Optimal. Leaf size=341 \[ \frac{\left (2 \sqrt{a} \sqrt{c}+b\right ) \left (\sqrt{a}+\sqrt{c} x\right ) \left (\sqrt{a} B+3 A \sqrt{c}\right ) \sqrt{\frac{a+b x+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{3 \sqrt [4]{a} c^{3/4} \sqrt{a+b x+c x^2}}-\frac{2 \sqrt [4]{a} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+b x+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} (6 A c+b B) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{3 c^{3/4} \sqrt{a+b x+c x^2}}+\frac{2 \sqrt{x} \sqrt{a+b x+c x^2} (6 A c+b B)}{3 \sqrt{c} \left (\sqrt{a}+\sqrt{c} x\right )}-\frac{2 (3 A-B x) \sqrt{a+b x+c x^2}}{3 \sqrt{x}} \]
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Rubi [A] time = 0.644395, antiderivative size = 341, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{\left (2 \sqrt{a} \sqrt{c}+b\right ) \left (\sqrt{a}+\sqrt{c} x\right ) \left (\sqrt{a} B+3 A \sqrt{c}\right ) \sqrt{\frac{a+b x+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{3 \sqrt [4]{a} c^{3/4} \sqrt{a+b x+c x^2}}-\frac{2 \sqrt [4]{a} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+b x+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} (6 A c+b B) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{3 c^{3/4} \sqrt{a+b x+c x^2}}+\frac{2 \sqrt{x} \sqrt{a+b x+c x^2} (6 A c+b B)}{3 \sqrt{c} \left (\sqrt{a}+\sqrt{c} x\right )}-\frac{2 (3 A-B x) \sqrt{a+b x+c x^2}}{3 \sqrt{x}} \]
Warning: Unable to verify antiderivative.
[In] Int[((A + B*x)*Sqrt[a + b*x + c*x^2])/x^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 94.0299, size = 318, normalized size = 0.93 \[ - \frac{2 \sqrt [4]{a} \sqrt{\frac{a + b x + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) \left (6 A c + B b\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2} - \frac{b}{4 \sqrt{a} \sqrt{c}}\right )}{3 c^{\frac{3}{4}} \sqrt{a + b x + c x^{2}}} - \frac{4 \left (\frac{3 A}{2} - \frac{B x}{2}\right ) \sqrt{a + b x + c x^{2}}}{3 \sqrt{x}} + \frac{2 \sqrt{x} \left (6 A c + B b\right ) \sqrt{a + b x + c x^{2}}}{3 \sqrt{c} \left (\sqrt{a} + \sqrt{c} x\right )} + \frac{\sqrt{\frac{a + b x + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) \left (\sqrt{a} \left (6 A c + B b\right ) + \sqrt{c} \left (3 A b + 2 B a\right )\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2} - \frac{b}{4 \sqrt{a} \sqrt{c}}\right )}{3 \sqrt [4]{a} c^{\frac{3}{4}} \sqrt{a + b x + c x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(c*x**2+b*x+a)**(1/2)/x**(3/2),x)
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Mathematica [C] time = 4.31631, size = 491, normalized size = 1.44 \[ \frac{\frac{i x \sqrt{\frac{4 a}{x \left (\sqrt{b^2-4 a c}+b\right )}+2} \sqrt{\frac{-x \sqrt{b^2-4 a c}+2 a+b x}{b x-x \sqrt{b^2-4 a c}}} \left (6 A c \sqrt{b^2-4 a c}+b B \sqrt{b^2-4 a c}+4 a B c+b^2 (-B)\right ) F\left (i \sinh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{a}{b+\sqrt{b^2-4 a c}}}}{\sqrt{x}}\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )}{c \sqrt{\frac{a}{\sqrt{b^2-4 a c}+b}}}-\frac{i x \left (\sqrt{b^2-4 a c}-b\right ) \sqrt{\frac{4 a}{x \left (\sqrt{b^2-4 a c}+b\right )}+2} \sqrt{\frac{-x \sqrt{b^2-4 a c}+2 a+b x}{b x-x \sqrt{b^2-4 a c}}} (6 A c+b B) E\left (i \sinh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{a}{b+\sqrt{b^2-4 a c}}}}{\sqrt{x}}\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )}{c \sqrt{\frac{a}{\sqrt{b^2-4 a c}+b}}}+\frac{4 (B x-3 A) (a+x (b+c x))}{\sqrt{x}}+\frac{4 (a+x (b+c x)) (6 A c+b B)}{c \sqrt{x}}}{6 \sqrt{a+x (b+c x)}} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*Sqrt[a + b*x + c*x^2])/x^(3/2),x]
[Out]
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Maple [B] time = 0.053, size = 1652, normalized size = 4.8 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(c*x^2+b*x+a)^(1/2)/x^(3/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x^{2} + b x + a}{\left (B x + A\right )}}{x^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + b*x + a)*(B*x + A)/x^(3/2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{c x^{2} + b x + a}{\left (B x + A\right )}}{x^{\frac{3}{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + b*x + a)*(B*x + A)/x^(3/2),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (A + B x\right ) \sqrt{a + b x + c x^{2}}}{x^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(c*x**2+b*x+a)**(1/2)/x**(3/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x^{2} + b x + a}{\left (B x + A\right )}}{x^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + b*x + a)*(B*x + A)/x^(3/2),x, algorithm="giac")
[Out]