Optimal. Leaf size=581 \[ \frac{2 \sqrt{d+e x} \sqrt{a+b x+c x^2} \left (c e (9 b d-5 a e)-4 b^2 e^2+12 c e x (2 c d-b e)+3 c^2 d^2\right )}{105 c^2 e}+\frac{4 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 a e+3 b d)+2 b^2 e^2+3 c^2 d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{105 c^3 e^2 \sqrt{d+e x} \sqrt{a+b x+c x^2}}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (-c e (29 a e+3 b d)+8 b^2 e^2+3 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{105 c^3 e^2 \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}+\frac{2 e \sqrt{d+e x} \left (a+b x+c x^2\right )^{3/2}}{7 c} \]
[Out]
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Rubi [A] time = 2.22615, antiderivative size = 581, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{2 \sqrt{d+e x} \sqrt{a+b x+c x^2} \left (c e (9 b d-5 a e)-4 b^2 e^2+12 c e x (2 c d-b e)+3 c^2 d^2\right )}{105 c^2 e}+\frac{4 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 a e+3 b d)+2 b^2 e^2+3 c^2 d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{105 c^3 e^2 \sqrt{d+e x} \sqrt{a+b x+c x^2}}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (-c e (29 a e+3 b d)+8 b^2 e^2+3 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{105 c^3 e^2 \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}+\frac{2 e \sqrt{d+e x} \left (a+b x+c x^2\right )^{3/2}}{7 c} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^(3/2)*Sqrt[a + b*x + c*x^2],x]
[Out]
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**(3/2)*(c*x**2+b*x+a)**(1/2),x)
[Out]
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Mathematica [C] time = 13.6195, size = 5328, normalized size = 9.17 \[ \text{Result too large to show} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^(3/2)*Sqrt[a + b*x + c*x^2],x]
[Out]
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Maple [B] time = 0.132, size = 6517, normalized size = 11.2 \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^(3/2)*(c*x^2+b*x+a)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{c x^{2} + b x + a}{\left (e x + d\right )}^{\frac{3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + b*x + a)*(e*x + d)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\sqrt{c x^{2} + b x + a}{\left (e x + d\right )}^{\frac{3}{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + b*x + a)*(e*x + d)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \left (d + e x\right )^{\frac{3}{2}} \sqrt{a + b x + c x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**(3/2)*(c*x**2+b*x+a)**(1/2),x)
[Out]
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GIAC/XCAS [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + b*x + a)*(e*x + d)^(3/2),x, algorithm="giac")
[Out]