Optimal. Leaf size=128 \[ -\frac{5 b^{3/2} (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{d^{7/2}}+\frac{5 b^2 \sqrt{a+b x} \sqrt{c+d x}}{d^3}-\frac{10 b (a+b x)^{3/2}}{3 d^2 \sqrt{c+d x}}-\frac{2 (a+b x)^{5/2}}{3 d (c+d x)^{3/2}} \]
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Rubi [A] time = 0.146591, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ -\frac{5 b^{3/2} (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{d^{7/2}}+\frac{5 b^2 \sqrt{a+b x} \sqrt{c+d x}}{d^3}-\frac{10 b (a+b x)^{3/2}}{3 d^2 \sqrt{c+d x}}-\frac{2 (a+b x)^{5/2}}{3 d (c+d x)^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^(5/2)/(c + d*x)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 19.9603, size = 119, normalized size = 0.93 \[ \frac{5 b^{\frac{3}{2}} \left (a d - b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{d} \sqrt{a + b x}} \right )}}{d^{\frac{7}{2}}} + \frac{5 b^{2} \sqrt{a + b x} \sqrt{c + d x}}{d^{3}} - \frac{10 b \left (a + b x\right )^{\frac{3}{2}}}{3 d^{2} \sqrt{c + d x}} - \frac{2 \left (a + b x\right )^{\frac{5}{2}}}{3 d \left (c + d x\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**(5/2)/(d*x+c)**(5/2),x)
[Out]
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Mathematica [A] time = 0.185716, size = 136, normalized size = 1.06 \[ \frac{\sqrt{a+b x} \left (-2 a^2 d^2-2 a b d (5 c+7 d x)+b^2 \left (15 c^2+20 c d x+3 d^2 x^2\right )\right )}{3 d^3 (c+d x)^{3/2}}-\frac{5 b^{3/2} (b c-a d) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{2 d^{7/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)^(5/2)/(c + d*x)^(5/2),x]
[Out]
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Maple [F] time = 0.046, size = 0, normalized size = 0. \[ \int{1 \left ( bx+a \right ) ^{{\frac{5}{2}}} \left ( dx+c \right ) ^{-{\frac{5}{2}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((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)/(d*x + c)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.498003, size = 1, normalized size = 0.01 \[ \left [-\frac{15 \,{\left (b^{2} c^{3} - a b c^{2} d +{\left (b^{2} c d^{2} - a b d^{3}\right )} x^{2} + 2 \,{\left (b^{2} c^{2} d - a b c d^{2}\right )} x\right )} \sqrt{\frac{b}{d}} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \,{\left (2 \, b d^{2} x + b c d + a d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} \sqrt{\frac{b}{d}} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \,{\left (3 \, b^{2} d^{2} x^{2} + 15 \, b^{2} c^{2} - 10 \, a b c d - 2 \, a^{2} d^{2} + 2 \,{\left (10 \, b^{2} c d - 7 \, a b d^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{12 \,{\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}}, -\frac{15 \,{\left (b^{2} c^{3} - a b c^{2} d +{\left (b^{2} c d^{2} - a b d^{3}\right )} x^{2} + 2 \,{\left (b^{2} c^{2} d - a b c d^{2}\right )} x\right )} \sqrt{-\frac{b}{d}} \arctan \left (\frac{2 \, b d x + b c + a d}{2 \, \sqrt{b x + a} \sqrt{d x + c} d \sqrt{-\frac{b}{d}}}\right ) - 2 \,{\left (3 \, b^{2} d^{2} x^{2} + 15 \, b^{2} c^{2} - 10 \, a b c d - 2 \, a^{2} d^{2} + 2 \,{\left (10 \, b^{2} c d - 7 \, a b d^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{6 \,{\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^(5/2)/(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((b*x+a)**(5/2)/(d*x+c)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.259903, size = 373, normalized size = 2.91 \[ \frac{{\left ({\left (b x + a\right )}{\left (\frac{3 \,{\left (b^{6} c d^{4} - a b^{5} d^{5}\right )}{\left (b x + a\right )}}{b^{2} c d^{5}{\left | b \right |} - a b d^{6}{\left | b \right |}} + \frac{20 \,{\left (b^{7} c^{2} d^{3} - 2 \, a b^{6} c d^{4} + a^{2} b^{5} d^{5}\right )}}{b^{2} c d^{5}{\left | b \right |} - a b d^{6}{\left | b \right |}}\right )} + \frac{15 \,{\left (b^{8} c^{3} d^{2} - 3 \, a b^{7} c^{2} d^{3} + 3 \, a^{2} b^{6} c d^{4} - a^{3} b^{5} d^{5}\right )}}{b^{2} c d^{5}{\left | b \right |} - a b d^{6}{\left | b \right |}}\right )} \sqrt{b x + a}}{3 \,{\left (b^{2} c +{\left (b x + a\right )} b d - a b d\right )}^{\frac{3}{2}}} + \frac{5 \,{\left (b^{4} c - a b^{3} d\right )}{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt{b d} d^{3}{\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^(5/2)/(d*x + c)^(5/2),x, algorithm="giac")
[Out]