Optimal. Leaf size=457 \[ -\frac{16 \sqrt{-a} \sqrt{c} \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (21 a^2 e^4+57 a c d^2 e^2+32 c^2 d^4\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{63 e^6 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{16 \sqrt{-a} \sqrt{c} d \sqrt{\frac{c x^2}{a}+1} \left (a e^2+c d^2\right ) \left (33 a e^2+32 c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{63 e^6 \sqrt{a+c x^2} \sqrt{d+e x}}-\frac{8 c \sqrt{a+c x^2} \sqrt{d+e x} \left (d \left (33 a e^2+32 c d^2\right )-3 e x \left (7 a e^2+8 c d^2\right )\right )}{63 e^5}-\frac{20 c \left (a+c x^2\right )^{3/2} (8 d-7 e x) \sqrt{d+e x}}{63 e^3}-\frac{2 \left (a+c x^2\right )^{5/2}}{e \sqrt{d+e x}} \]
[Out]
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Rubi [A] time = 1.27863, antiderivative size = 457, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ -\frac{16 \sqrt{-a} \sqrt{c} \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (21 a^2 e^4+57 a c d^2 e^2+32 c^2 d^4\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{63 e^6 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{16 \sqrt{-a} \sqrt{c} d \sqrt{\frac{c x^2}{a}+1} \left (a e^2+c d^2\right ) \left (33 a e^2+32 c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{63 e^6 \sqrt{a+c x^2} \sqrt{d+e x}}-\frac{8 c \sqrt{a+c x^2} \sqrt{d+e x} \left (d \left (33 a e^2+32 c d^2\right )-3 e x \left (7 a e^2+8 c d^2\right )\right )}{63 e^5}-\frac{20 c \left (a+c x^2\right )^{3/2} (8 d-7 e x) \sqrt{d+e x}}{63 e^3}-\frac{2 \left (a+c x^2\right )^{5/2}}{e \sqrt{d+e x}} \]
Antiderivative was successfully verified.
[In] Int[(a + c*x^2)^(5/2)/(d + e*x)^(3/2),x]
[Out]
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+a)**(5/2)/(e*x+d)**(3/2),x)
[Out]
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Mathematica [C] time = 7.35665, size = 684, normalized size = 1.5 \[ \frac{\sqrt{d+e x} \left (-\frac{2 \left (a+c x^2\right ) \left (63 a^2 e^4+2 a c e^2 \left (106 d^2+29 d e x-14 e^2 x^2\right )+c^2 \left (128 d^4+32 d^3 e x-16 d^2 e^2 x^2+10 d e^3 x^3-7 e^4 x^4\right )\right )}{e^5 (d+e x)}+\frac{16 \left (e^2 \left (a+c x^2\right ) \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left (21 a^2 e^4+57 a c d^2 e^2+32 c^2 d^4\right )-\sqrt{a} \sqrt{c} e (d+e x)^{3/2} \left (12 i a^{3/2} \sqrt{c} d e^3+21 a^2 e^4+8 i \sqrt{a} c^{3/2} d^3 e+57 a c d^2 e^2+32 c^2 d^4\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} F\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )+\sqrt{c} (d+e x)^{3/2} \left (57 a^{3/2} c d^2 e^3+21 a^{5/2} e^5-21 i a^2 \sqrt{c} d e^4-57 i a c^{3/2} d^3 e^2+32 \sqrt{a} c^2 d^4 e-32 i c^{5/2} d^5\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )\right )}{e^7 (d+e x) \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}\right )}{63 \sqrt{a+c x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + c*x^2)^(5/2)/(d + e*x)^(3/2),x]
[Out]
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Maple [B] time = 0.044, size = 1736, normalized size = 3.8 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+a)^(5/2)/(e*x+d)^(3/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + a\right )}^{\frac{5}{2}}}{{\left (e x + d\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^(5/2)/(e*x + d)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (c^{2} x^{4} + 2 \, a c x^{2} + a^{2}\right )} \sqrt{c x^{2} + a}}{{\left (e x + d\right )}^{\frac{3}{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^(5/2)/(e*x + d)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + c x^{2}\right )^{\frac{5}{2}}}{\left (d + e x\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+a)**(5/2)/(e*x+d)**(3/2),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)^(5/2)/(e*x + d)^(3/2),x, algorithm="giac")
[Out]