Optimal. Leaf size=275 \[ \frac{3 \sqrt{a+b x} \sqrt{c+d x} \left (a^2 d^2+6 a b c d+b^2 c^2\right )}{4 c}-\frac{3 \sqrt{a} \left (a^2 d^2+10 a b c d+5 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 \sqrt{c}}+\frac{3 \sqrt{b} \left (5 a^2 d^2+10 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 \sqrt{d}}-\frac{(a+b x)^{5/2} (c+d x)^{3/2}}{2 x^2}-\frac{(a+b x)^{3/2} (c+d x)^{3/2} (3 a d+5 b c)}{4 c x}+\frac{3 b \sqrt{a+b x} (c+d x)^{3/2} (a d+3 b c)}{4 c} \]
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Rubi [A] time = 0.905715, antiderivative size = 275, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318 \[ \frac{3 \sqrt{a+b x} \sqrt{c+d x} \left (a^2 d^2+6 a b c d+b^2 c^2\right )}{4 c}-\frac{3 \sqrt{a} \left (a^2 d^2+10 a b c d+5 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 \sqrt{c}}+\frac{3 \sqrt{b} \left (5 a^2 d^2+10 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 \sqrt{d}}-\frac{(a+b x)^{5/2} (c+d x)^{3/2}}{2 x^2}-\frac{(a+b x)^{3/2} (c+d x)^{3/2} (3 a d+5 b c)}{4 c x}+\frac{3 b \sqrt{a+b x} (c+d x)^{3/2} (a d+3 b c)}{4 c} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x)^(5/2)*(c + d*x)^(3/2))/x^3,x]
[Out]
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Rubi in Sympy [A] time = 140.032, size = 245, normalized size = 0.89 \[ - \frac{3 \sqrt{a} \left (a^{2} d^{2} + 10 a b c d + 5 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{4 \sqrt{c}} + \frac{3 \sqrt{b} \left (5 a^{2} d^{2} + 10 a b c d + b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{4 \sqrt{d}} + 3 b \sqrt{a + b x} \sqrt{c + d x} \left (a d + b c\right ) - \frac{\left (a + b x\right )^{\frac{5}{2}} \left (c + d x\right )^{\frac{3}{2}}}{2 x^{2}} + \frac{3 d \left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (a d + 3 b c\right )}{4 c} - \frac{\left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}} \left (3 a d + 5 b c\right )}{4 c x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**(5/2)*(d*x+c)**(3/2)/x**3,x)
[Out]
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Mathematica [A] time = 0.214794, size = 259, normalized size = 0.94 \[ \frac{1}{8} \left (\frac{3 \sqrt{a} \log (x) \left (a^2 d^2+10 a b c d+5 b^2 c^2\right )}{\sqrt{c}}-\frac{3 \sqrt{a} \left (a^2 d^2+10 a b c d+5 b^2 c^2\right ) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{\sqrt{c}}+\frac{3 \sqrt{b} \left (5 a^2 d^2+10 a b c d+b^2 c^2\right ) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{\sqrt{d}}-\frac{2 \sqrt{a+b x} \sqrt{c+d x} \left (a^2 (2 c+5 d x)+9 a b x (c-d x)-b^2 x^2 (5 c+2 d x)\right )}{x^2}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x)^(5/2)*(c + d*x)^(3/2))/x^3,x]
[Out]
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Maple [B] time = 0.026, size = 650, normalized size = 2.4 \[{\frac{1}{8\,{x}^{2}}\sqrt{bx+a}\sqrt{dx+c} \left ( 15\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){x}^{2}{a}^{2}b{d}^{2}\sqrt{ac}+30\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){x}^{2}a{b}^{2}cd\sqrt{ac}+3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){x}^{2}{b}^{3}{c}^{2}\sqrt{ac}-3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{2}{a}^{3}{d}^{2}\sqrt{bd}-30\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{2}{a}^{2}bcd\sqrt{bd}-15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{2}a{b}^{2}{c}^{2}\sqrt{bd}+4\,{x}^{3}{b}^{2}d\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+18\,{x}^{2}abd\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+10\,{x}^{2}{b}^{2}c\sqrt{bd}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}-10\,x{a}^{2}d\sqrt{bd}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}-18\,xabc\sqrt{bd}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}-4\,{a}^{2}c\sqrt{bd}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac} \right ){\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ac}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^(5/2)*(d*x+c)^(3/2)/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((b*x + a)^(5/2)*(d*x + c)^(3/2)/x^3,x, algorithm="maxima")
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Fricas [A] time = 2.94736, 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)*(d*x + c)^(3/2)/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((b*x+a)**(5/2)*(d*x+c)**(3/2)/x**3,x)
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GIAC/XCAS [A] time = 0.706793, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^(5/2)*(d*x + c)^(3/2)/x^3,x, algorithm="giac")
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