Optimal. Leaf size=691 \[ \frac{4 \sqrt{a+b x+c x^2} (2 c d-b e) \left (-4 c e (b d-2 a e)-b^2 e^2+4 c^2 d^2\right )}{15 e^2 \sqrt{d+e x} \left (a e^2-b d e+c d^2\right )^2}+\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-4 c e (4 b d-5 a e)-b^2 e^2+16 c^2 d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{15 e^3 \sqrt{d+e x} \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (-4 c e (b d-2 a e)-b^2 e^2+4 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{15 e^3 \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^2 \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}-\frac{2 \sqrt{a+b x+c x^2} \left (e x \left (-2 c e (7 b d-5 a e)+b^2 e^2+14 c^2 d^2\right )-c d e (5 b d-4 a e)-b e^2 (2 b d-3 a e)+8 c^2 d^3\right )}{15 e^2 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )} \]
[Out]
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Rubi [A] time = 1.98476, antiderivative size = 691, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{4 \sqrt{a+b x+c x^2} (2 c d-b e) \left (-4 c e (b d-2 a e)-b^2 e^2+4 c^2 d^2\right )}{15 e^2 \sqrt{d+e x} \left (a e^2-b d e+c d^2\right )^2}+\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-4 c e (4 b d-5 a e)-b^2 e^2+16 c^2 d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{15 e^3 \sqrt{d+e x} \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (-4 c e (b d-2 a e)-b^2 e^2+4 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{15 e^3 \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^2 \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}-\frac{2 \sqrt{a+b x+c x^2} \left (e x \left (-2 c e (7 b d-5 a e)+b^2 e^2+14 c^2 d^2\right )-c d e (5 b d-4 a e)-b e^2 (2 b d-3 a e)+8 c^2 d^3\right )}{15 e^2 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )} \]
Antiderivative was successfully verified.
[In] Int[((b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(d + e*x)^(7/2),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((2*c*x+b)*(c*x**2+b*x+a)**(1/2)/(e*x+d)**(7/2),x)
[Out]
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Mathematica [C] time = 6.64712, size = 5427, normalized size = 7.85 \[ \text{Result too large to show} \]
Antiderivative was successfully verified.
[In] Integrate[((b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(d + e*x)^(7/2),x]
[Out]
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Maple [B] time = 0.093, size = 19265, normalized size = 27.9 \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2*c*x+b)*(c*x^2+b*x+a)^(1/2)/(e*x+d)^(7/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x^{2} + b x + a}{\left (2 \, c x + b\right )}}{{\left (e x + d\right )}^{\frac{7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + b*x + a)*(2*c*x + b)/(e*x + d)^(7/2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{c x^{2} + b x + a}{\left (2 \, c x + b\right )}}{{\left (e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}\right )} \sqrt{e x + d}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + b*x + a)*(2*c*x + b)/(e*x + d)^(7/2),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((2*c*x+b)*(c*x**2+b*x+a)**(1/2)/(e*x+d)**(7/2),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)*(2*c*x + b)/(e*x + d)^(7/2),x, algorithm="giac")
[Out]