Optimal. Leaf size=198 \[ \frac{16 b^2 \sqrt{a+b x} (-7 a B e+6 A b e+b B d)}{105 e \sqrt{d+e x} (b d-a e)^4}+\frac{8 b \sqrt{a+b x} (-7 a B e+6 A b e+b B d)}{105 e (d+e x)^{3/2} (b d-a e)^3}+\frac{2 \sqrt{a+b x} (-7 a B e+6 A b e+b B d)}{35 e (d+e x)^{5/2} (b d-a e)^2}-\frac{2 \sqrt{a+b x} (B d-A e)}{7 e (d+e x)^{7/2} (b d-a e)} \]
[Out]
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Rubi [A] time = 0.364643, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{16 b^2 \sqrt{a+b x} (-7 a B e+6 A b e+b B d)}{105 e \sqrt{d+e x} (b d-a e)^4}+\frac{8 b \sqrt{a+b x} (-7 a B e+6 A b e+b B d)}{105 e (d+e x)^{3/2} (b d-a e)^3}+\frac{2 \sqrt{a+b x} (-7 a B e+6 A b e+b B d)}{35 e (d+e x)^{5/2} (b d-a e)^2}-\frac{2 \sqrt{a+b x} (B d-A e)}{7 e (d+e x)^{7/2} (b d-a e)} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)/(Sqrt[a + b*x]*(d + e*x)^(9/2)),x]
[Out]
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Rubi in Sympy [A] time = 36.4267, size = 187, normalized size = 0.94 \[ \frac{16 b^{2} \sqrt{a + b x} \left (6 A b e - 7 B a e + B b d\right )}{105 e \sqrt{d + e x} \left (a e - b d\right )^{4}} - \frac{8 b \sqrt{a + b x} \left (6 A b e - 7 B a e + B b d\right )}{105 e \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{3}} + \frac{2 \sqrt{a + b x} \left (6 A b e - 7 B a e + B b d\right )}{35 e \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )^{2}} - \frac{2 \sqrt{a + b x} \left (A e - B d\right )}{7 e \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)/(e*x+d)**(9/2)/(b*x+a)**(1/2),x)
[Out]
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Mathematica [A] time = 0.499984, size = 148, normalized size = 0.75 \[ \frac{2 \sqrt{a+b x} \left (8 b^2 (d+e x)^3 (-7 a B e+6 A b e+b B d)+3 (d+e x) (b d-a e)^2 (-7 a B e+6 A b e+b B d)+4 b (d+e x)^2 (b d-a e) (-7 a B e+6 A b e+b B d)-15 (b d-a e)^3 (B d-A e)\right )}{105 e (d+e x)^{7/2} (b d-a e)^4} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)/(Sqrt[a + b*x]*(d + e*x)^(9/2)),x]
[Out]
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Maple [A] time = 0.013, size = 322, normalized size = 1.6 \[ -{\frac{-96\,A{b}^{3}{e}^{3}{x}^{3}+112\,Ba{b}^{2}{e}^{3}{x}^{3}-16\,B{b}^{3}d{e}^{2}{x}^{3}+48\,Aa{b}^{2}{e}^{3}{x}^{2}-336\,A{b}^{3}d{e}^{2}{x}^{2}-56\,B{a}^{2}b{e}^{3}{x}^{2}+400\,Ba{b}^{2}d{e}^{2}{x}^{2}-56\,B{b}^{3}{d}^{2}e{x}^{2}-36\,A{a}^{2}b{e}^{3}x+168\,Aa{b}^{2}d{e}^{2}x-420\,A{b}^{3}{d}^{2}ex+42\,B{a}^{3}{e}^{3}x-202\,B{a}^{2}bd{e}^{2}x+518\,Ba{b}^{2}{d}^{2}ex-70\,B{b}^{3}{d}^{3}x+30\,A{a}^{3}{e}^{3}-126\,A{a}^{2}bd{e}^{2}+210\,Aa{b}^{2}{d}^{2}e-210\,A{b}^{3}{d}^{3}+12\,B{a}^{3}d{e}^{2}-56\,B{a}^{2}b{d}^{2}e+140\,Ba{b}^{2}{d}^{3}}{105\,{e}^{4}{a}^{4}-420\,b{e}^{3}d{a}^{3}+630\,{b}^{2}{e}^{2}{d}^{2}{a}^{2}-420\,a{b}^{3}{d}^{3}e+105\,{b}^{4}{d}^{4}}\sqrt{bx+a} \left ( ex+d \right ) ^{-{\frac{7}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)/(e*x+d)^(9/2)/(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)^(9/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 1.09473, size = 738, normalized size = 3.73 \[ -\frac{2 \,{\left (15 \, A a^{3} e^{3} + 35 \,{\left (2 \, B a b^{2} - 3 \, A b^{3}\right )} d^{3} - 7 \,{\left (4 \, B a^{2} b - 15 \, A a b^{2}\right )} d^{2} e + 3 \,{\left (2 \, B a^{3} - 21 \, A a^{2} b\right )} d e^{2} - 8 \,{\left (B b^{3} d e^{2} -{\left (7 \, B a b^{2} - 6 \, A b^{3}\right )} e^{3}\right )} x^{3} - 4 \,{\left (7 \, B