Optimal. Leaf size=94 \[ \frac{2 \sqrt{a+b x} (-3 a B e+2 A b e+b B d)}{3 e \sqrt{d+e x} (b d-a e)^2}-\frac{2 \sqrt{a+b x} (B d-A e)}{3 e (d+e x)^{3/2} (b d-a e)} \]
[Out]
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Rubi [A] time = 0.168818, antiderivative size = 94, 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 \[ \frac{2 \sqrt{a+b x} (-3 a B e+2 A b e+b B d)}{3 e \sqrt{d+e x} (b d-a e)^2}-\frac{2 \sqrt{a+b x} (B d-A e)}{3 e (d+e x)^{3/2} (b d-a e)} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)/(Sqrt[a + b*x]*(d + e*x)^(5/2)),x]
[Out]
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Rubi in Sympy [A] time = 12.4328, size = 83, normalized size = 0.88 \[ - \frac{4 \sqrt{a + b x} \left (- A b e + \frac{B \left (3 a e - b d\right )}{2}\right )}{3 e \sqrt{d + e x} \left (a e - b d\right )^{2}} - \frac{2 \sqrt{a + b x} \left (A e - B d\right )}{3 e \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)/(e*x+d)**(5/2)/(b*x+a)**(1/2),x)
[Out]
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Mathematica [A] time = 0.12656, size = 65, normalized size = 0.69 \[ \frac{2 \sqrt{a+b x} (A (-a e+3 b d+2 b e x)+B (-2 a d-3 a e x+b d x))}{3 (d+e x)^{3/2} (b d-a e)^2} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)/(Sqrt[a + b*x]*(d + e*x)^(5/2)),x]
[Out]
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Maple [A] time = 0.01, size = 73, normalized size = 0.8 \[ -{\frac{-4\,Abex+6\,Baex-2\,Bbdx+2\,Aae-6\,Abd+4\,Bad}{3\,{a}^{2}{e}^{2}-6\,bead+3\,{b}^{2}{d}^{2}}\sqrt{bx+a} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)/(e*x+d)^(5/2)/(b*x+a)^(1/2),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)^(5/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.45801, size = 189, normalized size = 2.01 \[ -\frac{2 \,{\left (A a e +{\left (2 \, B a - 3 \, A b\right )} d -{\left (B b d -{\left (3 \, B a - 2 \, A b\right )} e\right )} x\right )} \sqrt{b x + a} \sqrt{e x + d}}{3 \,{\left (b^{2} d^{4} - 2 \, a b d^{3} e + a^{2} d^{2} e^{2} +{\left (b^{2} d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )} x^{2} + 2 \,{\left (b^{2} d^{3} e - 2 \, a b d^{2} e^{2} + a^{2} d e^{3}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/(sqrt(b*x + a)*(e*x + d)^(5/2)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{A + B x}{\sqrt{a + b x} \left (d + e x\right )^{\frac{5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)/(e*x+d)**(5/2)/(b*x+a)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.236492, size = 242, normalized size = 2.57 \[ -\frac{\sqrt{b x + a}{\left (\frac{{\left (B b^{4} d{\left | b \right |} e - 3 \, B a b^{3}{\left | b \right |} e^{2} + 2 \, A b^{4}{\left | b \right |} e^{2}\right )}{\left (b x + a\right )}}{b^{8} d^{2} e^{4} - 2 \, a b^{7} d e^{5} + a^{2} b^{6} e^{6}} - \frac{3 \,{\left (B a b^{4} d{\left | b \right |} e - A b^{5} d{\left | b \right |} e - B a^{2} b^{3}{\left | b \right |} e^{2} + A a b^{4}{\left | b \right |} e^{2}\right )}}{b^{8} d^{2} e^{4} - 2 \, a b^{7} d e^{5} + a^{2} b^{6} e^{6}}\right )}}{48 \,{\left (b^{2} d +{\left (b x + a\right )} b e - a b e\right )}^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/(sqrt(b*x + a)*(e*x + d)^(5/2)),x, algorithm="giac")
[Out]