt{c + d x^{2}}}{3 a x^{3}} - \frac{\sqrt{c + d x^{2}} \left (4 a d - 3 b c\right )}{3 a^{2} x} + \frac{\left (a d - b c\right )^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{x \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{2}}} \right )}}{a^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x**2+c)**(3/2)/x**4/(b*x**2+a),x)
[Out]
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Mathematica [A] time = 0.180249, size = 90, normalized size = 0.88 \[ \frac{(b c-a d)^{3/2} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{a^{5/2}}+\frac{\sqrt{c+d x^2} \left (3 b c x^2-a \left (c+4 d x^2\right )\right )}{3 a^2 x^3} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x^2)^(3/2)/(x^4*(a + b*x^2)),x]
[Out]
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Maple [B] time = 0.021, size = 2089, normalized size = 20.5 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x^2+c)^(3/2)/x^4/(b*x^2+a),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x^{2} + c\right )}^{\frac{3}{2}}}{{\left (b x^{2} + a\right )} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)^(3/2)/((b*x^2 + a)*x^4),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.287334, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (b c - a d\right )} x^{3} \sqrt{-\frac{b c - a d}{a}} \log \left (\frac{{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} + 4 \,{\left (a^{2} c x -{\left (a b c - 2 \, a^{2} d\right )} x^{3}\right )} \sqrt{d x^{2} + c} \sqrt{-\frac{b c - a d}{a}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - 4 \,{\left ({\left (3 \, b c - 4 \, a d\right )} x^{2} - a c\right )} \sqrt{d x^{2} + c}}{12 \, a^{2} x^{3}}, -\frac{3 \,{\left (b c - a d\right )} x^{3} \sqrt{\frac{b c - a d}{a}} \arctan \left (-\frac{{\left (b c - 2 \, a d\right )} x^{2} - a c}{2 \, \sqrt{d x^{2} + c} a x \sqrt{\frac{b c - a d}{a}}}\right ) - 2 \,{\left ({\left (3 \, b c - 4 \, a d\right )} x^{2} - a c\right )} \sqrt{d x^{2} + c}}{6 \, a^{2} x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)^(3/2)/((b*x^2 + a)*x^4),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c + d x^{2}\right )^{\frac{3}{2}}}{x^{4} \left (a + b x^{2}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x**2+c)**(3/2)/x**4/(b*x**2+a),x)
[Out]
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GIAC/XCAS [A] time = 0.643643, size = 346, normalized size = 3.39 \[ -\frac{{\left (b^{2} c^{2} \sqrt{d} - 2 \, a b c d^{\frac{3}{2}} + a^{2} d^{\frac{5}{2}}\right )} \arctan \left (\frac{{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt{a b c d - a^{2} d^{2}}}\right )}{\sqrt{a b c d - a^{2} d^{2}} a^{2}} - \frac{2 \,{\left (3 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} b c^{2} \sqrt{d} - 6 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} a c d^{\frac{3}{2}} - 6 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b c^{3} \sqrt{d} + 6 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a c^{2} d^{\frac{3}{2}} + 3 \, b c^{4} \sqrt{d} - 4 \, a c^{3} d^{\frac{3}{2}}\right )}}{3 \,{\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} - c\right )}^{3} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)^(3/2)/((b*x^2 + a)*x^4),x, algorithm="giac")
[Out]