Optimal. Leaf size=154 \[ -\frac {\left (c \sqrt {a+b x^2}\right )^{3/2}}{3 x^3}-\frac {b \left (c \sqrt {a+b x^2}\right )^{3/2}}{2 a x}+\frac {b^2 x \left (c \sqrt {a+b x^2}\right )^{3/2}}{2 a \left (a+b x^2\right )}-\frac {b^{3/2} \left (c \sqrt {a+b x^2}\right )^{3/2} E\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{2 a^{3/2} \left (1+\frac {b x^2}{a}\right )^{3/4}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {1973, 283, 331,
233, 202} \begin {gather*} -\frac {b^{3/2} \left (c \sqrt {a+b x^2}\right )^{3/2} E\left (\left .\frac {1}{2} \text {ArcTan}\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} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 202
Rule 233
Rule 283
Rule 331
Rule 1973
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] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.03, size = 57, normalized size = 0.37 \begin {gather*} -\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 (1+\frac {b x^2}{a}\right )^{3/4}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (c \sqrt {b \,x^{2}+a}\right )^{\frac {3}{2}}}{x^{4}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c \sqrt {a + b x^{2}}\right )^{\frac {3}{2}}}{x^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (c\,\sqrt {b\,x^2+a}\right )}^{3/2}}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________