Optimal. Leaf size=287 \[ \frac{6 \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{5 c d^{7/2} \left (b^2-4 a c\right )^{5/4} \sqrt{a+b x+c x^2}}-\frac{6 \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{5 c d^{7/2} \left (b^2-4 a c\right )^{5/4} \sqrt{a+b x+c x^2}}+\frac{12 \sqrt{a+b x+c x^2}}{5 d^3 \left (b^2-4 a c\right )^2 \sqrt{b d+2 c d x}}+\frac{4 \sqrt{a+b x+c x^2}}{5 d \left (b^2-4 a c\right ) (b d+2 c d x)^{5/2}} \]
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Rubi [A] time = 0.843015, antiderivative size = 287, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{6 \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{5 c d^{7/2} \left (b^2-4 a c\right )^{5/4} \sqrt{a+b x+c x^2}}-\frac{6 \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{5 c d^{7/2} \left (b^2-4 a c\right )^{5/4} \sqrt{a+b x+c x^2}}+\frac{12 \sqrt{a+b x+c x^2}}{5 d^3 \left (b^2-4 a c\right )^2 \sqrt{b d+2 c d x}}+\frac{4 \sqrt{a+b x+c x^2}}{5 d \left (b^2-4 a c\right ) (b d+2 c d x)^{5/2}} \]
Antiderivative was successfully verified.
[In] Int[1/((b*d + 2*c*d*x)^(7/2)*Sqrt[a + b*x + c*x^2]),x]
[Out]
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Rubi in Sympy [A] time = 152.422, size = 274, normalized size = 0.95 \[ \frac{4 \sqrt{a + b x + c x^{2}}}{5 d \left (- 4 a c + b^{2}\right ) \left (b d + 2 c d x\right )^{\frac{5}{2}}} + \frac{12 \sqrt{a + b x + c x^{2}}}{5 d^{3} \left (- 4 a c + b^{2}\right )^{2} \sqrt{b d + 2 c d x}} - \frac{6 \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} E\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{5 c d^{\frac{7}{2}} \left (- 4 a c + b^{2}\right )^{\frac{5}{4}} \sqrt{a + b x + c x^{2}}} + \frac{6 \sqrt{\frac{c \left (a + b x + c x^{2}\right )}{4 a c - b^{2}}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}\middle | -1\right )}{5 c d^{\frac{7}{2}} \left (- 4 a c + b^{2}\right )^{\frac{5}{4}} \sqrt{a + b x + c x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(2*c*d*x+b*d)**(7/2)/(c*x**2+b*x+a)**(1/2),x)
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Mathematica [C] time = 1.54924, size = 206, normalized size = 0.72 \[ \frac{2 \left (8 (a+x (b+c x)) \left (c \left (3 c x^2-a\right )+b^2+3 b c x\right )-\frac{3 i (b+2 c x)^4 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \left (E\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{b+2 c x}{\sqrt{b^2-4 a c}}}\right )\right |-1\right )\right )}{c \left (-\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )^{3/2}}\right )}{5 d \left (b^2-4 a c\right )^2 \sqrt{a+x (b+c x)} (d (b+2 c x))^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((b*d + 2*c*d*x)^(7/2)*Sqrt[a + b*x + c*x^2]),x]
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Maple [B] time = 0.038, size = 874, normalized size = 3.1 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(2*c*d*x+b*d)^(7/2)/(c*x^2+b*x+a)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (2 \, c d x + b d\right )}^{\frac{7}{2}} \sqrt{c x^{2} + b x + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((2*c*d*x + b*d)^(7/2)*sqrt(c*x^2 + b*x + a)),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (8 \, c^{3} d^{3} x^{3} + 12 \, b c^{2} d^{3} x^{2} + 6 \, b^{2} c d^{3} x + b^{3} d^{3}\right )} \sqrt{2 \, c d x + b d} \sqrt{c x^{2} + b x + a}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((2*c*d*x + b*d)^(7/2)*sqrt(c*x^2 + b*x + a)),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(1/(2*c*d*x+b*d)**(7/2)/(c*x**2+b*x+a)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (2 \, c d x + b d\right )}^{\frac{7}{2}} \sqrt{c x^{2} + b x + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((2*c*d*x + b*d)^(7/2)*sqrt(c*x^2 + b*x + a)),x, algorithm="giac")
[Out]