Optimal. Leaf size=362 \[ -\frac{6 a^{3/2} \sqrt{b} \sqrt [4]{\frac{b x^2}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{5 d \sqrt [4]{a+b x^2}}+\frac{\sqrt [4]{a} \sqrt{-\frac{b x^2}{a}} (a d-b c)^{3/2} \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{d^{5/2} x}-\frac{\sqrt [4]{a} \sqrt{-\frac{b x^2}{a}} (a d-b c)^{3/2} \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{d^{5/2} x}-\frac{2 b x (b c-a d)}{d^2 \sqrt [4]{a+b x^2}}+\frac{2 \sqrt{a} \sqrt{b} \sqrt [4]{\frac{b x^2}{a}+1} (b c-a d) E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{d^2 \sqrt [4]{a+b x^2}}+\frac{6 a b x}{5 d \sqrt [4]{a+b x^2}}+\frac{2 b x \left (a+b x^2\right )^{3/4}}{5 d} \]
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Rubi [A] time = 0.683797, antiderivative size = 362, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381 \[ -\frac{6 a^{3/2} \sqrt{b} \sqrt [4]{\frac{b x^2}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{5 d \sqrt [4]{a+b x^2}}+\frac{\sqrt [4]{a} \sqrt{-\frac{b x^2}{a}} (a d-b c)^{3/2} \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{d^{5/2} x}-\frac{\sqrt [4]{a} \sqrt{-\frac{b x^2}{a}} (a d-b c)^{3/2} \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{d^{5/2} x}-\frac{2 b x (b c-a d)}{d^2 \sqrt [4]{a+b x^2}}+\frac{2 \sqrt{a} \sqrt{b} \sqrt [4]{\frac{b x^2}{a}+1} (b c-a d) E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{d^2 \sqrt [4]{a+b x^2}}+\frac{6 a b x}{5 d \sqrt [4]{a+b x^2}}+\frac{2 b x \left (a+b x^2\right )^{3/4}}{5 d} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^2)^(7/4)/(c + d*x^2),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{\sqrt [4]{a} \sqrt{- \frac{b x^{2}}{a}} \left (a d - b c\right )^{\frac{3}{2}} \Pi \left (- \frac{\sqrt{a} \sqrt{d}}{\sqrt{a d - b c}}; \operatorname{asin}{\left (\frac{\sqrt [4]{a + b x^{2}}}{\sqrt [4]{a}} \right )}\middle | -1\right )}{d^{\frac{5}{2}} x} - \frac{\sqrt [4]{a} \sqrt{- \frac{b x^{2}}{a}} \left (a d - b c\right )^{\frac{3}{2}} \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d - b c}}; \operatorname{asin}{\left (\frac{\sqrt [4]{a + b x^{2}}}{\sqrt [4]{a}} \right )}\middle | -1\right )}{d^{\frac{5}{2}} x} - \frac{3 a^{2} b \int \frac{1}{\left (a + b x^{2}\right )^{\frac{5}{4}}}\, dx}{5 d} + \frac{6 a b x}{5 d \sqrt [4]{a + b x^{2}}} - \frac{a b \left (a d - b c\right ) \int \frac{1}{\left (a + b x^{2}\right )^{\frac{5}{4}}}\, dx}{d^{2}} + \frac{2 b x \left (a + b x^{2}\right )^{\frac{3}{4}}}{5 d} + \frac{2 b x \left (a d - b c\right )}{d^{2} \sqrt [4]{a + b x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)**(7/4)/(d*x**2+c),x)
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Mathematica [C] time = 1.10477, size = 431, normalized size = 1.19 \[ \frac{2 x \left (\frac{b \left (3 x^2 \left (a+b x^2\right ) \left (c+d x^2\right ) \left (4 a d F_1\left (\frac{5}{2};\frac{1}{4},2;\frac{7}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+b c F_1\left (\frac{5}{2};\frac{5}{4},1;\frac{7}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )-5 a c \left (6 a c+14 a d x^2+b c x^2+6 b d x^4\right ) F_1\left (\frac{3}{2};\frac{1}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )}{x^2 \left (4 a d F_1\left (\frac{5}{2};\frac{1}{4},2;\frac{7}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+b c F_1\left (\frac{5}{2};\frac{5}{4},1;\frac{7}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )-10 a c F_1\left (\frac{3}{2};\frac{1}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}-\frac{9 a^2 c (5 a d-2 b c) F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{x^2 \left (4 a d F_1\left (\frac{3}{2};\frac{1}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+b c F_1\left (\frac{3}{2};\frac{5}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )-6 a c F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}\right )}{15 d \sqrt [4]{a+b x^2} \left (c+d x^2\right )} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(a + b*x^2)^(7/4)/(c + d*x^2),x]
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Maple [F] time = 0.097, size = 0, normalized size = 0. \[ \int{\frac{1}{d{x}^{2}+c} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{4}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)^(7/4)/(d*x^2+c),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{\frac{7}{4}}}{d x^{2} + c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^(7/4)/(d*x^2 + c),x, algorithm="maxima")
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^(7/4)/(d*x^2 + c),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x^{2}\right )^{\frac{7}{4}}}{c + d x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)**(7/4)/(d*x**2+c),x)
[Out]
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^(7/4)/(d*x^2 + c),x, algorithm="giac")
[Out]