Optimal. Leaf size=227 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} (b c-5 a d) (a d+b c)}{8 a^2 x}-\frac{\left (5 a^3 d^3+15 a^2 b c d^2-5 a b^2 c^2 d+b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{8 a^{5/2} \sqrt{c}}+2 \sqrt{b} d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )-\frac{\sqrt{a+b x} (c+d x)^{5/2}}{3 x^3}-\frac{\sqrt{a+b x} (c+d x)^{3/2} (5 a d+b c)}{12 a x^2} \]
[Out]
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Rubi [A] time = 0.667691, antiderivative size = 227, 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 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} (b c-5 a d) (a d+b c)}{8 a^2 x}-\frac{\left (5 a^3 d^3+15 a^2 b c d^2-5 a b^2 c^2 d+b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{8 a^{5/2} \sqrt{c}}+2 \sqrt{b} d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )-\frac{\sqrt{a+b x} (c+d x)^{5/2}}{3 x^3}-\frac{\sqrt{a+b x} (c+d x)^{3/2} (5 a d+b c)}{12 a x^2} \]
Antiderivative was successfully verified.
[In] Int[(Sqrt[a + b*x]*(c + d*x)^(5/2))/x^4,x]
[Out]
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Rubi in Sympy [A] time = 101.529, size = 212, normalized size = 0.93 \[ 2 \sqrt{b} d^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )} - \frac{\sqrt{a + b x} \left (c + d x\right )^{\frac{5}{2}}}{3 x^{3}} - \frac{\sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}} \left (5 a d + b c\right )}{12 a x^{2}} - \frac{\sqrt{a + b x} \sqrt{c + d x} \left (a d + b c\right ) \left (5 a d - b c\right )}{8 a^{2} x} - \frac{\left (5 a^{3} d^{3} + 15 a^{2} b c d^{2} - 5 a b^{2} c^{2} d + b^{3} c^{3}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{8 a^{\frac{5}{2}} \sqrt{c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x+c)**(5/2)*(b*x+a)**(1/2)/x**4,x)
[Out]
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Mathematica [A] time = 0.5408, size = 283, normalized size = 1.25 \[ \sqrt{a+b x} \sqrt{c+d x} \left (\frac{-33 a^2 d^2-14 a b c d+3 b^2 c^2}{24 a^2 x}-\frac{c (13 a d+b c)}{12 a x^2}-\frac{c^2}{3 x^3}\right )+\frac{\log (x) \left (5 a^3 d^3+15 a^2 b c d^2-5 a b^2 c^2 d+b^3 c^3\right )}{16 a^{5/2} \sqrt{c}}-\frac{\left (5 a^3 d^3+15 a^2 b c d^2-5 a b^2 c^2 d+b^3 c^3\right ) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{16 a^{5/2} \sqrt{c}}+\sqrt{b} d^{5/2} \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(Sqrt[a + b*x]*(c + d*x)^(5/2))/x^4,x]
[Out]
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Maple [B] time = 0.024, size = 601, normalized size = 2.7 \[{\frac{1}{48\,{a}^{2}{x}^{3}}\sqrt{bx+a}\sqrt{dx+c} \left ( 48\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){x}^{3}{a}^{2}b{d}^{3}\sqrt{ac}-15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{3}{a}^{3}{d}^{3}\sqrt{bd}-45\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{3}{a}^{2}bc{d}^{2}\sqrt{bd}+15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{3}a{b}^{2}{c}^{2}d\sqrt{bd}-3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{3}{b}^{3}{c}^{3}\sqrt{bd}-66\,\sqrt{d{x}^{2}b+adx+bcx+ac}{d}^{2}\sqrt{bd}{a}^{2}\sqrt{ac}{x}^{2}-28\,\sqrt{d{x}^{2}b+adx+bcx+ac}db\sqrt{bd}a\sqrt{ac}{x}^{2}c+6\,{c}^{2}\sqrt{d{x}^{2}b+adx+bcx+ac}{b}^{2}\sqrt{bd}\sqrt{ac}{x}^{2}-52\,\sqrt{d{x}^{2}b+adx+bcx+ac}d\sqrt{bd}{a}^{2}\sqrt{ac}xc-4\,{c}^{2}\sqrt{d{x}^{2}b+adx+bcx+ac}b\sqrt{bd}a\sqrt{ac}x-16\,{c}^{2}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}{a}^{2}\sqrt{ac} \right ){\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ac}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x+c)^(5/2)*(b*x+a)^(1/2)/x^4,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^4,x, algorithm="maxima")
[Out]
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Fricas [A] time = 2.25791, 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^4,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((d*x+c)**(5/2)*(b*x+a)**(1/2)/x**4,x)
[Out]
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GIAC/XCAS [A] time = 0.674676, 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^4,x, algorithm="giac")
[Out]