Optimal. Leaf size=362 \[ -\frac{6 a^{3/2} \sqrt{b} \sqrt [4]{\frac{b x^2}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{5 d \sqrt [4]{a+b x^2}}-\frac{2 b x (b c-a d)}{d^2 \sqrt [4]{a+b x^2}}+\frac{2 \sqrt{a} \sqrt{b} \sqrt [4]{\frac{b x^2}{a}+1} (b c-a d) E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{d^2 \sqrt [4]{a+b x^2}}+\frac{\sqrt [4]{a} \sqrt{-\frac{b x^2}{a}} (a d-b c)^{3/2} \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{d^{5/2} x}-\frac{\sqrt [4]{a} \sqrt{-\frac{b x^2}{a}} (a d-b c)^{3/2} \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{d^{5/2} x}+\frac{6 a b x}{5 d \sqrt [4]{a+b x^2}}+\frac{2 b x \left (a+b x^2\right )^{3/4}}{5 d} \]
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Rubi [A] time = 0.265366, 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, Rules used = {402, 195, 229, 227, 196, 399, 490, 1218} \[ -\frac{6 a^{3/2} \sqrt{b} \sqrt [4]{\frac{b x^2}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{5 d \sqrt [4]{a+b x^2}}-\frac{2 b x (b c-a d)}{d^2 \sqrt [4]{a+b x^2}}+\frac{2 \sqrt{a} \sqrt{b} \sqrt [4]{\frac{b x^2}{a}+1} (b c-a d) E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{d^2 \sqrt [4]{a+b x^2}}+\frac{\sqrt [4]{a} \sqrt{-\frac{b x^2}{a}} (a d-b c)^{3/2} \Pi \left (-\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{d^{5/2} x}-\frac{\sqrt [4]{a} \sqrt{-\frac{b x^2}{a}} (a d-b c)^{3/2} \Pi \left (\frac{\sqrt{a} \sqrt{d}}{\sqrt{a d-b c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right )\right |-1\right )}{d^{5/2} x}+\frac{6 a b x}{5 d \sqrt [4]{a+b x^2}}+\frac{2 b x \left (a+b x^2\right )^{3/4}}{5 d} \]
Antiderivative was successfully verified.
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Rule 402
Rule 195
Rule 229
Rule 227
Rule 196
Rule 399
Rule 490
Rule 1218
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^{7/4}}{c+d x^2} \, dx &=\frac{b \int \left (a+b x^2\right )^{3/4} \, dx}{d}-\frac{(b c-a d) \int \frac{\left (a+b x^2\right )^{3/4}}{c+d x^2} \, dx}{d}\\ &=\frac{2 b x \left (a+b x^2\right )^{3/4}}{5 d}+\frac{(3 a b) \int \frac{1}{\sqrt [4]{a+b x^2}} \, dx}{5 d}-\frac{(b (b c-a d)) \int \frac{1}{\sqrt [4]{a+b x^2}} \, dx}{d^2}+\frac{(b c-a d)^2 \int \frac{1}{\sqrt [4]{a+b x^2} \left (c+d x^2\right )} \, dx}{d^2}\\ &=\frac{2 b x \left (a+b x^2\right )^{3/4}}{5 d}+\frac{\left (2 (b c-a d)^2 \sqrt{-\frac{b x^2}{a}}\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1-\frac{x^4}{a}} \left (b c-a d+d x^4\right )} \, dx,x,\sqrt [4]{a+b x^2}\right )}{d^2 x}+\frac{\left (3 a b \sqrt [4]{1+\frac{b x^2}{a}}\right ) \int \frac{1}{\sqrt [4]{1+\frac{b x^2}{a}}} \, dx}{5 d \sqrt [4]{a+b x^2}}-\frac{\left (b (b c-a d) \sqrt [4]{1+\frac{b x^2}{a}}\right ) \int \frac{1}{\sqrt [4]{1+\frac{b x^2}{a}}} \, dx}{d^2 \sqrt [4]{a+b x^2}}\\ &=\frac{6 a b x}{5 d \sqrt [4]{a+b x^2}}-\frac{2 b (b c-a d) x}{d^2 \sqrt [4]{a+b x^2}}+\frac{2 b x \left (a+b x^2\right )^{3/4}}{5 d}-\frac{\left ((b c-a d)^2 \sqrt{-\frac{b x^2}{a}}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (\sqrt{-b c+a d}-\sqrt{d} x^2\right ) \sqrt{1-\frac{x^4}{a}}} \, dx,x,\sqrt [4]{a+b x^2}\right )}{d^{5/2} x}+\frac{\left ((b c-a d)^2 \sqrt{-\frac{b x^2}{a}}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (\sqrt{-b c+a d}+\sqrt{d} x^2\right ) \sqrt{1-\frac{x^4}{a}}} \, dx,x,\sqrt [4]{a+b x^2}\right )}{d^{5/2} x}-\frac{\left (3 a b \sqrt [4]{1+\frac{b x^2}{a}}\right ) \int \frac{1}{\left (1+\frac{b x^2}{a}\right )^{5/4}} \, dx}{5 d \sqrt [4]{a+b x^2}}+\frac{\left (b (b c-a d) \sqrt [4]{1+\frac{b x^2}{a}}\right ) \int \frac{1}{\left (1+\frac{b x^2}{a}\right )^{5/4}} \, dx}{d^2 \sqrt [4]{a+b x^2}}\\ &=\frac{6 a b x}{5 d \sqrt [4]{a+b x^2}}-\frac{2 b (b c-a d) x}{d^2 \sqrt [4]{a+b x^2}}+\frac{2 b x \left (a+b x^2\right )^{3/4}}{5 d}-\frac{6 a^{3/2} \sqrt{b} \sqrt [4]{1+\frac{b x^2}{a}} 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{2 \sqrt{a} \sqrt{b} (b c-a d) \sqrt [4]{1+\frac{b x^2}{a}} 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{\sqrt [4]{a} (-b c+a d)^{3/2} \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 )}{d^{5/2} x}-\frac{\sqrt [4]{a} (-b c+a d)^{3/2} \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 )}{d^{5/2} x}\\ \end{align*}
Mathematica [C] time = 0.465512, size = 346, normalized size = 0.96 \[ \frac{x \left (\frac{6 \left (b x^2 \left (a+b x^2\right ) \left (c+d x^2\right ) \left (4 a d F_1\left (\frac{3}{2};\frac{1}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+b c F_1\left (\frac{3}{2};\frac{5}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )-3 a c \left (5 a^2 d+2 a b d x^2+2 b^2 x^2 \left (c+d x^2\right )\right ) F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )}{\left (c+d x^2\right ) \left (x^2 \left (4 a d F_1\left (\frac{3}{2};\frac{1}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )+b c F_1\left (\frac{3}{2};\frac{5}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )-6 a c F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )\right )}+\frac{b x^2 \sqrt [4]{\frac{b x^2}{a}+1} (8 a d-5 b c) F_1\left (\frac{3}{2};\frac{1}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{d x^2}{c}\right )}{c}\right )}{15 d \sqrt [4]{a+b x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.064, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{d{x}^{2}+c} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{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{{\left (b x^{2} + a\right )}^{\frac{7}{4}}}{d x^{2} + c}\,{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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b x^{2}\right )^{\frac{7}{4}}}{c + d x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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