Optimal. Leaf size=419 \[ \frac{\sqrt{b} \left (\frac{b x^2}{a}+1\right )^{3/4} \left (-21 a^2 d^2-76 a b c d+20 b^2 c^2\right ) \text{EllipticF}\left (\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ),2\right )}{42 a^{3/2} c \left (a+b x^2\right )^{3/4} (b c-a d)^3}+\frac{b x \left (-21 a^2 d^2-76 a b c d+20 b^2 c^2\right )}{42 a^2 c \left (a+b x^2\right )^{3/4} (b c-a d)^3}+\frac{\sqrt [4]{a} d^2 \sqrt{-\frac{b x^2}{a}} (13 b c-2 a d) \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c x (b c-a d)^4}+\frac{\sqrt [4]{a} d^2 \sqrt{-\frac{b x^2}{a}} (13 b c-2 a d) \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c x (b c-a d)^4}-\frac{d x}{2 c \left (a+b x^2\right )^{7/4} \left (c+d x^2\right ) (b c-a d)}+\frac{b x (7 a d+4 b c)}{14 a c \left (a+b x^2\right )^{7/4} (b c-a d)^2} \]
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Rubi [A] time = 0.484125, antiderivative size = 419, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {414, 527, 530, 233, 231, 401, 108, 409, 1218} \[ \frac{b x \left (-21 a^2 d^2-76 a b c d+20 b^2 c^2\right )}{42 a^2 c \left (a+b x^2\right )^{3/4} (b c-a d)^3}+\frac{\sqrt{b} \left (\frac{b x^2}{a}+1\right )^{3/4} \left (-21 a^2 d^2-76 a b c d+20 b^2 c^2\right ) F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{42 a^{3/2} c \left (a+b x^2\right )^{3/4} (b c-a d)^3}+\frac{\sqrt [4]{a} d^2 \sqrt{-\frac{b x^2}{a}} (13 b c-2 a d) \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c x (b c-a d)^4}+\frac{\sqrt [4]{a} d^2 \sqrt{-\frac{b x^2}{a}} (13 b c-2 a d) \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c x (b c-a d)^4}-\frac{d x}{2 c \left (a+b x^2\right )^{7/4} \left (c+d x^2\right ) (b c-a d)}+\frac{b x (7 a d+4 b c)}{14 a c \left (a+b x^2\right )^{7/4} (b c-a d)^2} \]
Antiderivative was successfully verified.
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Rule 414
Rule 527
Rule 530
Rule 233
Rule 231
Rule 401
Rule 108
Rule 409
Rule 1218
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b x^2\right )^{11/4} \left (c+d x^2\right )^2} \, dx &=-\frac{d x}{2 c (b c-a d) \left (a+b x^2\right )^{7/4} \left (c+d x^2\right )}+\frac{\int \frac{2 b c-a d-\frac{9}{2} b d x^2}{\left (a+b x^2\right )^{11/4} \left (c+d x^2\right )} \, dx}{2 c (b c-a d)}\\ &=\frac{b (4 b c+7 a d) x}{14 a c (b c-a d)^2 \left (a+b x^2\right )^{7/4}}-\frac{d x}{2 c (b c-a d) \left (a+b x^2\right )^{7/4} \left (c+d x^2\right )}-\frac{\int \frac{\frac{1}{2} \left (-10 b^2 c^2+28 a b c d-7 a^2 d^2\right )-\frac{5}{4} b d (4 b c+7 a d) x^2}{\left (a+b x^2\right )^{7/4} \left (c+d x^2\right )} \, dx}{7 a c (b c-a d)^2}\\ &=\frac{b (4 b c+7 a d) x}{14 a c (b c-a d)^2 \left (a+b x^2\right )^{7/4}}+\frac{b \left (20 b^2 c^2-76 a b c d-21 a^2 d^2\right ) x}{42 a^2 c (b c-a d)^3 \left (a+b x^2\right )^{3/4}}-\frac{d x}{2 c (b c-a d) \left (a+b x^2\right )^{7/4} \left (c+d x^2\right )}+\frac{2 \int \frac{\frac{1}{4} \left (10 b^3 c^3-38 a b^2 c^2 d+126 a^2 b c d^2-21 a^3 d^3\right )+\frac{1}{8} b d \left (20 b^2 c^2-76 a b c d-21 a^2 d^2\right ) x^2}{\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )} \, dx}{21 a^2 c (b c-a d)^3}\\ &=\frac{b (4 b c+7 a d) x}{14 a c (b c-a d)^2 \left (a+b x^2\right )^{7/4}}+\frac{b \left (20 b^2 c^2-76 a b c d-21 a^2 d^2\right ) x}{42 a^2 c (b c-a d)^3 \left (a+b x^2\right )^{3/4}}-\frac{d x}{2 c (b c-a d) \left (a+b x^2\right )^{7/4} \left (c+d x^2\right )}+\frac{\left (d^2 (13 b c-2 a d)\right ) \int \frac{1}{\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )} \, dx}{4 c (b c-a d)^3}+\frac{\left (b \left (20 b^2 c^2-76 a b c d-21 a^2 d^2\right )\right ) \int \frac{1}{\left (a+b x^2\right )^{3/4}} \, dx}{84 a^2 c (b c-a d)^3}\\ &=\frac{b (4 b c+7 a d) x}{14 a c (b c-a d)^2 \left (a+b x^2\right )^{7/4}}+\frac{b \left (20 b^2 c^2-76 a b c d-21 a^2 d^2\right ) x}{42 a^2 c (b c-a d)^3 \left (a+b x^2\right )^{3/4}}-\frac{d x}{2 c (b c-a d) \left (a+b x^2\right )^{7/4} \left (c+d x^2\right )}+\frac{\left (d^2 (13 b c-2 a d) \sqrt{-\frac{b x^2}{a}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-\frac{b x}{a}} (a+b x)^{3/4} (c+d x)} \, dx,x,x^2\right )}{8 c (b c-a d)^3 x}+\frac{\left (b \left (20 b^2 c^2-76 a b c d-21 a^2 d^2\right ) \left (1+\frac{b x^2}{a}\right )^{3/4}\right ) \int \frac{1}{\left (1+\frac{b x^2}{a}\right )^{3/4}} \, dx}{84 a^2 c (b c-a d)^3 \left (a+b x^2\right )^{3/4}}\\ &=\frac{b (4 b c+7 a d) x}{14 a c (b c-a d)^2 \left (a+b x^2\right )^{7/4}}+\frac{b \left (20 b^2 c^2-76 a b c d-21 a^2 d^2\right ) x}{42 a^2 c (b c-a d)^3 \left (a+b x^2\right )^{3/4}}-\frac{d x}{2 c (b c-a d) \left (a+b x^2\right )^{7/4} \left (c+d x^2\right )}+\frac{\sqrt{b} \left (20 b^2 c^2-76 a b c d-21 a^2 d^2\right ) \left (1+\frac{b x^2}{a}\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{42 a^{3/2} c (b c-a d)^3 \left (a+b x^2\right )^{3/4}}-\frac{\left (d^2 (13 b c-2 a d) \sqrt{-\frac{b x^2}{a}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^4}{a}} \left (-b c+a d-d x^4\right )} \, dx,x,\sqrt [4]{a+b x^2}\right )}{2 c (b c-a d)^3 x}\\ &=\frac{b (4 b c+7 a d) x}{14 a c (b c-a d)^2 \left (a+b x^2\right )^{7/4}}+\frac{b \left (20 b^2 c^2-76 a b c d-21 a^2 d^2\right ) x}{42 a^2 c (b c-a d)^3 \left (a+b x^2\right )^{3/4}}-\frac{d x}{2 c (b c-a d) \left (a+b x^2\right )^{7/4} \left (c+d x^2\right )}+\frac{\sqrt{b} \left (20 b^2 c^2-76 a b c d-21 a^2 d^2\right ) \left (1+\frac{b x^2}{a}\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{42 a^{3/2} c (b c-a d)^3 \left (a+b x^2\right )^{3/4}}+\frac{\left (d^2 (13 b c-2 a d) \sqrt{-\frac{b x^2}{a}}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-\frac{\sqrt{d} x^2}{\sqrt{-b c+a d}}\right ) \sqrt{1-\frac{x^4}{a}}} \, dx,x,\sqrt [4]{a+b x^2}\right )}{4 c (b c-a d)^4 x}+\frac{\left (d^2 (13 b c-2 a d) \sqrt{-\frac{b