Optimal. Leaf size=316 \[ \frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e^2 (d+e x)^{5/2}}+\frac{2 (2 c d-b e) (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^{3/2}}+\frac{2 (2 c d-b e)^2 (e f-d g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 \sqrt{d+e x}}-\frac{2 (2 c d-b e)^{5/2} (e f-d g) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{e^2}-\frac{2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{7 c e^2 (d+e x)^{7/2}} \]
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Rubi [A] time = 1.24924, antiderivative size = 316, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ \frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e^2 (d+e x)^{5/2}}+\frac{2 (2 c d-b e) (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^{3/2}}+\frac{2 (2 c d-b e)^2 (e f-d g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 \sqrt{d+e x}}-\frac{2 (2 c d-b e)^{5/2} (e f-d g) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{e^2}-\frac{2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{7 c e^2 (d+e x)^{7/2}} \]
Antiderivative was successfully verified.
[In] Int[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2))/(d + e*x)^(7/2),x]
[Out]
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Rubi in Sympy [A] time = 146.255, size = 282, normalized size = 0.89 \[ \frac{2 \left (b e - 2 c d\right )^{\frac{5}{2}} \left (d g - e f\right ) \operatorname{atan}{\left (\frac{\sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{\sqrt{d + e x} \sqrt{b e - 2 c d}} \right )}}{e^{2}} - \frac{2 \left (b e - 2 c d\right )^{2} \left (d g - e f\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{e^{2} \sqrt{d + e x}} + \frac{2 \left (b e - 2 c d\right ) \left (d g - e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}}{3 e^{2} \left (d + e x\right )^{\frac{3}{2}}} - \frac{2 \left (d g - e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{5}{2}}}{5 e^{2} \left (d + e x\right )^{\frac{5}{2}}} - \frac{2 g \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{7}{2}}}{7 c e^{2} \left (d + e x\right )^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2)/(e*x+d)**(7/2),x)
[Out]
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Mathematica [A] time = 1.61551, size = 255, normalized size = 0.81 \[ \frac{2 ((d+e x) (c (d-e x)-b e))^{5/2} \left (\frac{15 b^3 e^3 g+b^2 c e^2 (-206 d g+161 e f+45 e g x)+b c^2 e \left (612 d^2 g-d e (567 f+167 g x)+e^2 x (77 f+45 g x)\right )+c^3 \left (-526 d^3 g+d^2 e (511 f+157 g x)-2 d e^2 x (56 f+33 g x)+3 e^3 x^2 (7 f+5 g x)\right )}{c (b e-c d+c e x)^2}+\frac{105 (2 c d-b e)^{5/2} (d g-e f) \tanh ^{-1}\left (\frac{\sqrt{-b e+c d-c e x}}{\sqrt{2 c d-b e}}\right )}{(c (d-e x)-b e)^{5/2}}\right )}{105 e^2 (d+e x)^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2))/(d + e*x)^(7/2),x]
[Out]
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Maple [B] time = 0.03, size = 956, normalized size = 3. \[{\frac{2}{105\,c{e}^{2}}\sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}} \left ( 15\,{x}^{3}{c}^{3}{e}^{3}g\sqrt{-cex-be+cd}\sqrt{be-2\,cd}+45\,{x}^{2}b{c}^{2}{e}^{3}g\sqrt{-cex-be+cd}\sqrt{be-2\,cd}-66\,{x}^{2}{c}^{3}d{e}^{2}g\sqrt{-cex-be+cd}\sqrt{be-2\,cd}+21\,{x}^{2}{c}^{3}{e}^{3}f\sqrt{-cex-be+cd}\sqrt{be-2\,cd}+105\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ){b}^{3}cd{e}^{3}g-105\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ){b}^{3}c{e}^{4}f-630\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ){b}^{2}{c}^{2}{d}^{2}{e}^{2}g+630\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ){b}^{2}{c}^{2}d{e}^{3}f+1260\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) b{c}^{3}{d}^{3}eg-1260\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) b{c}^{3}{d}^{2}{e}^{2}f-840\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ){c}^{4}{d}^{4}g+840\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ){c}^{4}{d}^{3}ef+45\,x{b}^{2}c{e}^{3}g\sqrt{-cex-be+cd}\sqrt{be-2\,cd}-167\,xb{c}^{2}d{e}^{2}g\sqrt{-cex-be+cd}\sqrt{be-2\,cd}+77\,xb{c}^{2}{e}^{3}f\sqrt{-cex-be+cd}\sqrt{be-2\,cd}+157\,x{c}^{3}{d}^{2}eg\sqrt{-cex-be+cd}\sqrt{be-2\,cd}-112\,x{c}^{3}d{e}^{2}f\sqrt{-cex-be+cd}\sqrt{be-2\,cd}+15\,\sqrt{be-2\,cd}\sqrt{-cex-be+cd}{b}^{3}{e}^{3}g-206\,\sqrt{be-2\,cd}\sqrt{-cex-be+cd}{b}^{2}cd{e}^{2}g+161\,\sqrt{be-2\,cd}\sqrt{-cex-be+cd}{b}^{2}c{e}^{3}f+612\,\sqrt{be-2\,cd}\sqrt{-cex-be+cd}b{c}^{2}{d}^{2}eg-567\,\sqrt{be-2\,cd}\sqrt{-cex-be+cd}b{c}^{2}d{e}^{2}f-526\,\sqrt{be-2\,cd}\sqrt{-cex-be+cd}{c}^{3}{d}^{3}g+511\,\sqrt{be-2\,cd}\sqrt{-cex-be+cd}{c}^{3}{d}^{2}ef \right ){\frac{1}{\sqrt{ex+d}}}{\frac{1}{\sqrt{-cex-be+cd}}}{\frac{1}{\sqrt{be-2\,cd}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^(7/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(5/2)*(g*x + f)/(e*x + d)^(7/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.312666, size = 1, normalized size = 0. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(5/2)*(g*x + f)/(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((g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2)/(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((-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(5/2)*(g*x + f)/(e*x + d)^(7/2),x, algorithm="giac")
[Out]