Optimal. Leaf size=219 \[ -\frac {2 c}{9 a e (e x)^{9/2} \left (a+b x^2\right )^{5/4}}-\frac {2 (14 b c-9 a d)}{45 a^2 e^3 (e x)^{5/2} \left (a+b x^2\right )^{5/4}}+\frac {4 (14 b c-9 a d)}{45 a^3 e^3 (e x)^{5/2} \sqrt [4]{a+b x^2}}-\frac {8 b (14 b c-9 a d)}{15 a^4 e^5 \sqrt {e x} \sqrt [4]{a+b x^2}}+\frac {16 b^{3/2} (14 b c-9 a d) \sqrt [4]{1+\frac {a}{b x^2}} \sqrt {e x} E\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{15 a^{9/2} e^6 \sqrt [4]{a+b x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.08, antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {464, 296, 292,
290, 342, 202} \begin {gather*} \frac {16 b^{3/2} \sqrt {e x} \sqrt [4]{\frac {a}{b x^2}+1} (14 b c-9 a d) E\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{15 a^{9/2} e^6 \sqrt [4]{a+b x^2}}-\frac {8 b (14 b c-9 a d)}{15 a^4 e^5 \sqrt {e x} \sqrt [4]{a+b x^2}}+\frac {4 (14 b c-9 a d)}{45 a^3 e^3 (e x)^{5/2} \sqrt [4]{a+b x^2}}-\frac {2 (14 b c-9 a d)}{45 a^2 e^3 (e x)^{5/2} \left (a+b x^2\right )^{5/4}}-\frac {2 c}{9 a e (e x)^{9/2} \left (a+b x^2\right )^{5/4}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 202
Rule 290
Rule 292
Rule 296
Rule 342
Rule 464
Rubi steps
\begin {align*} \int \frac {c+d x^2}{(e x)^{11/2} \left (a+b x^2\right )^{9/4}} \, dx &=-\frac {2 c}{9 a e (e x)^{9/2} \left (a+b x^2\right )^{5/4}}-\frac {(14 b c-9 a d) \int \frac {1}{(e x)^{7/2} \left (a+b x^2\right )^{9/4}} \, dx}{9 a e^2}\\ &=-\frac {2 c}{9 a e (e x)^{9/2} \left (a+b x^2\right )^{5/4}}-\frac {2 (14 b c-9 a d)}{45 a^2 e^3 (e x)^{5/2} \left (a+b x^2\right )^{5/4}}-\frac {(2 (14 b c-9 a d)) \int \frac {1}{(e x)^{7/2} \left (a+b x^2\right )^{5/4}} \, dx}{9 a^2 e^2}\\ &=-\frac {2 c}{9 a e (e x)^{9/2} \left (a+b x^2\right )^{5/4}}-\frac {2 (14 b c-9 a d)}{45 a^2 e^3 (e x)^{5/2} \left (a+b x^2\right )^{5/4}}+\frac {4 (14 b c-9 a d)}{45 a^3 e^3 (e x)^{5/2} \sqrt [4]{a+b x^2}}+\frac {(4 b (14 b c-9 a d)) \int \frac {1}{(e x)^{3/2} \left (a+b x^2\right )^{5/4}} \, dx}{15 a^3 e^4}\\ &=-\frac {2 c}{9 a e (e x)^{9/2} \left (a+b x^2\right )^{5/4}}-\frac {2 (14 b c-9 a d)}{45 a^2 e^3 (e x)^{5/2} \left (a+b x^2\right )^{5/4}}+\frac {4 (14 b c-9 a d)}{45 a^3 e^3 (e x)^{5/2} \sqrt [4]{a+b x^2}}-\frac {8 b (14 b c-9 a d)}{15 a^4 e^5 \sqrt {e x} \sqrt [4]{a+b x^2}}-\frac {\left (8 b^2 (14 b c-9 a d)\right ) \int \frac {\sqrt {e x}}{\left (a+b x^2\right )^{5/4}} \, dx}{15 a^4 e^6}\\ &=-\frac {2 c}{9 a e (e x)^{9/2} \left (a+b x^2\right )^{5/4}}-\frac {2 (14 b c-9 a d)}{45 a^2 e^3 (e x)^{5/2} \left (a+b x^2\right )^{5/4}}+\frac {4 (14 b c-9 a d)}{45 a^3 e^3 (e x)^{5/2} \sqrt [4]{a+b x^2}}-\frac {8 b (14 b c-9 a d)}{15 a^4 e^5 \sqrt {e x} \sqrt [4]{a+b x^2}}-\frac {\left (8 b (14 b c-9 a d) \sqrt [4]{1+\frac {a}{b x^2}} \sqrt {e x}\right ) \int \frac {1}{\left (1+\frac {a}{b x^2}\right )^{5/4} x^2} \, dx}{15 a^4 e^6 \sqrt [4]{a+b x^2}}\\ &=-\frac {2 c}{9 a e (e x)^{9/2} \left (a+b x^2\right )^{5/4}}-\frac {2 (14 b c-9 a d)}{45 a^2 e^3 (e x)^{5/2} \left (a+b x^2\right )^{5/4}}+\frac {4 (14 b c-9 a d)}{45 a^3 e^3 (e x)^{5/2} \sqrt [4]{a+b x^2}}-\frac {8 b (14 b c-9 a d)}{15 a^4 e^5 \sqrt {e x} \sqrt [4]{a+b x^2}}+\frac {\left (8 b (14 b c-9 a d) \sqrt [4]{1+\frac {a}{b x^2}} \sqrt {e x}\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {a x^2}{b}\right )^{5/4}} \, dx,x,\frac {1}{x}\right )}{15 a^4 e^6 \sqrt [4]{a+b x^2}}\\ &=-\frac {2 c}{9 a e (e x)^{9/2} \left (a+b x^2\right )^{5/4}}-\frac {2 (14 b c-9 a d)}{45 a^2 e^3 (e x)^{5/2} \left (a+b x^2\right )^{5/4}}+\frac {4 (14 b c-9 a d)}{45 a^3 e^3 (e x)^{5/2} \sqrt [4]{a+b x^2}}-\frac {8 b (14 b c-9 a d)}{15 a^4 e^5 \sqrt {e x} \sqrt [4]{a+b x^2}}+\frac {16 b^{3/2} (14 b c-9 a d) \sqrt [4]{1+\frac {a}{b x^2}} \sqrt {e x} E\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{15 a^{9/2} e^6 \sqrt [4]{a+b x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.04, size = 87, normalized size = 0.40 \begin {gather*} \frac {2 x \left (-5 a^2 c-(-14 b c+9 a d) x^2 \left (a+b x^2\right ) \sqrt [4]{1+\frac {b x^2}{a}} \, _2F_1\left (-\frac {5}{4},\frac {9}{4};-\frac {1}{4};-\frac {b x^2}{a}\right )\right )}{45 a^3 (e x)^{11/2} \left (a+b x^2\right )^{5/4}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {d \,x^{2}+c}{\left (e x \right )^{\frac {11}{2}} \left (b \,x^{2}+a \right )^{\frac {9}{4}}}\, dx\]
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]
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]
________________________________________________________________________________________
Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 = -0.00 \begin {gather*} \int \frac {d\,x^2+c}{{\left (e\,x\right )}^{11/2}\,{\left (b\,x^2+a\right )}^{9/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________