Optimal. Leaf size=499 \[ \frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-4 c e (4 b d-a e)+3 b^2 e^2+16 c^2 d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{3 c e^4 \sqrt{d+e x} \sqrt{a+b x+c x^2}}-\frac{8 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{3 e^4 \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}+\frac{2 \sqrt{a+b x+c x^2} (-3 b e+8 c d+2 c e x)}{3 e^3 \sqrt{d+e x}}-\frac{2 \left (a+b x+c x^2\right )^{3/2}}{3 e (d+e x)^{3/2}} \]
[Out]
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Rubi [A] time = 1.39026, antiderivative size = 499, 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 \[ \frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-4 c e (4 b d-a e)+3 b^2 e^2+16 c^2 d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{3 c e^4 \sqrt{d+e x} \sqrt{a+b x+c x^2}}-\frac{8 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{3 e^4 \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}+\frac{2 \sqrt{a+b x+c x^2} (-3 b e+8 c d+2 c e x)}{3 e^3 \sqrt{d+e x}}-\frac{2 \left (a+b x+c x^2\right )^{3/2}}{3 e (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x + c*x^2)^(3/2)/(d + e*x)^(5/2),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((c*x**2+b*x+a)**(3/2)/(e*x+d)**(5/2),x)
[Out]
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Mathematica [C] time = 13.1204, size = 5751, normalized size = 11.53 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(a + b*x + c*x^2)^(3/2)/(d + e*x)^(5/2),x]
[Out]
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Maple [B] time = 0.056, size = 5874, normalized size = 11.8 \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x+a)^(3/2)/(e*x+d)^(5/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}}{{\left (e x + d\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^(3/2)/(e*x + d)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}}{{\left (e^{2} x^{2} + 2 \, d e x + d^{2}\right )} \sqrt{e x + d}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^(3/2)/(e*x + d)^(5/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{\left (d + e x\right )^{\frac{5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+b*x+a)**(3/2)/(e*x+d)**(5/2),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((c*x^2 + b*x + a)^(3/2)/(e*x + d)^(5/2),x, algorithm="giac")
[Out]