b^{3} d^{2} e - 2 \,{\left (25 \, B a b^{2} - 21 \, A b^{3}\right )} d e^{2} +{\left (7 \, B a^{2} b - 6 \, A a b^{2}\right )} e^{3}\right )} x^{2} -{\left (35 \, B b^{3} d^{3} - 7 \,{\left (37 \, B a b^{2} - 30 \, A b^{3}\right )} d^{2} e +{\left (101 \, B a^{2} b - 84 \, A a b^{2}\right )} d e^{2} - 3 \,{\left (7 \, B a^{3} - 6 \, A a^{2} b\right )} e^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{e x + d}}{105 \,{\left (b^{4} d^{8} - 4 \, a b^{3} d^{7} e + 6 \, a^{2} b^{2} d^{6} e^{2} - 4 \, a^{3} b d^{5} e^{3} + a^{4} d^{4} e^{4} +{\left (b^{4} d^{4} e^{4} - 4 \, a b^{3} d^{3} e^{5} + 6 \, a^{2} b^{2} d^{2} e^{6} - 4 \, a^{3} b d e^{7} + a^{4} e^{8}\right )} x^{4} + 4 \,{\left (b^{4} d^{5} e^{3} - 4 \, a b^{3} d^{4} e^{4} + 6 \, a^{2} b^{2} d^{3} e^{5} - 4 \, a^{3} b d^{2} e^{6} + a^{4} d e^{7}\right )} x^{3} + 6 \,{\left (b^{4} d^{6} e^{2} - 4 \, a b^{3} d^{5} e^{3} + 6 \, a^{2} b^{2} d^{4} e^{4} - 4 \, a^{3} b d^{3} e^{5} + a^{4} d^{2} e^{6}\right )} x^{2} + 4 \,{\left (b^{4} d^{7} e - 4 \, a b^{3} d^{6} e^{2} + 6 \, a^{2} b^{2} d^{5} e^{3} - 4 \, a^{3} b d^{4} e^{4} + a^{4} d^{3} e^{5}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/(sqrt(b*x + a)*(e*x + d)^(9/2)),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)**(9/2)/(b*x+a)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.288498, size = 782, normalized size = 3.95 \[ -\frac{{\left ({\left (4 \,{\left (b x + a\right )}{\left (\frac{2 \,{\left (B b^{8} d{\left | b \right |} e^{5} - 7 \, B a b^{7}{\left | b \right |} e^{6} + 6 \, A b^{8}{\left | b \right |} e^{6}\right )}{\left (b x + a\right )}}{b^{16} d^{4} e^{8} - 4 \, a b^{15} d^{3} e^{9} + 6 \, a^{2} b^{14} d^{2} e^{10} - 4 \, a^{3} b^{13} d e^{11} + a^{4} b^{12} e^{12}} + \frac{7 \,{\left (B b^{9} d^{2}{\left | b \right |} e^{4} - 8 \, B a b^{8} d{\left | b \right |} e^{5} + 6 \, A b^{9} d{\left | b \right |} e^{5} + 7 \, B a^{2} b^{7}{\left | b \right |} e^{6} - 6 \, A a b^{8}{\left | b \right |} e^{6}\right )}}{b^{16} d^{4} e^{8} - 4 \, a b^{15} d^{3} e^{9} + 6 \, a^{2} b^{14} d^{2} e^{10} - 4 \, a^{3} b^{13} d e^{11} + a^{4} b^{12} e^{12}}\right )} + \frac{35 \,{\left (B b^{10} d^{3}{\left | b \right |} e^{3} - 9 \, B a b^{9} d^{2}{\left | b \right |} e^{4} + 6 \, A b^{10} d^{2}{\left | b \right |} e^{4} + 15 \, B a^{2} b^{8} d{\left | b \right |} e^{5} - 12 \, A a b^{9} d{\left | b \right |} e^{5} - 7 \, B a^{3} b^{7}{\left | b \right |} e^{6} + 6 \, A a^{2} b^{8}{\left | b \right |} e^{6}\right )}}{b^{16} d^{4} e^{8} - 4 \, a b^{15} d^{3} e^{9} + 6 \, a^{2} b^{14} d^{2} e^{10} - 4 \, a^{3} b^{13} d e^{11} + a^{4} b^{12} e^{12}}\right )}{\left (b x + a\right )} - \frac{105 \,{\left (B a b^{10} d^{3}{\left | b \right |} e^{3} - A b^{11} d^{3}{\left | b \right |} e^{3} - 3 \, B a^{2} b^{9} d^{2}{\left | b \right |} e^{4} + 3 \, A a b^{10} d^{2}{\left | b \right |} e^{4} + 3 \, B a^{3} b^{8} d{\left | b \right |} e^{5} - 3 \, A a^{2} b^{9} d{\left | b \right |} e^{5} - B a^{4} b^{7}{\left | b \right |} e^{6} + A a^{3} b^{8}{\left | b \right |} e^{6}\right )}}{b^{16} d^{4} e^{8} - 4 \, a b^{15} d^{3} e^{9} + 6 \, a^{2} b^{14} d^{2} e^{10} - 4 \, a^{3} b^{13} d e^{11} + a^{4} b^{12} e^{12}}\right )} \sqrt{b x + a}}{80640 \,{\left (b^{2} d +{\left (b x + a\right )} b e - a b e\right )}^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/(sqrt(b*x + a)*(e*x + d)^(9/2)),x, algorithm="giac")
[Out]