Optimal. Leaf size=152 \[ \frac{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2}}{11 e^3 (a+b x)}-\frac{4 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)}{9 e^3 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^2}{7 e^3 (a+b x)} \]
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Rubi [A] time = 0.21209, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.086 \[ \frac{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2}}{11 e^3 (a+b x)}-\frac{4 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)}{9 e^3 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^2}{7 e^3 (a+b x)} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]
[Out]
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Rubi in Sympy [A] time = 25.0349, size = 129, normalized size = 0.85 \[ \frac{2 \left (a + b x\right ) \left (d + e x\right )^{\frac{7}{2}} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{11 e} + \frac{8 \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{99 e^{2}} + \frac{16 \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right )^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{693 e^{3} \left (a + b x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)*(e*x+d)**(5/2)*((b*x+a)**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.119331, size = 79, normalized size = 0.52 \[ \frac{2 \sqrt{(a+b x)^2} (d+e x)^{7/2} \left (99 a^2 e^2+22 a b e (7 e x-2 d)+b^2 \left (8 d^2-28 d e x+63 e^2 x^2\right )\right )}{693 e^3 (a+b x)} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]
[Out]
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Maple [A] time = 0.007, size = 79, normalized size = 0.5 \[{\frac{126\,{x}^{2}{b}^{2}{e}^{2}+308\,xab{e}^{2}-56\,x{b}^{2}de+198\,{a}^{2}{e}^{2}-88\,abde+16\,{b}^{2}{d}^{2}}{693\,{e}^{3} \left ( bx+a \right ) } \left ( ex+d \right ) ^{{\frac{7}{2}}}\sqrt{ \left ( bx+a \right ) ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)*(e*x+d)^(5/2)*((b*x+a)^2)^(1/2),x)
[Out]
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Maxima [A] time = 0.74149, size = 289, normalized size = 1.9 \[ \frac{2 \,{\left (7 \, b e^{4} x^{4} - 2 \, b d^{4} + 9 \, a d^{3} e +{\left (19 \, b d e^{3} + 9 \, a e^{4}\right )} x^{3} + 3 \,{\left (5 \, b d^{2} e^{2} + 9 \, a d e^{3}\right )} x^{2} +{\left (b d^{3} e + 27 \, a d^{2} e^{2}\right )} x\right )} \sqrt{e x + d} a}{63 \, e^{2}} + \frac{2 \,{\left (63 \, b e^{5} x^{5} + 8 \, b d^{5} - 22 \, a d^{4} e + 7 \,{\left (23 \, b d e^{4} + 11 \, a e^{5}\right )} x^{4} +{\left (113 \, b d^{2} e^{3} + 209 \, a d e^{4}\right )} x^{3} + 3 \,{\left (b d^{3} e^{2} + 55 \, a d^{2} e^{3}\right )} x^{2} -{\left (4 \, b d^{4} e - 11 \, a d^{3} e^{2}\right )} x\right )} \sqrt{e x + d} b}{693 \, e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x + a)^2)*(b*x + a)*(e*x + d)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.276693, size = 235, normalized size = 1.55 \[ \frac{2 \,{\left (63 \, b^{2} e^{5} x^{5} + 8 \, b^{2} d^{5} - 44 \, a b d^{4} e + 99 \, a^{2} d^{3} e^{2} + 7 \,{\left (23 \, b^{2} d e^{4} + 22 \, a b e^{5}\right )} x^{4} +{\left (113 \, b^{2} d^{2} e^{3} + 418 \, a b d e^{4} + 99 \, a^{2} e^{5}\right )} x^{3} + 3 \,{\left (b^{2} d^{3} e^{2} + 110 \, a b d^{2} e^{3} + 99 \, a^{2} d e^{4}\right )} x^{2} -{\left (4 \, b^{2} d^{4} e - 22 \, a b d^{3} e^{2} - 297 \, a^{2} d^{2} e^{3}\right )} x\right )} \sqrt{e x + d}}{693 \, e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x + a)^2)*(b*x + a)*(e*x + d)^(5/2),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)**(5/2)*((b*x+a)**2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.303946, size = 653, normalized size = 4.3 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x + a)^2)*(b*x + a)*(e*x + d)^(5/2),x, algorithm="giac")
[Out]