Optimal. Leaf size=95 \[ \frac{2 (a+b x)^{3/2} (-5 a B e+2 A b e+3 b B d)}{15 e (d+e x)^{3/2} (b d-a e)^2}-\frac{2 (a+b x)^{3/2} (B d-A e)}{5 e (d+e x)^{5/2} (b d-a e)} \]
[Out]
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Rubi [A] time = 0.166895, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{2 (a+b x)^{3/2} (-5 a B e+2 A b e+3 b B d)}{15 e (d+e x)^{3/2} (b d-a e)^2}-\frac{2 (a+b x)^{3/2} (B d-A e)}{5 e (d+e x)^{5/2} (b d-a e)} \]
Antiderivative was successfully verified.
[In] Int[(Sqrt[a + b*x]*(A + B*x))/(d + e*x)^(7/2),x]
[Out]
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Rubi in Sympy [A] time = 12.5278, size = 85, normalized size = 0.89 \[ - \frac{4 \left (a + b x\right )^{\frac{3}{2}} \left (- A b e + \frac{B \left (5 a e - 3 b d\right )}{2}\right )}{15 e \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{2}} - \frac{2 \left (a + b x\right )^{\frac{3}{2}} \left (A e - B d\right )}{5 e \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(b*x+a)**(1/2)/(e*x+d)**(7/2),x)
[Out]
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Mathematica [A] time = 0.152472, size = 66, normalized size = 0.69 \[ \frac{2 (a+b x)^{3/2} (A (-3 a e+5 b d+2 b e x)+B (-2 a d-5 a e x+3 b d x))}{15 (d+e x)^{5/2} (b d-a e)^2} \]
Antiderivative was successfully verified.
[In] Integrate[(Sqrt[a + b*x]*(A + B*x))/(d + e*x)^(7/2),x]
[Out]
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Maple [A] time = 0.011, size = 74, normalized size = 0.8 \[ -{\frac{-4\,Abex+10\,Baex-6\,Bbdx+6\,Aae-10\,Abd+4\,Bad}{15\,{a}^{2}{e}^{2}-30\,bead+15\,{b}^{2}{d}^{2}} \left ( bx+a \right ) ^{{\frac{3}{2}}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(b*x+a)^(1/2)/(e*x+d)^(7/2),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)*sqrt(b*x + a)/(e*x + d)^(7/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.531964, size = 298, normalized size = 3.14 \[ -\frac{2 \,{\left (3 \, A a^{2} e -{\left (3 \, B b^{2} d -{\left (5 \, B a b - 2 \, A b^{2}\right )} e\right )} x^{2} +{\left (2 \, B a^{2} - 5 \, A a b\right )} d -{\left ({\left (B a b + 5 \, A b^{2}\right )} d -{\left (5 \, B a^{2} + A a b\right )} e\right )} x\right )} \sqrt{b x + a} \sqrt{e x + d}}{15 \,{\left (b^{2} d^{5} - 2 \, a b d^{4} e + a^{2} d^{3} e^{2} +{\left (b^{2} d^{2} e^{3} - 2 \, a b d e^{4} + a^{2} e^{5}\right )} x^{3} + 3 \,{\left (b^{2} d^{3} e^{2} - 2 \, a b d^{2} e^{3} + a^{2} d e^{4}\right )} x^{2} + 3 \,{\left (b^{2} d^{4} e - 2 \, a b d^{3} e^{2} + a^{2} d^{2} e^{3}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*sqrt(b*x + a)/(e*x + d)^(7/2),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)*(b*x+a)**(1/2)/(e*x+d)**(7/2),x)
[Out]
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GIAC/XCAS [A] time = 0.25549, size = 281, normalized size = 2.96 \[ -\frac{{\left (b x + a\right )}^{\frac{3}{2}}{\left (\frac{{\left (3 \, B b^{6} d{\left | b \right |} e^{2} - 5 \, B a b^{5}{\left | b \right |} e^{3} + 2 \, A b^{6}{\left | b \right |} e^{3}\right )}{\left (b x + a\right )}}{b^{12} d^{3} e^{6} - 3 \, a b^{11} d^{2} e^{7} + 3 \, a^{2} b^{10} d e^{8} - a^{3} b^{9} e^{9}} - \frac{5 \,{\left (B a b^{6} d{\left | b \right |} e^{2} - A b^{7} d{\left | b \right |} e^{2} - B a^{2} b^{5}{\left | b \right |} e^{3} + A a b^{6}{\left | b \right |} e^{3}\right )}}{b^{12} d^{3} e^{6} - 3 \, a b^{11} d^{2} e^{7} + 3 \, a^{2} b^{10} d e^{8} - a^{3} b^{9} e^{9}}\right )}}{960 \,{\left (b^{2} d +{\left (b x + a\right )} b e - a b e\right )}^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*sqrt(b*x + a)/(e*x + d)^(7/2),x, algorithm="giac")
[Out]