Optimal. Leaf size=253 \[ -\frac{5 (b d-a e)^2 (-a B e-6 A b e+7 b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{8 \sqrt{b} e^{9/2}}+\frac{5 \sqrt{a+b x} \sqrt{d+e x} (b d-a e) (-a B e-6 A b e+7 b B d)}{8 e^4}-\frac{5 (a+b x)^{3/2} \sqrt{d+e x} (-a B e-6 A b e+7 b B d)}{12 e^3}+\frac{(a+b x)^{5/2} \sqrt{d+e x} (-a B e-6 A b e+7 b B d)}{3 e^2 (b d-a e)}-\frac{2 (a+b x)^{7/2} (B d-A e)}{e \sqrt{d+e x} (b d-a e)} \]
[Out]
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Rubi [A] time = 0.532623, antiderivative size = 253, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{5 (b d-a e)^2 (-a B e-6 A b e+7 b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{8 \sqrt{b} e^{9/2}}+\frac{5 \sqrt{a+b x} \sqrt{d+e x} (b d-a e) (-a B e-6 A b e+7 b B d)}{8 e^4}-\frac{5 (a+b x)^{3/2} \sqrt{d+e x} (-a B e-6 A b e+7 b B d)}{12 e^3}+\frac{(a+b x)^{5/2} \sqrt{d+e x} (-a B e-6 A b e+7 b B d)}{3 e^2 (b d-a e)}-\frac{2 (a+b x)^{7/2} (B d-A e)}{e \sqrt{d+e x} (b d-a e)} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x)^(5/2)*(A + B*x))/(d + e*x)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 50.4226, size = 241, normalized size = 0.95 \[ - \frac{2 \left (a + b x\right )^{\frac{7}{2}} \left (A e - B d\right )}{e \sqrt{d + e x} \left (a e - b d\right )} + \frac{\left (a + b x\right )^{\frac{5}{2}} \sqrt{d + e x} \left (6 A b e + B a e - 7 B b d\right )}{3 e^{2} \left (a e - b d\right )} + \frac{5 \left (a + b x\right )^{\frac{3}{2}} \sqrt{d + e x} \left (6 A b e + B a e - 7 B b d\right )}{12 e^{3}} + \frac{5 \sqrt{a + b x} \sqrt{d + e x} \left (a e - b d\right ) \left (6 A b e + B a e - 7 B b d\right )}{8 e^{4}} + \frac{5 \left (a e - b d\right )^{2} \left (6 A b e + B a e - 7 B b d\right ) \operatorname{atanh}{\left (\frac{\sqrt{e} \sqrt{a + b x}}{\sqrt{b} \sqrt{d + e x}} \right )}}{8 \sqrt{b} e^{\frac{9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**(5/2)*(B*x+A)/(e*x+d)**(3/2),x)
[Out]
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Mathematica [A] time = 0.430037, size = 230, normalized size = 0.91 \[ \frac{\sqrt{a+b x} \left (3 a^2 e^2 (-16 A e+27 B d+11 B e x)+2 a b e \left (3 A e (25 d+9 e x)+B \left (-95 d^2-34 d e x+13 e^2 x^2\right )\right )+b^2 \left (6 A e \left (-15 d^2-5 d e x+2 e^2 x^2\right )+B \left (105 d^3+35 d^2 e x-14 d e^2 x^2+8 e^3 x^3\right )\right )\right )}{24 e^4 \sqrt{d+e x}}+\frac{5 (b d-a e)^2 (a B e+6 A b e-7 b B d) \log \left (2 \sqrt{b} \sqrt{e} \sqrt{a+b x} \sqrt{d+e x}+a e+b d+2 b e x\right )}{16 \sqrt{b} e^{9/2}} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x)^(5/2)*(A + B*x))/(d + e*x)^(3/2),x]
[Out]
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Maple [B] time = 0.046, size = 1184, normalized size = 4.7 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^(5/2)*(B*x+A)/(e*x+d)^(3/2),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)*(b*x + a)^(5/2)/(e*x + d)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 1.01544, 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)*(b*x + a)^(5/2)/(e*x + d)^(3/2),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)*(B*x+A)/(e*x+d)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.265439, size = 564, normalized size = 2.23 \[ \frac{{\left ({\left (2 \,{\left (\frac{4 \,{\left (b x + a\right )} B b{\left | b \right |} e^{6}}{b^{10} d e^{8} - a b^{9} e^{9}} - \frac{7 \, B b^{2} d{\left | b \right |} e^{5} - B a b{\left | b \right |} e^{6} - 6 \, A b^{2}{\left | b \right |} e^{6}}{b^{10} d e^{8} - a b^{9} e^{9}}\right )}{\left (b x + a\right )} + \frac{5 \,{\left (7 \, B b^{3} d^{2}{\left | b \right |} e^{4} - 8 \, B a b^{2} d{\left | b \right |} e^{5} - 6 \, A b^{3} d{\left | b \right |} e^{5} + B a^{2} b{\left | b \right |} e^{6} + 6 \, A a b^{2}{\left | b \right |} e^{6}\right )}}{b^{10} d e^{8} - a b^{9} e^{9}}\right )}{\left (b x + a\right )} + \frac{15 \,{\left (7 \, B b^{4} d^{3}{\left | b \right |} e^{3} - 15 \, B a b^{3} d^{2}{\left | b \right |} e^{4} - 6 \, A b^{4} d^{2}{\left | b \right |} e^{4} + 9 \, B a^{2} b^{2} d{\left | b \right |} e^{5} + 12 \, A a b^{3} d{\left | b \right |} e^{5} - B a^{3} b{\left | b \right |} e^{6} - 6 \, A a^{2} b^{2}{\left | b \right |} e^{6}\right )}}{b^{10} d e^{8} - a b^{9} e^{9}}\right )} \sqrt{b x + a}}{184320 \, \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e}} + \frac{{\left (7 \, B b^{2} d^{2}{\left | b \right |} - 8 \, B a b d{\left | b \right |} e - 6 \, A b^{2} d{\left | b \right |} e + B a^{2}{\left | b \right |} e^{2} + 6 \, A a b{\left | b \right |} e^{2}\right )} e^{\left (-\frac{11}{2}\right )}{\rm ln}\left ({\left | -\sqrt{b x + a} \sqrt{b} e^{\frac{1}{2}} + \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e} \right |}\right )}{12288 \, b^{\frac{17}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)^(5/2)/(e*x + d)^(3/2),x, algorithm="giac")
[Out]