Optimal. Leaf size=198 \[ \frac{\sqrt{d} \left (-a^2 d^2+10 a b c d+15 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 b^{3/2}}-\frac{c^{3/2} (5 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{\sqrt{a}}-\frac{\sqrt{a+b x} (c+d x)^{5/2}}{x}+\frac{3}{2} d \sqrt{a+b x} (c+d x)^{3/2}+\frac{d \sqrt{a+b x} \sqrt{c+d x} (a d+11 b c)}{4 b} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.657025, antiderivative size = 198, 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 \[ \frac{\sqrt{d} \left (-a^2 d^2+10 a b c d+15 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 b^{3/2}}-\frac{c^{3/2} (5 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{\sqrt{a}}-\frac{\sqrt{a+b x} (c+d x)^{5/2}}{x}+\frac{3}{2} d \sqrt{a+b x} (c+d x)^{3/2}+\frac{d \sqrt{a+b x} \sqrt{c+d x} (a d+11 b c)}{4 b} \]
Antiderivative was successfully verified.
[In] Int[(Sqrt[a + b*x]*(c + d*x)^(5/2))/x^2,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 69.7545, size = 182, normalized size = 0.92 \[ \frac{3 d \sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}}}{2} - \frac{\sqrt{a + b x} \left (c + d x\right )^{\frac{5}{2}}}{x} + \frac{d \sqrt{a + b x} \sqrt{c + d x} \left (a d + 11 b c\right )}{4 b} - \frac{\sqrt{d} \left (a^{2} d^{2} - 10 a b c d - 15 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{4 b^{\frac{3}{2}}} - \frac{c^{\frac{3}{2}} \left (5 a d + b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{\sqrt{a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x+c)**(5/2)*(b*x+a)**(1/2)/x**2,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.553977, size = 214, normalized size = 1.08 \[ \frac{1}{8} \left (\frac{\sqrt{d} \left (-a^2 d^2+10 a b c d+15 b^2 c^2\right ) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{b^{3/2}}+\frac{4 c^{3/2} \log (x) (5 a d+b c)}{\sqrt{a}}-\frac{4 c^{3/2} (5 a d+b c) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{\sqrt{a}}+2 \sqrt{a+b x} \sqrt{c+d x} \left (\frac{d^2 (a+2 b x)}{b}-\frac{4 c^2}{x}+9 c d\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(Sqrt[a + b*x]*(c + d*x)^(5/2))/x^2,x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.023, size = 503, normalized size = 2.5 \[ -{\frac{1}{8\,bx}\sqrt{bx+a}\sqrt{dx+c} \left ( \ln \left ({\frac{1}{2} \left ( 2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc \right ){\frac{1}{\sqrt{bd}}}} \right ) x{a}^{2}{d}^{3}\sqrt{ac}-10\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) xabc{d}^{2}\sqrt{ac}-15\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) x{b}^{2}{c}^{2}d\sqrt{ac}+20\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ) xab{c}^{2}d\sqrt{bd}+4\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ) x{b}^{2}{c}^{3}\sqrt{bd}-4\,{x}^{2}b{d}^{2}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}-2\,xa{d}^{2}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}-18\,xbcd\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+8\,b{c}^{2}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd} \right ){\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ac}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x+c)^(5/2)*(b*x+a)^(1/2)/x^2,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(sqrt(b*x + a)*(d*x + c)^(5/2)/x^2,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 2.8966, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x + a)*(d*x + c)^(5/2)/x^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((d*x+c)**(5/2)*(b*x+a)**(1/2)/x**2,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.596493, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x + a)*(d*x + c)^(5/2)/x^2,x, algorithm="giac")
[Out]