Optimal. Leaf size=154 \[ -\frac{b^{3/2} \left (c \sqrt{a+b x^2}\right )^{3/2} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{2 a^{3/2} \left (\frac{b x^2}{a}+1\right )^{3/4}}+\frac{b^2 x \left (c \sqrt{a+b x^2}\right )^{3/2}}{2 a \left (a+b x^2\right )}-\frac{b \left (c \sqrt{a+b x^2}\right )^{3/2}}{2 a x}-\frac{\left (c \sqrt{a+b x^2}\right )^{3/2}}{3 x^3} \]
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Rubi [A] time = 0.157801, antiderivative size = 193, normalized size of antiderivative = 1.25, number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {6720, 277, 325, 229, 227, 196} \[ \frac{b^2 c x \sqrt{c \sqrt{a+b x^2}}}{2 a \sqrt{a+b x^2}}-\frac{b^{3/2} c \sqrt [4]{\frac{b x^2}{a}+1} \sqrt{c \sqrt{a+b x^2}} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{2 \sqrt{a} \sqrt{a+b x^2}}-\frac{b c \sqrt{a+b x^2} \sqrt{c \sqrt{a+b x^2}}}{2 a x}-\frac{c \sqrt{a+b x^2} \sqrt{c \sqrt{a+b x^2}}}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 6720
Rule 277
Rule 325
Rule 229
Rule 227
Rule 196
Rubi steps
\begin{align*} \int \frac{\left (c \sqrt{a+b x^2}\right )^{3/2}}{x^4} \, dx &=\frac{\left (c \sqrt{c \sqrt{a+b x^2}}\right ) \int \frac{\left (a+b x^2\right )^{3/4}}{x^4} \, dx}{\sqrt [4]{a+b x^2}}\\ &=-\frac{c \sqrt{c \sqrt{a+b x^2}} \sqrt{a+b x^2}}{3 x^3}+\frac{\left (b c \sqrt{c \sqrt{a+b x^2}}\right ) \int \frac{1}{x^2 \sqrt [4]{a+b x^2}} \, dx}{2 \sqrt [4]{a+b x^2}}\\ &=-\frac{c \sqrt{c \sqrt{a+b x^2}} \sqrt{a+b x^2}}{3 x^3}-\frac{b c \sqrt{c \sqrt{a+b x^2}} \sqrt{a+b x^2}}{2 a x}+\frac{\left (b^2 c \sqrt{c \sqrt{a+b x^2}}\right ) \int \frac{1}{\sqrt [4]{a+b x^2}} \, dx}{4 a \sqrt [4]{a+b x^2}}\\ &=-\frac{c \sqrt{c \sqrt{a+b x^2}} \sqrt{a+b x^2}}{3 x^3}-\frac{b c \sqrt{c \sqrt{a+b x^2}} \sqrt{a+b x^2}}{2 a x}+\frac{\left (b^2 c \sqrt{c \sqrt{a+b x^2}} \sqrt [4]{1+\frac{b x^2}{a}}\right ) \int \frac{1}{\sqrt [4]{1+\frac{b x^2}{a}}} \, dx}{4 a \sqrt{a+b x^2}}\\ &=\frac{b^2 c x \sqrt{c \sqrt{a+b x^2}}}{2 a \sqrt{a+b x^2}}-\frac{c \sqrt{c \sqrt{a+b x^2}} \sqrt{a+b x^2}}{3 x^3}-\frac{b c \sqrt{c \sqrt{a+b x^2}} \sqrt{a+b x^2}}{2 a x}-\frac{\left (b^2 c \sqrt{c \sqrt{a+b x^2}} \sqrt [4]{1+\frac{b x^2}{a}}\right ) \int \frac{1}{\left (1+\frac{b x^2}{a}\right )^{5/4}} \, dx}{4 a \sqrt{a+b x^2}}\\ &=\frac{b^2 c x \sqrt{c \sqrt{a+b x^2}}}{2 a \sqrt{a+b x^2}}-\frac{c \sqrt{c \sqrt{a+b x^2}} \sqrt{a+b x^2}}{3 x^3}-\frac{b c \sqrt{c \sqrt{a+b x^2}} \sqrt{a+b x^2}}{2 a x}-\frac{b^{3/2} c \sqrt{c \sqrt{a+b x^2}} \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 )}{2 \sqrt{a} \sqrt{a+b x^2}}\\ \end{align*}
Mathematica [C] time = 0.0123121, size = 57, normalized size = 0.37 \[ -\frac{\left (c \sqrt{a+b x^2}\right )^{3/2} \, _2F_1\left (-\frac{3}{2},-\frac{3}{4};-\frac{1}{2};-\frac{b x^2}{a}\right )}{3 x^3 \left (\frac{b x^2}{a}+1\right )^{3/4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.007, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{4}} \left ( c\sqrt{b{x}^{2}+a} \right ) ^{{\frac{3}{2}}}}\, 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 (\sqrt{b x^{2} + a} c\right )^{\frac{3}{2}}}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \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 (c \sqrt{a + b x^{2}}\right )^{\frac{3}{2}}}{x^{4}}\, 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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