Optimal. Leaf size=126 \[ -\frac{2 b (d+e x)^{5/2} (-2 a B e-A b e+3 b B d)}{5 e^4}+\frac{2 (d+e x)^{3/2} (b d-a e) (-a B e-2 A b e+3 b B d)}{3 e^4}-\frac{2 \sqrt{d+e x} (b d-a e)^2 (B d-A e)}{e^4}+\frac{2 b^2 B (d+e x)^{7/2}}{7 e^4} \]
[Out]
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Rubi [A] time = 0.155909, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ -\frac{2 b (d+e x)^{5/2} (-2 a B e-A b e+3 b B d)}{5 e^4}+\frac{2 (d+e x)^{3/2} (b d-a e) (-a B e-2 A b e+3 b B d)}{3 e^4}-\frac{2 \sqrt{d+e x} (b d-a e)^2 (B d-A e)}{e^4}+\frac{2 b^2 B (d+e x)^{7/2}}{7 e^4} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2))/Sqrt[d + e*x],x]
[Out]
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Rubi in Sympy [A] time = 53.7812, size = 124, normalized size = 0.98 \[ \frac{2 B b^{2} \left (d + e x\right )^{\frac{7}{2}}}{7 e^{4}} + \frac{2 b \left (d + e x\right )^{\frac{5}{2}} \left (A b e + 2 B a e - 3 B b d\right )}{5 e^{4}} + \frac{2 \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right ) \left (2 A b e + B a e - 3 B b d\right )}{3 e^{4}} + \frac{2 \sqrt{d + e x} \left (A e - B d\right ) \left (a e - b d\right )^{2}}{e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)/(e*x+d)**(1/2),x)
[Out]
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Mathematica [A] time = 0.188306, size = 137, normalized size = 1.09 \[ \frac{2 \sqrt{d+e x} \left (35 a^2 e^2 (3 A e-2 B d+B e x)+14 a b e \left (5 A e (e x-2 d)+B \left (8 d^2-4 d e x+3 e^2 x^2\right )\right )+b^2 \left (7 A e \left (8 d^2-4 d e x+3 e^2 x^2\right )-3 B \left (16 d^3-8 d^2 e x+6 d e^2 x^2-5 e^3 x^3\right )\right )\right )}{105 e^4} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2))/Sqrt[d + e*x],x]
[Out]
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Maple [A] time = 0.013, size = 169, normalized size = 1.3 \[{\frac{30\,B{x}^{3}{b}^{2}{e}^{3}+42\,A{b}^{2}{e}^{3}{x}^{2}+84\,Bab{e}^{3}{x}^{2}-36\,B{b}^{2}d{e}^{2}{x}^{2}+140\,Axab{e}^{3}-56\,Ax{b}^{2}d{e}^{2}+70\,Bx{a}^{2}{e}^{3}-112\,Bxabd{e}^{2}+48\,B{b}^{2}{d}^{2}ex+210\,A{a}^{2}{e}^{3}-280\,Aabd{e}^{2}+112\,A{b}^{2}{d}^{2}e-140\,Bd{e}^{2}{a}^{2}+224\,B{d}^{2}abe-96\,B{b}^{2}{d}^{3}}{105\,{e}^{4}}\sqrt{ex+d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)/(e*x+d)^(1/2),x)
[Out]
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Maxima [A] time = 0.732164, size = 215, normalized size = 1.71 \[ \frac{2 \,{\left (15 \,{\left (e x + d\right )}^{\frac{7}{2}} B b^{2} - 21 \,{\left (3 \, B b^{2} d -{\left (2 \, B a b + A b^{2}\right )} e\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 35 \,{\left (3 \, B b^{2} d^{2} - 2 \,{\left (2 \, B a b + A b^{2}\right )} d e +{\left (B a^{2} + 2 \, A a b\right )} e^{2}\right )}{\left (e x + d\right )}^{\frac{3}{2}} - 105 \,{\left (B b^{2} d^{3} - A a^{2} e^{3} -{\left (2 \, B a b + A b^{2}\right )} d^{2} e +{\left (B a^{2} + 2 \, A a b\right )} d e^{2}\right )} \sqrt{e x + d}\right )}}{105 \, e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)*(B*x + A)/sqrt(e*x + d),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.290406, size = 209, normalized size = 1.66 \[ \frac{2 \,{\left (15 \, B b^{2} e^{3} x^{3} - 48 \, B b^{2} d^{3} + 105 \, A a^{2} e^{3} + 56 \,{\left (2 \, B a b + A b^{2}\right )} d^{2} e - 70 \,{\left (B a^{2} + 2 \, A a b\right )} d e^{2} - 3 \,{\left (6 \, B b^{2} d e^{2} - 7 \,{\left (2 \, B a b + A b^{2}\right )} e^{3}\right )} x^{2} +{\left (24 \, B b^{2} d^{2} e - 28 \,{\left (2 \, B a b + A b^{2}\right )} d e^{2} + 35 \,{\left (B a^{2} + 2 \, A a b\right )} e^{3}\right )} x\right )} \sqrt{e x + d}}{105 \, e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)*(B*x + A)/sqrt(e*x + d),x, algorithm="fricas")
