Optimal. Leaf size=164 \[ -\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (-a B e-A b e+2 b B d)}{5 e^3 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e) (B d-A e)}{3 e^3 (a+b x)}+\frac{2 b B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2}}{7 e^3 (a+b x)} \]
[Out]
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Rubi [A] time = 0.253483, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057 \[ -\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (-a B e-A b e+2 b B d)}{5 e^3 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e) (B d-A e)}{3 e^3 (a+b x)}+\frac{2 b B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2}}{7 e^3 (a+b x)} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]
[Out]
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Rubi in Sympy [A] time = 27.6712, size = 170, normalized size = 1.04 \[ \frac{B \left (2 a + 2 b x\right ) \left (d + e x\right )^{\frac{3}{2}} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{7 b e} + \frac{2 \left (d + e x\right )^{\frac{3}{2}} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (7 A b e - 3 B a e - 4 B b d\right )}{35 b e^{2}} + \frac{4 \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (7 A b e - 3 B a e - 4 B b d\right )}{105 b e^{3} \left (a + b x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(e*x+d)**(1/2)*((b*x+a)**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.105634, size = 88, normalized size = 0.54 \[ \frac{2 \sqrt{(a+b x)^2} (d+e x)^{3/2} \left (7 a e (5 A e-2 B d+3 B e x)+7 A b e (3 e x-2 d)+b B \left (8 d^2-12 d e x+15 e^2 x^2\right )\right )}{105 e^3 (a+b x)} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]
[Out]
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Maple [A] time = 0.008, size = 89, normalized size = 0.5 \[{\frac{30\,B{x}^{2}b{e}^{2}+42\,Ab{e}^{2}x+42\,aB{e}^{2}x-24\,Bbdex+70\,A{e}^{2}a-28\,Abde-28\,aBde+16\,Bb{d}^{2}}{105\,{e}^{3} \left ( bx+a \right ) } \left ( ex+d \right ) ^{{\frac{3}{2}}}\sqrt{ \left ( bx+a \right ) ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(e*x+d)^(1/2)*((b*x+a)^2)^(1/2),x)
[Out]
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Maxima [A] time = 0.736378, size = 162, normalized size = 0.99 \[ \frac{2 \,{\left (3 \, b e^{2} x^{2} - 2 \, b d^{2} + 5 \, a d e +{\left (b d e + 5 \, a e^{2}\right )} x\right )} \sqrt{e x + d} A}{15 \, e^{2}} + \frac{2 \,{\left (15 \, b e^{3} x^{3} + 8 \, b d^{3} - 14 \, a d^{2} e + 3 \,{\left (b d e^{2} + 7 \, a e^{3}\right )} x^{2} -{\left (4 \, b d^{2} e - 7 \, a d e^{2}\right )} x\right )} \sqrt{e x + d} B}{105 \, e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x + a)^2)*(B*x + A)*sqrt(e*x + d),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.289161, size = 146, normalized size = 0.89 \[ \frac{2 \,{\left (15 \, B b e^{3} x^{3} + 8 \, B b d^{3} + 35 \, A a d e^{2} - 14 \,{\left (B a + A b\right )} d^{2} e + 3 \,{\left (B b d e^{2} + 7 \,{\left (B a + A b\right )} e^{3}\right )} x^{2} -{\left (4 \, B b d^{2} e - 35 \, A a e^{3} - 7 \,{\left (B a + A b\right )} d e^{2}\right )} x\right )} \sqrt{e x + d}}{105 \, e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x + a)^2)*(B*x + A)*sqrt(e*x + d),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \left (A + B x\right ) \sqrt{d + e x} \sqrt{\left (a + b x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(e*x+d)**(1/2)*((b*x+a)**2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.284535, size = 193, normalized size = 1.18 \[ \frac{2}{105} \,{\left (7 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} B a e^{\left (-1\right )}{\rm sign}\left (b x + a\right ) + 7 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} A b e^{\left (-1\right )}{\rm sign}\left (b x + a\right ) +{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{12} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{12} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{12}\right )} B b e^{\left (-14\right )}{\rm sign}\left (b x + a\right ) + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} A a{\rm sign}\left (b x + a\right )\right )} e^{\left (-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x + a)^2)*(B*x + A)*sqrt(e*x + d),x, algorithm="giac")
[Out]