Optimal. Leaf size=222 \[ \frac{5 \sqrt{b} (7 b c-3 a d) (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 d^{9/2}}-\frac{5 b \sqrt{a+b x} \sqrt{c+d x} (7 b c-3 a d)}{4 d^4}+\frac{5 b (a+b x)^{3/2} \sqrt{c+d x} (7 b c-3 a d)}{6 d^3 (b c-a d)}-\frac{2 (a+b x)^{5/2} (7 b c-3 a d)}{3 d^2 \sqrt{c+d x} (b c-a d)}-\frac{2 c (a+b x)^{7/2}}{3 d (c+d x)^{3/2} (b c-a d)} \]
[Out]
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Rubi [A] time = 0.321757, antiderivative size = 222, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{5 \sqrt{b} (7 b c-3 a d) (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 d^{9/2}}-\frac{5 b \sqrt{a+b x} \sqrt{c+d x} (7 b c-3 a d)}{4 d^4}+\frac{5 b (a+b x)^{3/2} \sqrt{c+d x} (7 b c-3 a d)}{6 d^3 (b c-a d)}-\frac{2 (a+b x)^{5/2} (7 b c-3 a d)}{3 d^2 \sqrt{c+d x} (b c-a d)}-\frac{2 c (a+b x)^{7/2}}{3 d (c+d x)^{3/2} (b c-a d)} \]
Antiderivative was successfully verified.
[In] Int[(x*(a + b*x)^(5/2))/(c + d*x)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 35.0257, size = 207, normalized size = 0.93 \[ \frac{5 \sqrt{b} \left (a d - b c\right ) \left (3 a d - 7 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{4 d^{\frac{9}{2}}} + \frac{5 b \left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (3 a d - 7 b c\right )}{6 d^{3} \left (a d - b c\right )} + \frac{5 b \sqrt{a + b x} \sqrt{c + d x} \left (3 a d - 7 b c\right )}{4 d^{4}} + \frac{2 c \left (a + b x\right )^{\frac{7}{2}}}{3 d \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )} - \frac{2 \left (a + b x\right )^{\frac{5}{2}} \left (3 a d - 7 b c\right )}{3 d^{2} \sqrt{c + d x} \left (a d - b c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(b*x+a)**(5/2)/(d*x+c)**(5/2),x)
[Out]
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Mathematica [A] time = 0.263988, size = 175, normalized size = 0.79 \[ \frac{\sqrt{a+b x} \left (-8 a^2 d^2 (2 c+3 d x)+a b d \left (115 c^2+158 c d x+27 d^2 x^2\right )+b^2 \left (-\left (105 c^3+140 c^2 d x+21 c d^2 x^2-6 d^3 x^3\right )\right )\right )}{12 d^4 (c+d x)^{3/2}}+\frac{5 \sqrt{b} (7 b c-3 a d) (b c-a d) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{8 d^{9/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(x*(a + b*x)^(5/2))/(c + d*x)^(5/2),x]
[Out]
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Maple [B] time = 0.034, size = 750, normalized size = 3.4 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(b*x+a)^(5/2)/(d*x+c)^(5/2),x)
[Out]
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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)*x/(d*x + c)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.701383, size = 1, normalized size = 0. \[ \left [\frac{15 \,{\left (7 \, b^{2} c^{4} - 10 \, a b c^{3} d + 3 \, a^{2} c^{2} d^{2} +{\left (7 \, b^{2} c^{2} d^{2} - 10 \, a b c d^{3} + 3 \, a^{2} d^{4}\right )} x^{2} + 2 \,{\left (7 \, b^{2} c^{3} d - 10 \, a b c^{2} d^{2} + 3 \, a^{2} c d^{3}\right )} x\right )} \sqrt{\frac{b}{d}} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \,{\left (2 \, b d^{2} x + b c d + a d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} \sqrt{\frac{b}{d}} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right ) + 4 \,{\left (6 \, b^{2} d^{3} x^{3} - 105 \, b^{2} c^{3} + 115 \, a b c^{2} d - 16 \, a^{2} c d^{2} - 3 \,{\left (7 \, b^{2} c d^{2} - 9 \, a b d^{3}\right )} x^{2} - 2 \,{\left (70 \, b^{2} c^{2} d - 79 \, a b c d^{2} + 12 \, a^{2} d^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{48 \,{\left (d^{6} x^{2} + 2 \, c d^{5} x + c^{2} d^{4}\right )}}, \frac{15 \,{\left (7 \, b^{2} c^{4} - 10 \, a b c^{3} d + 3 \, a^{2} c^{2} d^{2} +{\left (7 \, b^{2} c^{2} d^{2} - 10 \, a b c d^{3} + 3 \, a^{2} d^{4}\right )} x^{2} + 2 \,{\left (7 \, b^{2} c^{3} d - 10 \, a b c^{2} d^{2} + 3 \, a^{2} c d^{3}\right )} x\right )} \sqrt{-\frac{b}{d}} \arctan \left (\frac{2 \, b d x + b c + a d}{2 \, \sqrt{b x + a} \sqrt{d x + c} d \sqrt{-\frac{b}{d}}}\right ) + 2 \,{\left (6 \, b^{2} d^{3} x^{3} - 105 \, b^{2} c^{3} + 115 \, a b c^{2} d - 16 \, a^{2} c d^{2} - 3 \,{\left (7 \, b^{2} c d^{2} - 9 \, a b d^{3}\right )} x^{2} - 2 \,{\left (70 \, b^{2} c^{2} d - 79 \, a b c d^{2} + 12 \, a^{2} d^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{24 \,{\left (d^{6} x^{2} + 2 \, c d^{5} x + c^{2} d^{4}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^(5/2)*x/(d*x + c)^(5/2),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(x*(b*x+a)**(5/2)/(d*x+c)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.261446, size = 545, normalized size = 2.45 \[ \frac{{\left ({\left (3 \,{\left (b x + a\right )}{\left (\frac{2 \,{\left (b^{5} c d^{6}{\left | b \right |} - a b^{4} d^{7}{\left | b \right |}\right )}{\left (b x + a\right )}}{b^{4} c d^{7} - a b^{3} d^{8}} - \frac{7 \, b^{6} c^{2} d^{5}{\left | b \right |} - 10 \, a b^{5} c d^{6}{\left | b \right |} + 3 \, a^{2} b^{4} d^{7}{\left | b \right |}}{b^{4} c d^{7} - a b^{3} d^{8}}\right )} - \frac{20 \,{\left (7 \, b^{7} c^{3} d^{4}{\left | b \right |} - 17 \, a b^{6} c^{2} d^{5}{\left | b \right |} + 13 \, a^{2} b^{5} c d^{6}{\left | b \right |} - 3 \, a^{3} b^{4} d^{7}{\left | b \right |}\right )}}{b^{4} c d^{7} - a b^{3} d^{8}}\right )}{\left (b x + a\right )} - \frac{15 \,{\left (7 \, b^{8} c^{4} d^{3}{\left | b \right |} - 24 \, a b^{7} c^{3} d^{4}{\left | b \right |} + 30 \, a^{2} b^{6} c^{2} d^{5}{\left | b \right |} - 16 \, a^{3} b^{5} c d^{6}{\left | b \right |} + 3 \, a^{4} b^{4} d^{7}{\left | b \right |}\right )}}{b^{4} c d^{7} - a b^{3} d^{8}}\right )} \sqrt{b x + a}}{12 \,{\left (b^{2} c +{\left (b x + a\right )} b d - a b d\right )}^{\frac{3}{2}}} - \frac{5 \,{\left (7 \, b^{2} c^{2}{\left | b \right |} - 10 \, a b c d{\left | b \right |} + 3 \, a^{2} d^{2}{\left | b \right |}\right )}{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{4 \, \sqrt{b d} d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^(5/2)*x/(d*x + c)^(5/2),x, algorithm="giac")
[Out]