[Out]
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Sympy [A] time = 34.3929, size = 583, normalized size = 4.63 \[ \begin{cases} - \frac{\frac{2 A a^{2} d}{\sqrt{d + e x}} + 2 A a^{2} \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right ) + \frac{4 A a b d \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right )}{e} + \frac{4 A a b \left (\frac{d^{2}}{\sqrt{d + e x}} + 2 d \sqrt{d + e x} - \frac{\left (d + e x\right )^{\frac{3}{2}}}{3}\right )}{e} + \frac{2 A b^{2} d \left (\frac{d^{2}}{\sqrt{d + e x}} + 2 d \sqrt{d + e x} - \frac{\left (d + e x\right )^{\frac{3}{2}}}{3}\right )}{e^{2}} + \frac{2 A b^{2} \left (- \frac{d^{3}}{\sqrt{d + e x}} - 3 d^{2} \sqrt{d + e x} + d \left (d + e x\right )^{\frac{3}{2}} - \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e^{2}} + \frac{2 B a^{2} d \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right )}{e} + \frac{2 B a^{2} \left (\frac{d^{2}}{\sqrt{d + e x}} + 2 d \sqrt{d + e x} - \frac{\left (d + e x\right )^{\frac{3}{2}}}{3}\right )}{e} + \frac{4 B a b d \left (\frac{d^{2}}{\sqrt{d + e x}} + 2 d \sqrt{d + e x} - \frac{\left (d + e x\right )^{\frac{3}{2}}}{3}\right )}{e^{2}} + \frac{4 B a b \left (- \frac{d^{3}}{\sqrt{d + e x}} - 3 d^{2} \sqrt{d + e x} + d \left (d + e x\right )^{\frac{3}{2}} - \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e^{2}} + \frac{2 B b^{2} d \left (- \frac{d^{3}}{\sqrt{d + e x}} - 3 d^{2} \sqrt{d + e x} + d \left (d + e x\right )^{\frac{3}{2}} - \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e^{3}} + \frac{2 B b^{2} \left (\frac{d^{4}}{\sqrt{d + e x}} + 4 d^{3} \sqrt{d + e x} - 2 d^{2} \left (d + e x\right )^{\frac{3}{2}} + \frac{4 d \left (d + e x\right )^{\frac{5}{2}}}{5} - \frac{\left (d + e x\right )^{\frac{7}{2}}}{7}\right )}{e^{3}}}{e} & \text{for}\: e \neq 0 \\\frac{A a^{2} x + \frac{B b^{2} x^{4}}{4} + \frac{x^{3} \left (A b^{2} + 2 B a b\right )}{3} + \frac{x^{2} \left (2 A a b + B a^{2}\right )}{2}}{\sqrt{d}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)/(e*x+d)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.282209, size = 317, normalized size = 2.52 \[ \frac{2}{105} \,{\left (35 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{x e + d} d\right )} B a^{2} e^{\left (-1\right )} + 70 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{x e + d} d\right )} A a b e^{\left (-1\right )} + 14 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} e^{8} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} d e^{8} + 15 \, \sqrt{x e + d} d^{2} e^{8}\right )} B a b e^{\left (-10\right )} + 7 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} e^{8} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} d e^{8} + 15 \, \sqrt{x e + d} d^{2} e^{8}\right )} A b^{2} e^{\left (-10\right )} + 3 \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{18} - 21 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{18} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{18} - 35 \, \sqrt{x e + d} d^{3} e^{18}\right )} B b^{2} e^{\left (-21\right )} + 105 \, \sqrt{x e + d} A a^{2}\right )} e^{\left (-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)*(B*x + A)/sqrt(e*x + d),x, algorithm="giac")
[Out]