Optimal. Leaf size=358 \[ \frac{32 \sqrt{-a} c^{3/2} d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{3 e^4 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}-\frac{8 \sqrt{-a} \sqrt{c} \sqrt{\frac{c x^2}{a}+1} \left (a e^2+4 c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{3 e^4 \sqrt{a+c x^2} \sqrt{d+e x}}+\frac{4 c \sqrt{a+c x^2} (4 d+e x)}{3 e^3 \sqrt{d+e x}}-\frac{2 \left (a+c x^2\right )^{3/2}}{3 e (d+e x)^{3/2}} \]
[Out]
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Rubi [A] time = 0.849304, antiderivative size = 358, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ \frac{32 \sqrt{-a} c^{3/2} d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{3 e^4 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}-\frac{8 \sqrt{-a} \sqrt{c} \sqrt{\frac{c x^2}{a}+1} \left (a e^2+4 c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{3 e^4 \sqrt{a+c x^2} \sqrt{d+e x}}+\frac{4 c \sqrt{a+c x^2} (4 d+e x)}{3 e^3 \sqrt{d+e x}}-\frac{2 \left (a+c x^2\right )^{3/2}}{3 e (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[(a + c*x^2)^(3/2)/(d + e*x)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 130.253, size = 340, normalized size = 0.95 \[ \frac{32 c^{\frac{3}{2}} d \sqrt{- a} \sqrt{1 + \frac{c x^{2}}{a}} \sqrt{d + e x} E\left (\operatorname{asin}{\left (\sqrt{- \frac{\sqrt{c} x}{2 \sqrt{- a}} + \frac{1}{2}} \right )}\middle | \frac{2 a e}{a e - \sqrt{c} d \sqrt{- a}}\right )}{3 e^{4} \sqrt{\frac{\sqrt{c} \sqrt{- a} \left (- d - e x\right )}{a e - \sqrt{c} d \sqrt{- a}}} \sqrt{a + c x^{2}}} - \frac{8 \sqrt{c} \sqrt{- a} \sqrt{\frac{\sqrt{c} \sqrt{- a} \left (- d - e x\right )}{a e - \sqrt{c} d \sqrt{- a}}} \sqrt{1 + \frac{c x^{2}}{a}} \left (a e^{2} + 4 c d^{2}\right ) F\left (\operatorname{asin}{\left (\sqrt{- \frac{\sqrt{c} x}{2 \sqrt{- a}} + \frac{1}{2}} \right )}\middle | \frac{2 a e}{a e - \sqrt{c} d \sqrt{- a}}\right )}{3 e^{4} \sqrt{a + c x^{2}} \sqrt{d + e x}} + \frac{8 c \sqrt{a + c x^{2}} \left (2 d + \frac{e x}{2}\right )}{3 e^{3} \sqrt{d + e x}} - \frac{2 \left (a + c x^{2}\right )^{\frac{3}{2}}}{3 e \left (d + e x\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+a)**(3/2)/(e*x+d)**(5/2),x)
[Out]
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Mathematica [C] time = 4.36621, size = 494, normalized size = 1.38 \[ \frac{2 \sqrt{a+c x^2} \left (c \left (8 d^2+10 d e x+e^2 x^2\right )-a e^2\right )}{3 e^3 (d+e x)^{3/2}}-\frac{8 c \left (4 d e^2 \left (a+c x^2\right ) \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}-\sqrt{a} e (d+e x)^{3/2} \left (4 \sqrt{c} d+i \sqrt{a} e\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} F\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )+4 \sqrt{c} d (d+e x)^{3/2} \left (\sqrt{a} e-i \sqrt{c} d\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )\right )}{3 e^5 \sqrt{a+c x^2} \sqrt{d+e x} \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + c*x^2)^(3/2)/(d + e*x)^(5/2),x]
[Out]
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Maple [B] time = 0.041, size = 1597, normalized size = 4.5 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+a)^(3/2)/(e*x+d)^(5/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + a\right )}^{\frac{3}{2}}}{{\left (e x + d\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + 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. \[{\rm integral}\left (\frac{{\left (c x^{2} + a\right )}^{\frac{3}{2}}}{{\left (e^{2} x^{2} + 2 \, d e x + d^{2}\right )} \sqrt{e x + d}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + 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. \[ \int \frac{\left (a + c x^{2}\right )^{\frac{3}{2}}}{\left (d + e x\right )^{\frac{5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+a)**(3/2)/(e*x+d)**(5/2),x)
[Out]
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^(3/2)/(e*x + d)^(5/2),x, algorithm="giac")
[Out]