Optimal. Leaf size=379 \[ -\frac{2 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) \left (24 b^2 e^2-71 b c d e+71 c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{105 c^{7/2} \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{16 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) \left (6 b^2 e^2-11 b c d e+11 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{105 c^{7/2} \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 e \sqrt{b x+c x^2} \sqrt{d+e x} \left (24 b^2 e^2-71 b c d e+71 c^2 d^2\right )}{105 c^3}+\frac{12 e \sqrt{b x+c x^2} (d+e x)^{3/2} (2 c d-b e)}{35 c^2}+\frac{2 e \sqrt{b x+c x^2} (d+e x)^{5/2}}{7 c} \]
[Out]
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Rubi [A] time = 1.38719, antiderivative size = 379, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348 \[ -\frac{2 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) \left (24 b^2 e^2-71 b c d e+71 c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{105 c^{7/2} \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{16 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) \left (6 b^2 e^2-11 b c d e+11 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{105 c^{7/2} \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 e \sqrt{b x+c x^2} \sqrt{d+e x} \left (24 b^2 e^2-71 b c d e+71 c^2 d^2\right )}{105 c^3}+\frac{12 e \sqrt{b x+c x^2} (d+e x)^{3/2} (2 c d-b e)}{35 c^2}+\frac{2 e \sqrt{b x+c x^2} (d+e x)^{5/2}}{7 c} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^(7/2)/Sqrt[b*x + c*x^2],x]
[Out]
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Rubi in Sympy [A] time = 156.414, size = 359, normalized size = 0.95 \[ \frac{2 e \left (d + e x\right )^{\frac{5}{2}} \sqrt{b x + c x^{2}}}{7 c} - \frac{12 e \left (d + e x\right )^{\frac{3}{2}} \left (b e - 2 c d\right ) \sqrt{b x + c x^{2}}}{35 c^{2}} + \frac{2 e \sqrt{d + e x} \sqrt{b x + c x^{2}} \left (24 b^{2} e^{2} - 71 b c d e + 71 c^{2} d^{2}\right )}{105 c^{3}} + \frac{2 d \sqrt{x} \sqrt{- b} \sqrt{1 + \frac{c x}{b}} \sqrt{1 + \frac{e x}{d}} \left (b e - c d\right ) \left (24 b^{2} e^{2} - 71 b c d e + 71 c^{2} d^{2}\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{- b}} \right )}\middle | \frac{b e}{c d}\right )}{105 c^{\frac{7}{2}} \sqrt{d + e x} \sqrt{b x + c x^{2}}} - \frac{16 \sqrt{x} \sqrt{- b} \sqrt{1 + \frac{c x}{b}} \sqrt{d + e x} \left (b e - 2 c d\right ) \left (6 b^{2} e^{2} - 11 b c d e + 11 c^{2} d^{2}\right ) E\left (\operatorname{asin}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{- b}} \right )}\middle | \frac{b e}{c d}\right )}{105 c^{\frac{7}{2}} \sqrt{1 + \frac{e x}{d}} \sqrt{b x + c x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**(7/2)/(c*x**2+b*x)**(1/2),x)
[Out]
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Mathematica [C] time = 3.99003, size = 388, normalized size = 1.02 \[ \frac{2 \sqrt{x} \left (e \sqrt{x} (b+c x) (d+e x) \left (24 b^2 e^2-b c e (89 d+18 e x)+c^2 \left (122 d^2+66 d e x+15 e^2 x^2\right )\right )+\frac{8 (b+c x) (d+e x) \left (-6 b^3 e^3+23 b^2 c d e^2-33 b c^2 d^2 e+22 c^3 d^3\right )}{c \sqrt{x}}+8 i e x \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (-6 b^3 e^3+23 b^2 c d e^2-33 b c^2 d^2 e+22 c^3 d^3\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+\frac{i x \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (48 b^4 e^4-208 b^3 c d e^3+353 b^2 c^2 d^2 e^2-298 b c^3 d^3 e+105 c^4 d^4\right ) F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )}{b}\right )}{105 c^3 \sqrt{x (b+c x)} \sqrt{d+e x}} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^(7/2)/Sqrt[b*x + c*x^2],x]
[Out]
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Maple [B] time = 0.063, size = 918, normalized size = 2.4 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^(7/2)/(c*x^2+b*x)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{\frac{7}{2}}}{\sqrt{c x^{2} + b x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^(7/2)/sqrt(c*x^2 + b*x),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}\right )} \sqrt{e x + d}}{\sqrt{c x^{2} + b x}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^(7/2)/sqrt(c*x^2 + b*x),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((e*x+d)**(7/2)/(c*x**2+b*x)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{\frac{7}{2}}}{\sqrt{c x^{2} + b x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^(7/2)/sqrt(c*x^2 + b*x),x, algorithm="giac")
[Out]