Optimal. Leaf size=249 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{3} \sqrt{c} (c+2 d x)}{\sqrt{c^3+4 d^3 x^3}}\right )}{3 \sqrt{3} c^{3/2} d}+\frac{2 \sqrt [3]{2} \sqrt{2+\sqrt{3}} \left (c+2^{2/3} d x\right ) \sqrt{\frac{c^2-2^{2/3} c d x+2 \sqrt [3]{2} d^2 x^2}{\left (\left (1+\sqrt{3}\right ) c+2^{2/3} d x\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) c+2^{2/3} d x}{\left (1+\sqrt{3}\right ) c+2^{2/3} d x}\right )|-7-4 \sqrt{3}\right )}{3 \sqrt [4]{3} c d \sqrt{\frac{c \left (c+2^{2/3} d x\right )}{\left (\left (1+\sqrt{3}\right ) c+2^{2/3} d x\right )^2}} \sqrt{c^3+4 d^3 x^3}} \]
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Rubi [A] time = 0.469822, antiderivative size = 249, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{3} \sqrt{c} (c+2 d x)}{\sqrt{c^3+4 d^3 x^3}}\right )}{3 \sqrt{3} c^{3/2} d}+\frac{2 \sqrt [3]{2} \sqrt{2+\sqrt{3}} \left (c+2^{2/3} d x\right ) \sqrt{\frac{c^2-2^{2/3} c d x+2 \sqrt [3]{2} d^2 x^2}{\left (\left (1+\sqrt{3}\right ) c+2^{2/3} d x\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) c+2^{2/3} d x}{\left (1+\sqrt{3}\right ) c+2^{2/3} d x}\right )|-7-4 \sqrt{3}\right )}{3 \sqrt [4]{3} c d \sqrt{\frac{c \left (c+2^{2/3} d x\right )}{\left (\left (1+\sqrt{3}\right ) c+2^{2/3} d x\right )^2}} \sqrt{c^3+4 d^3 x^3}} \]
Antiderivative was successfully verified.
[In] Int[1/((c + d*x)*Sqrt[c^3 + 4*d^3*x^3]),x]
[Out]
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(d*x+c)/(4*d**3*x**3+c**3)**(1/2),x)
[Out]
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Mathematica [C] time = 0.333141, size = 169, normalized size = 0.68 \[ -\frac{i 2^{5/6} \sqrt{\frac{\sqrt [3]{2} c+2 d x}{\left (1+\sqrt [3]{-1}\right ) c}} \sqrt{\frac{4 d^2 x^2}{c^2}-\frac{2 \sqrt [3]{2} d x}{c}+2^{2/3}} \Pi \left (\frac{i \sqrt [3]{2} \sqrt{3}}{2+\sqrt [3]{-2}};\sin ^{-1}\left (\frac{\sqrt{\frac{\sqrt [3]{2} c+2 (-1)^{2/3} d x}{\left (1+\sqrt [3]{-1}\right ) c}}}{\sqrt [6]{2}}\right )|\sqrt [3]{-1}\right )}{\left (2+\sqrt [3]{-2}\right ) d \sqrt{c^3+4 d^3 x^3}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[1/((c + d*x)*Sqrt[c^3 + 4*d^3*x^3]),x]
[Out]
