Optimal. Leaf size=323 \[ \frac{\sqrt{d} \sqrt{a+b x^2} \left (-3 a^2 d^2-7 a b c d+2 b^2 c^2\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 a^2 \sqrt{c} \sqrt{c+d x^2} (b c-a d)^3 \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{2 b x (b c-3 a d)}{3 a^2 \sqrt{a+b x^2} \sqrt{c+d x^2} (b c-a d)^2}-\frac{b \sqrt{c} \sqrt{d} \sqrt{a+b x^2} (b c-9 a d) F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 a^2 \sqrt{c+d x^2} (b c-a d)^3 \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{b x}{3 a \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2} (b c-a d)} \]
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Rubi [A] time = 0.674622, antiderivative size = 323, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217 \[ \frac{\sqrt{d} \sqrt{a+b x^2} \left (-3 a^2 d^2-7 a b c d+2 b^2 c^2\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 a^2 \sqrt{c} \sqrt{c+d x^2} (b c-a d)^3 \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{2 b x (b c-3 a d)}{3 a^2 \sqrt{a+b x^2} \sqrt{c+d x^2} (b c-a d)^2}-\frac{b \sqrt{c} \sqrt{d} \sqrt{a+b x^2} (b c-9 a d) F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 a^2 \sqrt{c+d x^2} (b c-a d)^3 \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{b x}{3 a \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2} (b c-a d)} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b*x^2)^(5/2)*(c + d*x^2)^(3/2)),x]
[Out]
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Rubi in Sympy [A] time = 96.8982, size = 284, normalized size = 0.88 \[ \frac{d x}{c \left (a + b x^{2}\right )^{\frac{3}{2}} \sqrt{c + d x^{2}} \left (a d - b c\right )} + \frac{b x \sqrt{c + d x^{2}} \left (3 a d + b c\right )}{3 a c \left (a + b x^{2}\right )^{\frac{3}{2}} \left (a d - b c\right )^{2}} - \frac{b \sqrt{c} \sqrt{d} \sqrt{a + b x^{2}} \left (9 a d - b c\right ) F\left (\operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | 1 - \frac{b c}{a d}\right )}{3 a^{2} \sqrt{\frac{c \left (a + b x^{2}\right )}{a \left (c + d x^{2}\right )}} \sqrt{c + d x^{2}} \left (a d - b c\right )^{3}} + \frac{\sqrt{b} \sqrt{c + d x^{2}} \left (3 a^{2} d^{2} + 7 a b c d - 2 b^{2} c^{2}\right ) E\left (\operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}\middle | - \frac{a d}{b c} + 1\right )}{3 a^{\frac{3}{2}} c \sqrt{\frac{a \left (c + d x^{2}\right )}{c \left (a + b x^{2}\right )}} \sqrt{a + b x^{2}} \left (a d - b c\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x**2+a)**(5/2)/(d*x**2+c)**(3/2),x)
[Out]
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Mathematica [C] time = 1.84406, size = 337, normalized size = 1.04 \[ \frac{2 i b c \left (a+b x^2\right ) \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left (3 a^2 d^2-4 a b c d+b^2 c^2\right ) F\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )+i b c \left (a+b x^2\right ) \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left (3 a^2 d^2+7 a b c d-2 b^2 c^2\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )+x \sqrt{\frac{b}{a}} \left (3 a^4 d^3+6 a^3 b d^3 x^2+a^2 b^2 d \left (8 c^2+8 c d x^2+3 d^2 x^4\right )+a b^3 c \left (-3 c^2+4 c d x^2+7 d^2 x^4\right )-2 b^4 c^2 x^2 \left (c+d x^2\right )\right )}{3 a^2 c \sqrt{\frac{b}{a}} \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2} (a d-b c)^3} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b*x^2)^(5/2)*(c + d*x^2)^(3/2)),x]
[Out]
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Maple [B] time = 0.044, size = 964, normalized size = 3. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b*x^2+a)^(5/2)/(d*x^2+c)^(3/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{5}{2}}{\left (d x^{2} + c\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)^(5/2)*(d*x^2 + c)^(3/2)),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b^{2} d x^{6} +{\left (b^{2} c + 2 \, a b d\right )} x^{4} + a^{2} c +{\left (2 \, a b c + a^{2} d\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)^(5/2)*(d*x^2 + c)^(3/2)),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + b x^{2}\right )^{\frac{5}{2}} \left (c + d x^{2}\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*x**2+a)**(5/2)/(d*x**2+c)**(3/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{5}{2}}{\left (d x^{2} + c\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)^(5/2)*(d*x^2 + c)^(3/2)),x, algorithm="giac")
[Out]