Optimal. Leaf size=218 \[ -2 a^{5/2} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )+\frac{\left (5 a^3 d^3+15 a^2 b c d^2-5 a b^2 c^2 d+b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 \sqrt{b} d^{5/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} (b c-5 a d) (a d+b c)}{8 d^2}+\frac{1}{3} (a+b x)^{5/2} \sqrt{c+d x}+\frac{(a+b x)^{3/2} \sqrt{c+d x} (5 a d+b c)}{12 d} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.759867, antiderivative size = 218, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -2 a^{5/2} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )+\frac{\left (5 a^3 d^3+15 a^2 b c d^2-5 a b^2 c^2 d+b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 \sqrt{b} d^{5/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} (b c-5 a d) (a d+b c)}{8 d^2}+\frac{1}{3} (a+b x)^{5/2} \sqrt{c+d x}+\frac{(a+b x)^{3/2} \sqrt{c+d x} (5 a d+b c)}{12 d} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x)^(5/2)*Sqrt[c + d*x])/x,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 63.4354, size = 199, normalized size = 0.91 \[ - 2 a^{\frac{5}{2}} \sqrt{c} \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )} + \frac{\left (a + b x\right )^{\frac{5}{2}} \sqrt{c + d x}}{3} + \frac{\left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (5 a d + b c\right )}{12 d} + \frac{\sqrt{a + b x} \sqrt{c + d x} \left (a d + b c\right ) \left (5 a d - b c\right )}{8 d^{2}} + \frac{\left (16 a^{2} b c d^{2} + \left (a d - b c\right ) \left (a d + b c\right ) \left (5 a d - b c\right )\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{d} \sqrt{a + b x}} \right )}}{8 \sqrt{b} d^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**(5/2)*(d*x+c)**(1/2)/x,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.146356, size = 233, normalized size = 1.07 \[ -a^{5/2} \sqrt{c} \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )+a^{5/2} \sqrt{c} \log (x)+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (33 a^2 d^2+2 a b d (7 c+13 d x)+b^2 \left (-3 c^2+2 c d x+8 d^2 x^2\right )\right )}{24 d^2}+\frac{\left (5 a^3 d^3+15 a^2 b c d^2-5 a b^2 c^2 d+b^3 c^3\right ) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{16 \sqrt{b} d^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x)^(5/2)*Sqrt[c + d*x])/x,x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.02, size = 583, normalized size = 2.7 \[{\frac{1}{48\,{d}^{2}}\sqrt{bx+a}\sqrt{dx+c} \left ( 16\,{x}^{2}{b}^{2}{d}^{2}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+15\,{d}^{3}\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{3}\sqrt{ac}+45\,{d}^{2}\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{2}c\sqrt{ac}b-15\,{c}^{2}\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) ad\sqrt{ac}{b}^{2}+3\,{c}^{3}\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){b}^{3}\sqrt{ac}-48\,{a}^{3}c\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ) \sqrt{bd}{d}^{2}+52\,{d}^{2}\sqrt{d{x}^{2}b+adx+bcx+ac}xa\sqrt{bd}\sqrt{ac}b+4\,d\sqrt{d{x}^{2}b+adx+bcx+ac}xc\sqrt{bd}\sqrt{ac}{b}^{2}+66\,{d}^{2}\sqrt{d{x}^{2}b+adx+bcx+ac}{a}^{2}\sqrt{bd}\sqrt{ac}+28\,d\sqrt{d{x}^{2}b+adx+bcx+ac}ac\sqrt{bd}\sqrt{ac}b-6\,{c}^{2}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}\sqrt{ac}{b}^{2} \right ){\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ac}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^(5/2)*(d*x+c)^(1/2)/x,x)
[Out]
_______________________________________________________________________________________
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((b*x + a)^(5/2)*sqrt(d*x + c)/x,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 5.92914, size = 1, normalized size = 0. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^(5/2)*sqrt(d*x + c)/x,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x\right )^{\frac{5}{2}} \sqrt{c + d x}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**(5/2)*(d*x+c)**(1/2)/x,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.302352, size = 443, normalized size = 2.03 \[ -\frac{2 \, \sqrt{b d} a^{3} c{\left | b \right |} \arctan \left (-\frac{b^{2} c + a b d -{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt{-a b c d} b}\right )}{\sqrt{-a b c d} b} + \frac{1}{24} \, \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \sqrt{b x + a}{\left (2 \,{\left (b x + a\right )}{\left (\frac{4 \,{\left (b x + a\right )}{\left | b \right |}}{b^{2}} + \frac{b^{3} c d^{3}{\left | b \right |} + 5 \, a b^{2} d^{4}{\left | b \right |}}{b^{4} d^{4}}\right )} - \frac{3 \,{\left (b^{4} c^{2} d^{2}{\left | b \right |} - 4 \, a b^{3} c d^{3}{\left | b \right |} - 5 \, a^{2} b^{2} d^{4}{\left | b \right |}\right )}}{b^{4} d^{4}}\right )} - \frac{{\left (\sqrt{b d} b^{3} c^{3}{\left | b \right |} - 5 \, \sqrt{b d} a b^{2} c^{2} d{\left | b \right |} + 15 \, \sqrt{b d} a^{2} b c d^{2}{\left | b \right |} + 5 \, \sqrt{b d} a^{3} d^{3}{\left | b \right |}\right )}{\rm ln}\left ({\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{16 \, b^{2} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^(5/2)*sqrt(d*x + c)/x,x, algorithm="giac")
[Out]