Optimal. Leaf size=257 \[ \frac{5 \sqrt{b} (b d-a e) (-3 a B e-4 A b e+7 b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{4 e^{9/2}}-\frac{5 b \sqrt{a+b x} \sqrt{d+e x} (-3 a B e-4 A b e+7 b B d)}{4 e^4}+\frac{5 b (a+b x)^{3/2} \sqrt{d+e x} (-3 a B e-4 A b e+7 b B d)}{6 e^3 (b d-a e)}-\frac{2 (a+b x)^{5/2} (-3 a B e-4 A b e+7 b B d)}{3 e^2 \sqrt{d+e x} (b d-a e)}-\frac{2 (a+b x)^{7/2} (B d-A e)}{3 e (d+e x)^{3/2} (b d-a e)} \]
[Out]
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Rubi [A] time = 0.508585, antiderivative size = 257, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{5 \sqrt{b} (b d-a e) (-3 a B e-4 A b e+7 b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{4 e^{9/2}}-\frac{5 b \sqrt{a+b x} \sqrt{d+e x} (-3 a B e-4 A b e+7 b B d)}{4 e^4}+\frac{5 b (a+b x)^{3/2} \sqrt{d+e x} (-3 a B e-4 A b e+7 b B d)}{6 e^3 (b d-a e)}-\frac{2 (a+b x)^{5/2} (-3 a B e-4 A b e+7 b B d)}{3 e^2 \sqrt{d+e x} (b d-a e)}-\frac{2 (a+b x)^{7/2} (B d-A e)}{3 e (d+e x)^{3/2} (b d-a e)} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x)^(5/2)*(A + B*x))/(d + e*x)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 50.2489, size = 253, normalized size = 0.98 \[ \frac{5 \sqrt{b} \left (a e - b d\right ) \left (4 A b e + 3 B a e - 7 B b d\right ) \operatorname{atanh}{\left (\frac{\sqrt{e} \sqrt{a + b x}}{\sqrt{b} \sqrt{d + e x}} \right )}}{4 e^{\frac{9}{2}}} + \frac{5 b \left (a + b x\right )^{\frac{3}{2}} \sqrt{d + e x} \left (4 A b e + 3 B a e - 7 B b d\right )}{6 e^{3} \left (a e - b d\right )} + \frac{5 b \sqrt{a + b x} \sqrt{d + e x} \left (4 A b e + 3 B a e - 7 B b d\right )}{4 e^{4}} - \frac{2 \left (a + b x\right )^{\frac{7}{2}} \left (A e - B d\right )}{3 e \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )} - \frac{2 \left (a + b x\right )^{\frac{5}{2}} \left (4 A b e + 3 B a e - 7 B b d\right )}{3 e^{2} \sqrt{d + e x} \left (a e - b d\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**(5/2)*(B*x+A)/(e*x+d)**(5/2),x)
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Mathematica [A] time = 0.569231, size = 228, normalized size = 0.89 \[ -\frac{\sqrt{a+b x} \left (8 a^2 e^2 (A e+2 B d+3 B e x)+a b e \left (8 A e (5 d+7 e x)-B \left (115 d^2+158 d e x+27 e^2 x^2\right )\right )+b^2 \left (B \left (105 d^3+140 d^2 e x+21 d e^2 x^2-6 e^3 x^3\right )-4 A e \left (15 d^2+20 d e x+3 e^2 x^2\right )\right )\right )}{12 e^4 (d+e x)^{3/2}}-\frac{5 \sqrt{b} (b d-a e) (3 a B e+4 A b e-7 b B d) \log \left (2 \sqrt{b} \sqrt{e} \sqrt{a+b x} \sqrt{d+e x}+a e+b d+2 b e x\right )}{8 e^{9/2}} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x)^(5/2)*(A + B*x))/(d + e*x)^(5/2),x]
[Out]
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Maple [B] time = 0.038, size = 1250, normalized size = 4.9 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^(5/2)*(B*x+A)/(e*x+d)^(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)*(b*x + a)^(5/2)/(e*x + d)^(5/2),x, algorithm="maxima")
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Fricas [A] time = 1.12278, 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)^(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)*(B*x+A)/(e*x+d)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.293697, size = 721, normalized size = 2.81 \[ -\frac{5 \,{\left (7 \, B b^{2} d^{2}{\left | b \right |} - 10 \, B a b d{\left | b \right |} e - 4 \, A b^{2} d{\left | b \right |} e + 3 \, B a^{2}{\left | b \right |} e^{2} + 4 \, A a b{\left | b \right |} e^{2}\right )} e^{\left (-\frac{9}{2}\right )}{\rm ln}\left ({\left | -\sqrt{b x + a} \sqrt{b} e^{\frac{1}{2}} + \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e} \right |}\right )}{4 \, \sqrt{b}} + \frac{{\left ({\left (3 \,{\left (b x + a\right )}{\left (\frac{2 \,{\left (B b^{5} d{\left | b \right |} e^{6} - B a b^{4}{\left | b \right |} e^{7}\right )}{\left (b x + a\right )}}{b^{4} d e^{7} - a b^{3} e^{8}} - \frac{7 \, B b^{6} d^{2}{\left | b \right |} e^{5} - 10 \, B a b^{5} d{\left | b \right |} e^{6} - 4 \, A b^{6} d{\left | b \right |} e^{6} + 3 \, B a^{2} b^{4}{\left | b \right |} e^{7} + 4 \, A a b^{5}{\left | b \right |} e^{7}}{b^{4} d e^{7} - a b^{3} e^{8}}\right )} - \frac{20 \,{\left (7 \, B b^{7} d^{3}{\left | b \right |} e^{4} - 17 \, B a b^{6} d^{2}{\left | b \right |} e^{5} - 4 \, A b^{7} d^{2}{\left | b \right |} e^{5} + 13 \, B a^{2} b^{5} d{\left | b \right |} e^{6} + 8 \, A a b^{6} d{\left | b \right |} e^{6} - 3 \, B a^{3} b^{4}{\left | b \right |} e^{7} - 4 \, A a^{2} b^{5}{\left | b \right |} e^{7}\right )}}{b^{4} d e^{7} - a b^{3} e^{8}}\right )}{\left (b x + a\right )} - \frac{15 \,{\left (7 \, B b^{8} d^{4}{\left | b \right |} e^{3} - 24 \, B a b^{7} d^{3}{\left | b \right |} e^{4} - 4 \, A b^{8} d^{3}{\left | b \right |} e^{4} + 30 \, B a^{2} b^{6} d^{2}{\left | b \right |} e^{5} + 12 \, A a b^{7} d^{2}{\left | b \right |} e^{5} - 16 \, B a^{3} b^{5} d{\left | b \right |} e^{6} - 12 \, A a^{2} b^{6} d{\left | b \right |} e^{6} + 3 \, B a^{4} b^{4}{\left | b \right |} e^{7} + 4 \, A a^{3} b^{5}{\left | b \right |} e^{7}\right )}}{b^{4} d e^{7} - a b^{3} e^{8}}\right )} \sqrt{b x + a}}{12 \,{\left (b^{2} d +{\left (b x + a\right )} b e - a b e\right )}^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)^(5/2)/(e*x + d)^(5/2),x, algorithm="giac")
[Out]