Optimal. Leaf size=211 \[ \frac{\sqrt{c} \left (-15 a^2 d^2-10 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{3/2}}+\frac{d^{3/2} (a d+5 b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{\sqrt{b}}-\frac{\sqrt{a+b x} (c+d x)^{5/2}}{2 x^2}-\frac{\sqrt{a+b x} (c+d x)^{3/2} (5 a d+b c)}{4 a x}+\frac{d \sqrt{a+b x} \sqrt{c+d x} (11 a d+b c)}{4 a} \]
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Rubi [A] time = 0.663822, antiderivative size = 211, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318 \[ \frac{\sqrt{c} \left (-15 a^2 d^2-10 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{3/2}}+\frac{d^{3/2} (a d+5 b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{\sqrt{b}}-\frac{\sqrt{a+b x} (c+d x)^{5/2}}{2 x^2}-\frac{\sqrt{a+b x} (c+d x)^{3/2} (5 a d+b c)}{4 a x}+\frac{d \sqrt{a+b x} \sqrt{c+d x} (11 a d+b c)}{4 a} \]
Antiderivative was successfully verified.
[In] Int[(Sqrt[a + b*x]*(c + d*x)^(5/2))/x^3,x]
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Rubi in Sympy [A] time = 88.6344, size = 194, normalized size = 0.92 \[ - \frac{\sqrt{a + b x} \left (c + d x\right )^{\frac{5}{2}}}{2 x^{2}} + \frac{d^{\frac{3}{2}} \left (a d + 5 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{\sqrt{b}} + \frac{d \sqrt{a + b x} \sqrt{c + d x} \left (11 a d + b c\right )}{4 a} - \frac{\sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}} \left (5 a d + b c\right )}{4 a x} - \frac{\sqrt{c} \left (15 a^{2} d^{2} + 10 a b c d - b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{4 a^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x+c)**(5/2)*(b*x+a)**(1/2)/x**3,x)
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Mathematica [A] time = 0.612001, size = 236, normalized size = 1.12 \[ -\frac{\sqrt{c} \log (x) \left (-15 a^2 d^2-10 a b c d+b^2 c^2\right )}{8 a^{3/2}}+\frac{\sqrt{c} \left (-15 a^2 d^2-10 a b c d+b^2 c^2\right ) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{8 a^{3/2}}+\sqrt{a+b x} \sqrt{c+d x} \left (-\frac{c (9 a d+b c)}{4 a x}-\frac{c^2}{2 x^2}+d^2\right )+\frac{d^{3/2} (a d+5 b c) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{2 \sqrt{b}} \]
Antiderivative was successfully verified.
[In] Integrate[(Sqrt[a + b*x]*(c + d*x)^(5/2))/x^3,x]
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Maple [B] time = 0.023, size = 512, normalized size = 2.4 \[ -{\frac{1}{8\,a{x}^{2}}\sqrt{bx+a}\sqrt{dx+c} \left ( 15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{2}{a}^{2}c{d}^{2}\sqrt{bd}+10\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{2}ab{c}^{2}d\sqrt{bd}-\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac \right ) } \right ){x}^{2}{b}^{2}{c}^{3}\sqrt{bd}-4\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){x}^{2}{a}^{2}{d}^{3}\sqrt{ac}-20\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){x}^{2}abc{d}^{2}\sqrt{ac}-8\,{x}^{2}a{d}^{2}\sqrt{bd}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+18\,xacd\sqrt{bd}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,xb{c}^{2}\sqrt{bd}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+4\,a{c}^{2}\sqrt{bd}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac} \right ){\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ac}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x+c)^(5/2)*(b*x+a)^(1/2)/x^3,x)
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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(b*x + a)*(d*x + c)^(5/2)/x^3,x, algorithm="maxima")
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Fricas [A] time = 2.32573, size = 1, normalized size = 0. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x + a)*(d*x + c)^(5/2)/x^3,x, algorithm="fricas")
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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((d*x+c)**(5/2)*(b*x+a)**(1/2)/x**3,x)
[Out]
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GIAC/XCAS [A] time = 0.626101, 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^3,x, algorithm="giac")
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