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Maple [B] time = 0.259, size = 495, normalized size = 2. \[ 2\,{\frac{1}{d\sqrt{4\,{d}^{3}{x}^{3}+{c}^{3}}} \left ({\frac{ \left ( 1/4\,\sqrt [3]{2}-i/4\sqrt{3}\sqrt [3]{2} \right ) c}{d}}-{\frac{ \left ( 1/4\,\sqrt [3]{2}+i/4\sqrt{3}\sqrt [3]{2} \right ) c}{d}} \right ) \sqrt{{1 \left ( x-{\frac{ \left ( 1/4\,\sqrt [3]{2}+i/4\sqrt{3}\sqrt [3]{2} \right ) c}{d}} \right ) \left ({\frac{ \left ( 1/4\,\sqrt [3]{2}-i/4\sqrt{3}\sqrt [3]{2} \right ) c}{d}}-{\frac{ \left ( 1/4\,\sqrt [3]{2}+i/4\sqrt{3}\sqrt [3]{2} \right ) c}{d}} \right ) ^{-1}}}\sqrt{{1 \left ( x+1/2\,{\frac{\sqrt [3]{2}c}{d}} \right ) \left ({\frac{ \left ( 1/4\,\sqrt [3]{2}+i/4\sqrt{3}\sqrt [3]{2} \right ) c}{d}}+1/2\,{\frac{\sqrt [3]{2}c}{d}} \right ) ^{-1}}}\sqrt{{1 \left ( x-{\frac{ \left ( 1/4\,\sqrt [3]{2}-i/4\sqrt{3}\sqrt [3]{2} \right ) c}{d}} \right ) \left ({\frac{ \left ( 1/4\,\sqrt [3]{2}+i/4\sqrt{3}\sqrt [3]{2} \right ) c}{d}}-{\frac{ \left ( 1/4\,\sqrt [3]{2}-i/4\sqrt{3}\sqrt [3]{2} \right ) c}{d}} \right ) ^{-1}}}{\it EllipticPi} \left ( \sqrt{{1 \left ( x-{\frac{ \left ( 1/4\,\sqrt [3]{2}+i/4\sqrt{3}\sqrt [3]{2} \right ) c}{d}} \right ) \left ({\frac{ \left ( 1/4\,\sqrt [3]{2}-i/4\sqrt{3}\sqrt [3]{2} \right ) c}{d}}-{\frac{ \left ( 1/4\,\sqrt [3]{2}+i/4\sqrt{3}\sqrt [3]{2} \right ) c}{d}} \right ) ^{-1}}},{1 \left ({\frac{ \left ( 1/4\,\sqrt [3]{2}+i/4\sqrt{3}\sqrt [3]{2} \right ) c}{d}}-{\frac{ \left ( 1/4\,\sqrt [3]{2}-i/4\sqrt{3}\sqrt [3]{2} \right ) c}{d}} \right ) \left ({\frac{ \left ( 1/4\,\sqrt [3]{2}+i/4\sqrt{3}\sqrt [3]{2} \right ) c}{d}}+{\frac{c}{d}} \right ) ^{-1}},\sqrt{{1 \left ({\frac{ \left ( 1/4\,\sqrt [3]{2}+i/4\sqrt{3}\sqrt [3]{2} \right ) c}{d}}-{\frac{ \left ( 1/4\,\sqrt [3]{2}-i/4\sqrt{3}\sqrt [3]{2} \right ) c}{d}} \right ) \left ({\frac{ \left ( 1/4\,\sqrt [3]{2}+i/4\sqrt{3}\sqrt [3]{2} \right ) c}{d}}+1/2\,{\frac{\sqrt [3]{2}c}{d}} \right ) ^{-1}}} \right ) \left ({\frac{ \left ( 1/4\,\sqrt [3]{2}+i/4\sqrt{3}\sqrt [3]{2} \right ) c}{d}}+{\frac{c}{d}} \right ) ^{-1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(d*x+c)/(4*d^3*x^3+c^3)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{4 \, d^{3} x^{3} + c^{3}}{\left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(4*d^3*x^3 + c^3)*(d*x + c)),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{4 \, d^{3} x^{3} + c^{3}}{\left (d x + c\right )}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(4*d^3*x^3 + c^3)*(d*x + c)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (c + d x\right ) \sqrt{c^{3} + 4 d^{3} x^{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(d*x+c)/(4*d**3*x**3+c**3)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{4 \, d^{3} x^{3} + c^{3}}{\left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(4*d^3*x^3 + c^3)*(d*x + c)),x, algorithm="giac")
[Out]