Optimal. Leaf size=154 \[ -\frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \left (-8 b c (A e+B d)+16 A c^2 d+5 b^2 B e\right )}{64 c^{7/2}}+\frac{(b+2 c x) \sqrt{b x+c x^2} \left (-8 b c (A e+B d)+16 A c^2 d+5 b^2 B e\right )}{64 c^3}-\frac{\left (b x+c x^2\right )^{3/2} (-8 c (A e+B d)+5 b B e-6 B c e x)}{24 c^2} \]
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Rubi [A] time = 0.306905, antiderivative size = 154, 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 \[ -\frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \left (-8 b c (A e+B d)+16 A c^2 d+5 b^2 B e\right )}{64 c^{7/2}}+\frac{(b+2 c x) \sqrt{b x+c x^2} \left (-8 b c (A e+B d)+16 A c^2 d+5 b^2 B e\right )}{64 c^3}-\frac{\left (b x+c x^2\right )^{3/2} (-8 c (A e+B d)+5 b B e-6 B c e x)}{24 c^2} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)*(d + e*x)*Sqrt[b*x + c*x^2],x]
[Out]
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Rubi in Sympy [A] time = 20.3593, size = 151, normalized size = 0.98 \[ \frac{b^{2} \left (- 5 B b^{2} e + 8 c \left (- 2 A c d + b \left (A e + B d\right )\right )\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{64 c^{\frac{7}{2}}} + \frac{\left (b x + c x^{2}\right )^{\frac{3}{2}} \left (- \frac{5 B b e}{2} + 3 B c e x + 4 c \left (A e + B d\right )\right )}{12 c^{2}} - \frac{\left (b + 2 c x\right ) \sqrt{b x + c x^{2}} \left (- 5 B b^{2} e + 8 c \left (- 2 A c d + b \left (A e + B d\right )\right )\right )}{64 c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(e*x+d)*(c*x**2+b*x)**(1/2),x)
[Out]
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Mathematica [A] time = 0.360254, size = 179, normalized size = 1.16 \[ \frac{\sqrt{x (b+c x)} \left (\sqrt{c} \left (-2 b^2 c (12 A e+12 B d+5 B e x)+8 b c^2 (2 A (3 d+e x)+B x (2 d+e x))+16 c^3 x (A (6 d+4 e x)+B x (4 d+3 e x))+15 b^3 B e\right )-\frac{3 b^2 \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right ) \left (-8 b c (A e+B d)+16 A c^2 d+5 b^2 B e\right )}{\sqrt{x} \sqrt{b+c x}}\right )}{192 c^{7/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)*(d + e*x)*Sqrt[b*x + c*x^2],x]
[Out]
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Maple [B] time = 0.01, size = 372, normalized size = 2.4 \[{\frac{Adx}{2}\sqrt{c{x}^{2}+bx}}+{\frac{Abd}{4\,c}\sqrt{c{x}^{2}+bx}}-{\frac{Ad{b}^{2}}{8}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{3}{2}}}}+{\frac{Ae}{3\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{Bd}{3\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{Abex}{4\,c}\sqrt{c{x}^{2}+bx}}-{\frac{Bbdx}{4\,c}\sqrt{c{x}^{2}+bx}}-{\frac{{b}^{2}Ae}{8\,{c}^{2}}\sqrt{c{x}^{2}+bx}}-{\frac{{b}^{2}Bd}{8\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{A{b}^{3}e}{16}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{5}{2}}}}+{\frac{B{b}^{3}d}{16}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{5}{2}}}}+{\frac{Bex}{4\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{5\,bBe}{24\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{b}^{2}Bex}{32\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{5\,{b}^{3}Be}{64\,{c}^{3}}\sqrt{c{x}^{2}+bx}}-{\frac{5\,Be{b}^{4}}{128}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{7}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(e*x+d)*(c*x^2+b*x)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + b*x)*(B*x + A)*(e*x + d),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.328271, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (48 \, B c^{3} e x^{3} + 8 \,{\left (8 \, B c^{3} d +{\left (B b c^{2} + 8 \, A c^{3}\right )} e\right )} x^{2} - 24 \,{\left (B b^{2} c - 2 \, A b c^{2}\right )} d + 3 \,{\left (5 \, B b^{3} - 8 \, A b^{2} c\right )} e + 2 \,{\left (8 \,{\left (B b c^{2} + 6 \, A c^{3}\right )} d -{\left (5 \, B b^{2} c - 8 \, A b c^{2}\right )} e\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{c} + 3 \,{\left (8 \,{\left (B b^{3} c - 2 \, A b^{2} c^{2}\right )} d -{\left (5 \, B b^{4} - 8 \, A b^{3} c\right )} e\right )} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} + 2 \, \sqrt{c x^{2} + b x} c\right )}{384 \, c^{\frac{7}{2}}}, \frac{{\left (48 \, B c^{3} e x^{3} + 8 \,{\left (8 \, B c^{3} d +{\left (B b c^{2} + 8 \, A c^{3}\right )} e\right )} x^{2} - 24 \,{\left (B b^{2} c - 2 \, A b c^{2}\right )} d + 3 \,{\left (5 \, B b^{3} - 8 \, A b^{2} c\right )} e + 2 \,{\left (8 \,{\left (B b c^{2} + 6 \, A c^{3}\right )} d -{\left (5 \, B b^{2} c - 8 \, A b c^{2}\right )} e\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{-c} + 3 \,{\left (8 \,{\left (B b^{3} c - 2 \, A b^{2} c^{2}\right )} d -{\left (5 \, B b^{4} - 8 \, A b^{3} c\right )} e\right )} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right )}{192 \, \sqrt{-c} c^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + b*x)*(B*x + A)*(e*x + d),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{x \left (b + c x\right )} \left (A + B x\right ) \left (d + e x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(e*x+d)*(c*x**2+b*x)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.292745, size = 277, normalized size = 1.8 \[ \frac{1}{192} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \,{\left (6 \, B x e + \frac{8 \, B c^{3} d + B b c^{2} e + 8 \, A c^{3} e}{c^{3}}\right )} x + \frac{8 \, B b c^{2} d + 48 \, A c^{3} d - 5 \, B b^{2} c e + 8 \, A b c^{2} e}{c^{3}}\right )} x - \frac{3 \,{\left (8 \, B b^{2} c d - 16 \, A b c^{2} d - 5 \, B b^{3} e + 8 \, A b^{2} c e\right )}}{c^{3}}\right )} - \frac{{\left (8 \, B b^{3} c d - 16 \, A b^{2} c^{2} d - 5 \, B b^{4} e + 8 \, A b^{3} c e\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{128 \, c^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + b*x)*(B*x + A)*(e*x + d),x, algorithm="giac")
[Out]