Optimal. Leaf size=209 \[ -\frac{\sqrt{\sqrt{c} d-\sqrt{a} e} \left (\sqrt{a} e+2 \sqrt{c} d\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{4 a^{3/2} c^{5/4}}+\frac{\left (2 \sqrt{c} d-\sqrt{a} e\right ) \sqrt{\sqrt{a} e+\sqrt{c} d} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{4 a^{3/2} c^{5/4}}+\frac{\sqrt{d+e x} (a e+c d x)}{2 a c \left (a-c x^2\right )} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.67898, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{\sqrt{\sqrt{c} d-\sqrt{a} e} \left (\sqrt{a} e+2 \sqrt{c} d\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{4 a^{3/2} c^{5/4}}+\frac{\left (2 \sqrt{c} d-\sqrt{a} e\right ) \sqrt{\sqrt{a} e+\sqrt{c} d} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{4 a^{3/2} c^{5/4}}+\frac{\sqrt{d+e x} (a e+c d x)}{2 a c \left (a-c x^2\right )} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^(3/2)/(a - c*x^2)^2,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 148.502, size = 201, normalized size = 0.96 \[ \frac{\sqrt{d + e x} \left (a e + c d x\right )}{2 a c \left (a - c x^{2}\right )} - \frac{\left (- \sqrt{a} \sqrt{c} d e + a e^{2} - 2 c d^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{c} \sqrt{d + e x}}{\sqrt{\sqrt{a} e + \sqrt{c} d}} \right )}}{4 a^{\frac{3}{2}} c^{\frac{5}{4}} \sqrt{\sqrt{a} e + \sqrt{c} d}} - \frac{\left (\sqrt{a} \sqrt{c} d e + a e^{2} - 2 c d^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{d + e x}}{\sqrt{\sqrt{a} e - \sqrt{c} d}} \right )}}{4 a^{\frac{3}{2}} c^{\frac{5}{4}} \sqrt{\sqrt{a} e - \sqrt{c} d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**(3/2)/(-c*x**2+a)**2,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.335529, size = 218, normalized size = 1.04 \[ \frac{-\frac{\left (-\sqrt{a} \sqrt{c} d e-a e^2+2 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-\sqrt{a} \sqrt{c} e}}\right )}{\sqrt{c d-\sqrt{a} \sqrt{c} e}}+\frac{\left (\sqrt{a} \sqrt{c} d e-a e^2+2 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} \sqrt{c} e+c d}}\right )}{\sqrt{\sqrt{a} \sqrt{c} e+c d}}-\frac{2 \sqrt{a} \sqrt{d+e x} (a e+c d x)}{c x^2-a}}{4 a^{3/2} c} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^(3/2)/(a - c*x^2)^2,x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.049, size = 590, normalized size = 2.8 \[ -{\frac{de}{ \left ( 2\,c{e}^{2}{x}^{2}-2\,a{e}^{2} \right ) a} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{{e}^{3}}{ \left ( 2\,c{e}^{2}{x}^{2}-2\,a{e}^{2} \right ) c}\sqrt{ex+d}}+{\frac{e{d}^{2}}{ \left ( 2\,c{e}^{2}{x}^{2}-2\,a{e}^{2} \right ) a}\sqrt{ex+d}}-{\frac{{e}^{6}{a}^{2}c}{4}{\it Artanh} \left ({ace\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( a{c}^{2}{e}^{2}d+\sqrt{{a}^{3}{c}^{3}{e}^{6}} \right ) a}}}} \right ){\frac{1}{\sqrt{{a}^{3}{c}^{3}{e}^{6}}}}{\frac{1}{\sqrt{ \left ( a{c}^{2}{e}^{2}d+\sqrt{{a}^{3}{c}^{3}{e}^{6}} \right ) a}}}}+{\frac{{e}^{4}a{c}^{2}{d}^{2}}{2}{\it Artanh} \left ({ace\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( a{c}^{2}{e}^{2}d+\sqrt{{a}^{3}{c}^{3}{e}^{6}} \right ) a}}}} \right ){\frac{1}{\sqrt{{a}^{3}{c}^{3}{e}^{6}}}}{\frac{1}{\sqrt{ \left ( a{c}^{2}{e}^{2}d+\sqrt{{a}^{3}{c}^{3}{e}^{6}} \right ) a}}}}+{\frac{d{e}^{2}}{4}{\it Artanh} \left ({ace\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( a{c}^{2}{e}^{2}d+\sqrt{{a}^{3}{c}^{3}{e}^{6}} \right ) a}}}} \right ){\frac{1}{\sqrt{ \left ( a{c}^{2}{e}^{2}d+\sqrt{{a}^{3}{c}^{3}{e}^{6}} \right ) a}}}}-{\frac{{e}^{6}{a}^{2}c}{4}\arctan \left ({ace\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( -a{c}^{2}{e}^{2}d+\sqrt{{a}^{3}{c}^{3}{e}^{6}} \right ) a}}}} \right ){\frac{1}{\sqrt{{a}^{3}{c}^{3}{e}^{6}}}}{\frac{1}{\sqrt{ \left ( -a{c}^{2}{e}^{2}d+\sqrt{{a}^{3}{c}^{3}{e}^{6}} \right ) a}}}}+{\frac{{e}^{4}a{c}^{2}{d}^{2}}{2}\arctan \left ({ace\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( -a{c}^{2}{e}^{2}d+\sqrt{{a}^{3}{c}^{3}{e}^{6}} \right ) a}}}} \right ){\frac{1}{\sqrt{{a}^{3}{c}^{3}{e}^{6}}}}{\frac{1}{\sqrt{ \left ( -a{c}^{2}{e}^{2}d+\sqrt{{a}^{3}{c}^{3}{e}^{6}} \right ) a}}}}-{\frac{d{e}^{2}}{4}\arctan \left ({ace\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( -a{c}^{2}{e}^{2}d+\sqrt{{a}^{3}{c}^{3}{e}^{6}} \right ) a}}}} \right ){\frac{1}{\sqrt{ \left ( -a{c}^{2}{e}^{2}d+\sqrt{{a}^{3}{c}^{3}{e}^{6}} \right ) a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^(3/2)/(-c*x^2+a)^2,x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{\frac{3}{2}}}{{\left (c x^{2} - a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^(3/2)/(c*x^2 - a)^2,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.230541, size = 917, normalized size = 4.39 \[ -\frac{{\left (a c^{2} x^{2} - a^{2} c\right )} \sqrt{\frac{a^{3} c^{2} \sqrt{\frac{e^{6}}{a^{3} c^{5}}} + 4 \, c d^{3} - 3 \, a d e^{2}}{a^{3} c^{2}}} \log \left (-{\left (4 \, c d^{2} e^{3} - a e^{5}\right )} \sqrt{e x + d} +{\left (2 \, a^{3} c^{4} d \sqrt{\frac{e^{6}}{a^{3} c^{5}}} + a^{2} c e^{4}\right )} \sqrt{\frac{a^{3} c^{2} \sqrt{\frac{e^{6}}{a^{3} c^{5}}} + 4 \, c d^{3} - 3 \, a d e^{2}}{a^{3} c^{2}}}\right ) -{\left (a c^{2} x^{2} - a^{2} c\right )} \sqrt{\frac{a^{3} c^{2} \sqrt{\frac{e^{6}}{a^{3} c^{5}}} + 4 \, c d^{3} - 3 \, a d e^{2}}{a^{3} c^{2}}} \log \left (-{\left (4 \, c d^{2} e^{3} - a e^{5}\right )} \sqrt{e x + d} -{\left (2 \, a^{3} c^{4} d \sqrt{\frac{e^{6}}{a^{3} c^{5}}} + a^{2} c e^{4}\right )} \sqrt{\frac{a^{3} c^{2} \sqrt{\frac{e^{6}}{a^{3} c^{5}}} + 4 \, c d^{3} - 3 \, a d e^{2}}{a^{3} c^{2}}}\right ) -{\left (a c^{2} x^{2} - a^{2} c\right )} \sqrt{-\frac{a^{3} c^{2} \sqrt{\frac{e^{6}}{a^{3} c^{5}}} - 4 \, c d^{3} + 3 \, a d e^{2}}{a^{3} c^{2}}} \log \left (-{\left (4 \, c d^{2} e^{3} - a e^{5}\right )} \sqrt{e x + d} +{\left (2 \, a^{3} c^{4} d \sqrt{\frac{e^{6}}{a^{3} c^{5}}} - a^{2} c e^{4}\right )} \sqrt{-\frac{a^{3} c^{2} \sqrt{\frac{e^{6}}{a^{3} c^{5}}} - 4 \, c d^{3} + 3 \, a d e^{2}}{a^{3} c^{2}}}\right ) +{\left (a c^{2} x^{2} - a^{2} c\right )} \sqrt{-\frac{a^{3} c^{2} \sqrt{\frac{e^{6}}{a^{3} c^{5}}} - 4 \, c d^{3} + 3 \, a d e^{2}}{a^{3} c^{2}}} \log \left (-{\left (4 \, c d^{2} e^{3} - a e^{5}\right )} \sqrt{e x + d} -{\left (2 \, a^{3} c^{4} d \sqrt{\frac{e^{6}}{a^{3} c^{5}}} - a^{2} c e^{4}\right )} \sqrt{-\frac{a^{3} c^{2} \sqrt{\frac{e^{6}}{a^{3} c^{5}}} - 4 \, c d^{3} + 3 \, a d e^{2}}{a^{3} c^{2}}}\right ) + 4 \,{\left (c d x + a e\right )} \sqrt{e x + d}}{8 \,{\left (a c^{2} x^{2} - a^{2} c\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^(3/2)/(c*x^2 - a)^2,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
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((e*x+d)**(3/2)/(-c*x**2+a)**2,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^(3/2)/(c*x^2 - a)^2,x, algorithm="giac")
[Out]