Optimal. Leaf size=356 \[ -\frac{\sqrt{d+e x} (a (B d-A e)-x (a B e+A c d))}{a \sqrt{a+c x^2} \left (a e^2+c d^2\right )}-\frac{\sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} (a B e+A c d) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{\sqrt{-a} \sqrt{c} \sqrt{a+c x^2} \left (a e^2+c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{A \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{\sqrt{-a} \sqrt{c} \sqrt{a+c x^2} \sqrt{d+e x}} \]
[Out]
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Rubi [A] time = 0.847887, antiderivative size = 356, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ -\frac{\sqrt{d+e x} (a (B d-A e)-x (a B e+A c d))}{a \sqrt{a+c x^2} \left (a e^2+c d^2\right )}-\frac{\sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} (a B e+A c d) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{\sqrt{-a} \sqrt{c} \sqrt{a+c x^2} \left (a e^2+c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{A \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{\sqrt{-a} \sqrt{c} \sqrt{a+c x^2} \sqrt{d+e x}} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)/(Sqrt[d + e*x]*(a + c*x^2)^(3/2)),x]
[Out]
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Rubi in Sympy [A] time = 140.013, size = 326, normalized size = 0.92 \[ \frac{A \sqrt{\frac{\sqrt{c} \sqrt{- a} \left (- d - e x\right )}{a e - \sqrt{c} d \sqrt{- a}}} \sqrt{1 + \frac{c x^{2}}{a}} F\left (\operatorname{asin}{\left (\sqrt{- \frac{\sqrt{c} x}{2 \sqrt{- a}} + \frac{1}{2}} \right )}\middle | \frac{2 a e}{a e - \sqrt{c} d \sqrt{- a}}\right )}{\sqrt{c} \sqrt{- a} \sqrt{a + c x^{2}} \sqrt{d + e x}} - \frac{\sqrt{1 + \frac{c x^{2}}{a}} \sqrt{d + e x} \left (A c d + B a e\right ) E\left (\operatorname{asin}{\left (\sqrt{- \frac{\sqrt{c} x}{2 \sqrt{- a}} + \frac{1}{2}} \right )}\middle | \frac{2 a e}{a e - \sqrt{c} d \sqrt{- a}}\right )}{\sqrt{c} \sqrt{- a} \sqrt{\frac{\sqrt{c} \sqrt{- a} \left (- d - e x\right )}{a e - \sqrt{c} d \sqrt{- a}}} \sqrt{a + c x^{2}} \left (a e^{2} + c d^{2}\right )} + \frac{\sqrt{d + e x} \left (a \left (A e - B d\right ) + x \left (A c d + B a e\right )\right )}{a \sqrt{a + c x^{2}} \left (a e^{2} + c d^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)/(c*x**2+a)**(3/2)/(e*x+d)**(1/2),x)
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Mathematica [C] time = 5.13513, size = 525, normalized size = 1.47 \[ \frac{\sqrt{d+e x} \left (2 (a (A e-B d+B e x)+A c d x)-\frac{2 \left (e^2 \left (a+c x^2\right ) \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} (a B e+A c d)+\sqrt{a} \sqrt{c} e (d+e x)^{3/2} \left (-A \sqrt{c}+i \sqrt{a} B\right ) \left (\sqrt{c} d+i \sqrt{a} e\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} F\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )+\sqrt{c} (d+e x)^{3/2} \left (\sqrt{a} e-i \sqrt{c} d\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} (a B e+A c d) E\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )\right )}{c e (d+e x) \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}\right )}{2 a \sqrt{a+c x^2} \left (a e^2+c d^2\right )} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)/(Sqrt[d + e*x]*(a + c*x^2)^(3/2)),x]
[Out]
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Maple [B] time = 0.088, size = 1317, normalized size = 3.7 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)/(c*x^2+a)^(3/2)/(e*x+d)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{B x + A}{{\left (c x^{2} + a\right )}^{\frac{3}{2}} \sqrt{e x + d}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((c*x^2 + a)^(3/2)*sqrt(e*x + d)),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{B x + A}{{\left (c x^{2} + a\right )}^{\frac{3}{2}} \sqrt{e x + d}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((c*x^2 + a)^(3/2)*sqrt(e*x + d)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{A + B x}{\left (a + c x^{2}\right )^{\frac{3}{2}} \sqrt{d + e x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)/(c*x**2+a)**(3/2)/(e*x+d)**(1/2),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((B*x + A)/((c*x^2 + a)^(3/2)*sqrt(e*x + d)),x, algorithm="giac")
[Out]