Optimal. Leaf size=145 \[ -\frac{2 \sqrt{a+b x} (B d-A e)}{5 e (d+e x)^{5/2} (b d-a e)}+\frac{4 b \sqrt{a+b x} (-5 a B e+4 A b e+b B d)}{15 e \sqrt{d+e x} (b d-a e)^3}+\frac{2 \sqrt{a+b x} (-5 a B e+4 A b e+b B d)}{15 e (d+e x)^{3/2} (b d-a e)^2} \]
[Out]
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Rubi [A] time = 0.268115, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{2 \sqrt{a+b x} (B d-A e)}{5 e (d+e x)^{5/2} (b d-a e)}+\frac{4 b \sqrt{a+b x} (-5 a B e+4 A b e+b B d)}{15 e \sqrt{d+e x} (b d-a e)^3}+\frac{2 \sqrt{a+b x} (-5 a B e+4 A b e+b B d)}{15 e (d+e x)^{3/2} (b d-a e)^2} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)/(Sqrt[a + b*x]*(d + e*x)^(7/2)),x]
[Out]
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Rubi in Sympy [A] time = 24.4646, size = 134, normalized size = 0.92 \[ - \frac{4 b \sqrt{a + b x} \left (4 A b e - 5 B a e + B b d\right )}{15 e \sqrt{d + e x} \left (a e - b d\right )^{3}} + \frac{2 \sqrt{a + b x} \left (4 A b e - 5 B a e + B b d\right )}{15 e \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{2}} - \frac{2 \sqrt{a + b x} \left (A e - B d\right )}{5 e \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)/(e*x+d)**(7/2)/(b*x+a)**(1/2),x)
[Out]
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Mathematica [A] time = 0.208749, size = 131, normalized size = 0.9 \[ \sqrt{a+b x} \sqrt{d+e x} \left (-\frac{2 (A e-B d)}{5 e (d+e x)^3 (a e-b d)}-\frac{4 b (-5 a B e+4 A b e+b B d)}{15 e (d+e x) (a e-b d)^3}+\frac{2 (-5 a B e+4 A b e+b B d)}{15 e (d+e x)^2 (a e-b d)^2}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)/(Sqrt[a + b*x]*(d + e*x)^(7/2)),x]
[Out]
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Maple [A] time = 0.013, size = 177, normalized size = 1.2 \[ -{\frac{16\,A{b}^{2}{e}^{2}{x}^{2}-20\,Bab{e}^{2}{x}^{2}+4\,B{b}^{2}de{x}^{2}-8\,Aab{e}^{2}x+40\,A{b}^{2}dex+10\,B{a}^{2}{e}^{2}x-52\,Babdex+10\,B{b}^{2}{d}^{2}x+6\,A{a}^{2}{e}^{2}-20\,Aabde+30\,A{b}^{2}{d}^{2}+4\,B{a}^{2}de-20\,Bab{d}^{2}}{15\,{a}^{3}{e}^{3}-45\,{a}^{2}bd{e}^{2}+45\,a{b}^{2}{d}^{2}e-15\,{b}^{3}{d}^{3}}\sqrt{bx+a} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)/(e*x+d)^(7/2)/(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)^(7/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.646446, size = 427, normalized size = 2.94 \[ \frac{2 \,{\left (3 \, A a^{2} e^{2} - 5 \,{\left (2 \, B a b - 3 \, A b^{2}\right )} d^{2} + 2 \,{\left (B a^{2} - 5 \, A a b\right )} d e + 2 \,{\left (B b^{2} d e -{\left (5 \, B a b - 4 \, A b^{2}\right )} e^{2}\right )} x^{2} +{\left (5 \, B b^{2} d^{2} - 2 \,{\left (13 \, B a b - 10 \, A b^{2}\right )} d e +{\left (5 \, B a^{2} - 4 \, A a b\right )} e^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{e x + d}}{15 \,{\left (b^{3} d^{6} - 3 \, a b^{2} d^{5} e + 3 \, a^{2} b d^{4} e^{2} - a^{3} d^{3} e^{3} +{\left (b^{3} d^{3} e^{3} - 3 \, a b^{2} d^{2} e^{4} + 3 \, a^{2} b d e^{5} - a^{3} e^{6}\right )} x^{3} + 3 \,{\left (b^{3} d^{4} e^{2} - 3 \, a b^{2} d^{3} e^{3} + 3 \, a^{2} b d^{2} e^{4} - a^{3} d e^{5}\right )} x^{2} + 3 \,{\left (b^{3} d^{5} e - 3 \, a b^{2} d^{4} e^{2} + 3 \, a^{2} b d^{3} e^{3} - a^{3} d^{2} e^{4}\right )} x\right )}} \]
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)/(e*x+d)**(7/2)/(b*x+a)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.263141, size = 477, normalized size = 3.29 \[ -\frac{{\left ({\left (b x + a\right )}{\left (\frac{2 \,{\left (B b^{6} d{\left | b \right |} e^{3} - 5 \, B a b^{5}{\left | b \right |} e^{4} + 4 \, A b^{6}{\left | b \right |} e^{4}\right )}{\left (b x + a\right )}}{b^{12} d^{3} e^{6} - 3 \, a b^{11} d^{2} e^{7} + 3 \, a^{2} b^{10} d e^{8} - a^{3} b^{9} e^{9}} + \frac{5 \,{\left (B b^{7} d^{2}{\left | b \right |} e^{2} - 6 \, B a b^{6} d{\left | b \right |} e^{3} + 4 \, A b^{7} d{\left | b \right |} e^{3} + 5 \, B a^{2} b^{5}{\left | b \right |} e^{4} - 4 \, A a b^{6}{\left | b \right |} e^{4}\right )}}{b^{12} d^{3} e^{6} - 3 \, a b^{11} d^{2} e^{7} + 3 \, a^{2} b^{10} d e^{8} - a^{3} b^{9} e^{9}}\right )} - \frac{15 \,{\left (B a b^{7} d^{2}{\left | b \right |} e^{2} - A b^{8} d^{2}{\left | b \right |} e^{2} - 2 \, B a^{2} b^{6} d{\left | b \right |} e^{3} + 2 \, A a b^{7} d{\left | b \right |} e^{3} + B a^{3} b^{5}{\left | b \right |} e^{4} - A a^{2} b^{6}{\left | b \right |} e^{4}\right )}}{b^{12} d^{3} e^{6} - 3 \, a b^{11} d^{2} e^{7} + 3 \, a^{2} b^{10} d e^{8} - a^{3} b^{9} e^{9}}\right )} \sqrt{b x + a}}{960 \,{\left (b^{2} d +{\left (b x + a\right )} b e - a b e\right )}^{\frac{5}{2}}} \]
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]