Optimal. Leaf size=208 \[ -\frac{2 \sqrt{c} e \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{a^3}+\frac{\sqrt{c+d x} \left (a^2 (-d) f-3 a b d e+4 b^2 c e\right )}{4 a^2 b (a+b x) (b c-a d)}+\frac{\left (a^3 d^2 f+3 a^2 b d^2 e-12 a b^2 c d e+8 b^3 c^2 e\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{4 a^3 b^{3/2} (b c-a d)^{3/2}}+\frac{\sqrt{c+d x} (b e-a f)}{2 a b (a+b x)^2} \]
[Out]
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Rubi [A] time = 0.833057, antiderivative size = 208, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ -\frac{2 \sqrt{c} e \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{a^3}+\frac{\sqrt{c+d x} \left (a^2 (-d) f-3 a b d e+4 b^2 c e\right )}{4 a^2 b (a+b x) (b c-a d)}+\frac{\left (a^3 d^2 f+3 a^2 b d^2 e-12 a b^2 c d e+8 b^3 c^2 e\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{4 a^3 b^{3/2} (b c-a d)^{3/2}}+\frac{\sqrt{c+d x} (b e-a f)}{2 a b (a+b x)^2} \]
Antiderivative was successfully verified.
[In] Int[(Sqrt[c + d*x]*(e + f*x))/(x*(a + b*x)^3),x]
[Out]
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Rubi in Sympy [A] time = 86.9637, size = 189, normalized size = 0.91 \[ - \frac{\sqrt{c + d x} \left (a f - b e\right )}{2 a b \left (a + b x\right )^{2}} + \frac{\sqrt{c + d x} \left (a d \left (a f + 3 b e\right ) - 4 b^{2} c e\right )}{4 a^{2} b \left (a + b x\right ) \left (a d - b c\right )} - \frac{2 \sqrt{c} e \operatorname{atanh}{\left (\frac{\sqrt{c + d x}}{\sqrt{c}} \right )}}{a^{3}} + \frac{\left (a^{3} d^{2} f + 3 a^{2} b d^{2} e - 12 a b^{2} c d e + 8 b^{3} c^{2} e\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{a d - b c}} \right )}}{4 a^{3} b^{\frac{3}{2}} \left (a d - b c\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((f*x+e)*(d*x+c)**(1/2)/x/(b*x+a)**3,x)
[Out]
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Mathematica [A] time = 0.571909, size = 184, normalized size = 0.88 \[ \frac{\frac{a \sqrt{c+d x} \left (\frac{(a+b x) \left (a^2 d f+3 a b d e-4 b^2 c e\right )}{a d-b c}+2 a (b e-a f)\right )}{b (a+b x)^2}+\frac{\left (a^3 d^2 f+3 a^2 b d^2 e-12 a b^2 c d e+8 b^3 c^2 e\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{3/2} (b c-a d)^{3/2}}-8 \sqrt{c} e \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{4 a^3} \]
Antiderivative was successfully verified.
[In] Integrate[(Sqrt[c + d*x]*(e + f*x))/(x*(a + b*x)^3),x]
[Out]
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Maple [B] time = 0.028, size = 424, normalized size = 2. \[ -2\,{\frac{e\sqrt{c}}{{a}^{3}}{\it Artanh} \left ({\frac{\sqrt{dx+c}}{\sqrt{c}}} \right ) }+{\frac{{d}^{2}f}{4\, \left ( bdx+ad \right ) ^{2} \left ( ad-bc \right ) } \left ( dx+c \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{d}^{2}be}{4\,a \left ( bdx+ad \right ) ^{2} \left ( ad-bc \right ) } \left ( dx+c \right ) ^{{\frac{3}{2}}}}-{\frac{{b}^{2}cde}{{a}^{2} \left ( bdx+ad \right ) ^{2} \left ( ad-bc \right ) } \left ( dx+c \right ) ^{{\frac{3}{2}}}}-{\frac{{d}^{2}f}{4\, \left ( bdx+ad \right ) ^{2}b}\sqrt{dx+c}}+{\frac{5\,{d}^{2}e}{4\,a \left ( bdx+ad \right ) ^{2}}\sqrt{dx+c}}-{\frac{bdce}{{a}^{2} \left ( bdx+ad \right ) ^{2}}\sqrt{dx+c}}+{\frac{{d}^{2}f}{ \left ( 4\,ad-4\,bc \right ) b}\arctan \left ({b\sqrt{dx+c}{\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}}+{\frac{3\,{d}^{2}e}{4\,a \left ( ad-bc \right ) }\arctan \left ({b\sqrt{dx+c}{\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}}-3\,{\frac{bdce}{{a}^{2} \left ( ad-bc \right ) \sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }+2\,{\frac{{b}^{2}{c}^{2}e}{{a}^{3} \left ( ad-bc \right ) \sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{\sqrt{dx+c}b}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((f*x+e)*(d*x+c)^(1/2)/x/(b*x+a)^3,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(d*x + c)*(f*x + e)/((b*x + a)^3*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.839792, size = 1, normalized size = 0. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x + c)*(f*x + e)/((b*x + a)^3*x),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((f*x+e)*(d*x+c)**(1/2)/x/(b*x+a)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.228061, size = 405, normalized size = 1.95 \[ -\frac{{\left (a^{3} d^{2} f + 8 \, b^{3} c^{2} e - 12 \, a b^{2} c d e + 3 \, a^{2} b d^{2} e\right )} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{4 \,{\left (a^{3} b^{2} c - a^{4} b d\right )} \sqrt{-b^{2} c + a b d}} + \frac{2 \, c \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c}}\right ) e}{a^{3} \sqrt{-c}} - \frac{{\left (d x + c\right )}^{\frac{3}{2}} a^{2} b d^{2} f + \sqrt{d x + c} a^{2} b c d^{2} f - \sqrt{d x + c} a^{3} d^{3} f - 4 \,{\left (d x + c\right )}^{\frac{3}{2}} b^{3} c d e + 4 \, \sqrt{d x + c} b^{3} c^{2} d e + 3 \,{\left (d x + c\right )}^{\frac{3}{2}} a b^{2} d^{2} e - 9 \, \sqrt{d x + c} a b^{2} c d^{2} e + 5 \, \sqrt{d x + c} a^{2} b d^{3} e}{4 \,{\left (a^{2} b^{2} c - a^{3} b d\right )}{\left ({\left (d x + c\right )} b - b c + a d\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x + c)*(f*x + e)/((b*x + a)^3*x),x, algorithm="giac")
[Out]