x^2}{a}}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+\frac{\sqrt{d} x^2}{\sqrt{-b c+a d}}\right ) \sqrt{1-\frac{x^4}{a}}} \, dx,x,\sqrt [4]{a+b x^2}\right )}{4 c (b c-a d)^4 x}\\ &=\frac{b (4 b c+7 a d) x}{14 a c (b c-a d)^2 \left (a+b x^2\right )^{7/4}}+\frac{b \left (20 b^2 c^2-76 a b c d-21 a^2 d^2\right ) x}{42 a^2 c (b c-a d)^3 \left (a+b x^2\right )^{3/4}}-\frac{d x}{2 c (b c-a d) \left (a+b x^2\right )^{7/4} \left (c+d x^2\right )}+\frac{\sqrt{b} \left (20 b^2 c^2-76 a b c d-21 a^2 d^2\right ) \left (1+\frac{b x^2}{a}\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{42 a^{3/2} c (b c-a d)^3 \left (a+b x^2\right )^{3/4}}+\frac{\sqrt [4]{a} d^2 (13 b c-2 a d) \sqrt{-\frac{b x^2}{a}} \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{-b c+a d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c (b c-a d)^4 x}+\frac{\sqrt [4]{a} d^2 (13 b c-2 a d) \sqrt{-\frac{b x^2}{a}} \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{-b c+a d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c (b c-a d)^4 x}\\ \end{align*}
Mathematica [C] time = 0.975074, size = 550, normalized size = 1.31 \[ \frac{\frac{b d x^3 \left (\frac{b x^2}{a}+1\right )^{3/4} \left (21 a^2 d^2+76 a b c d-20 b^2 c^2\right ) F_1\left (\frac{3}{2};\frac{3}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{(a d-b c)^3}+\frac{6 c \left (x^3 \left (a^2 b^2 d \left (88 c^2+88 c d x^2+21 d^2 x^4\right )+42 a^3 b d^3 x^2+21 a^4 d^3+4 a b^3 c \left (-8 c^2+11 c d x^2+19 d^2 x^4\right )-20 b^4 c^2 x^2 \left (c+d x^2\right )\right ) \left (4 a d F_1\left (\frac{3}{2};\frac{3}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+3 b c F_1\left (\frac{3}{2};\frac{7}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )-6 a c x \left (a^2 b^2 d \left (126 c^2-38 c d x^2+21 d^2 x^4\right )+63 a^3 b d^2 \left (d x^2-2 c\right )+42 a^4 d^3+2 a b^3 c \left (-21 c^2+41 c d x^2+38 d^2 x^4\right )-10 b^4 c^2 x^2 \left (3 c+2 d x^2\right )\right ) F_1\left (\frac{1}{2};\frac{3}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )}{\left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)^3 \left (6 a c F_1\left (\frac{1}{2};\frac{3}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )-x^2 \left (4 a d F_1\left (\frac{3}{2};\frac{3}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+3 b c F_1\left (\frac{3}{2};\frac{7}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )\right )}}{252 a^2 c^2 \left (a+b x^2\right )^{3/4}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.045, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( d{x}^{2}+c \right ) ^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{11}{4}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{11}{4}}{\left (d x^{2} + c\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{11}{4}}{\left (d x^{2} + c\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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