Optimal. Leaf size=200 \[ \frac {\sqrt {\frac {d x^4}{c}+1} (e x)^{m+1} \left (a^2 d^2 (m+3) (m+7)+b c (m+1) (b c (m+5)-2 a d (m+7))\right ) \, _2F_1\left (\frac {1}{2},\frac {m+1}{4};\frac {m+5}{4};-\frac {d x^4}{c}\right )}{d^2 e (m+1) (m+3) (m+7) \sqrt {c+d x^4}}-\frac {b \sqrt {c+d x^4} (e x)^{m+1} (b c (m+5)-2 a d (m+7))}{d^2 e (m+3) (m+7)}+\frac {b^2 \sqrt {c+d x^4} (e x)^{m+5}}{d e^5 (m+7)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.21, antiderivative size = 194, normalized size of antiderivative = 0.97, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {464, 459, 365, 364} \[ \frac {\sqrt {\frac {d x^4}{c}+1} (e x)^{m+1} \left (\frac {a^2 d^2 (m+7)}{m+1}+\frac {b c (b c (m+5)-2 a d (m+7))}{m+3}\right ) \, _2F_1\left (\frac {1}{2},\frac {m+1}{4};\frac {m+5}{4};-\frac {d x^4}{c}\right )}{d^2 e (m+7) \sqrt {c+d x^4}}-\frac {b \sqrt {c+d x^4} (e x)^{m+1} (b c (m+5)-2 a d (m+7))}{d^2 e (m+3) (m+7)}+\frac {b^2 \sqrt {c+d x^4} (e x)^{m+5}}{d e^5 (m+7)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 364
Rule 365
Rule 459
Rule 464
Rubi steps
\begin {align*} \int \frac {(e x)^m \left (a+b x^4\right )^2}{\sqrt {c+d x^4}} \, dx &=\frac {b^2 (e x)^{5+m} \sqrt {c+d x^4}}{d e^5 (7+m)}+\frac {\int \frac {(e x)^m \left (a^2 d (7+m)-b (b c (5+m)-2 a d (7+m)) x^4\right )}{\sqrt {c+d x^4}} \, dx}{d (7+m)}\\ &=-\frac {b (b c (5+m)-2 a d (7+m)) (e x)^{1+m} \sqrt {c+d x^4}}{d^2 e (3+m) (7+m)}+\frac {b^2 (e x)^{5+m} \sqrt {c+d x^4}}{d e^5 (7+m)}-\left (-a^2-\frac {b c (1+m) (b c (5+m)-2 a d (7+m))}{d^2 (3+m) (7+m)}\right ) \int \frac {(e x)^m}{\sqrt {c+d x^4}} \, dx\\ &=-\frac {b (b c (5+m)-2 a d (7+m)) (e x)^{1+m} \sqrt {c+d x^4}}{d^2 e (3+m) (7+m)}+\frac {b^2 (e x)^{5+m} \sqrt {c+d x^4}}{d e^5 (7+m)}-\frac {\left (\left (-a^2-\frac {b c (1+m) (b c (5+m)-2 a d (7+m))}{d^2 (3+m) (7+m)}\right ) \sqrt {1+\frac {d x^4}{c}}\right ) \int \frac {(e x)^m}{\sqrt {1+\frac {d x^4}{c}}} \, dx}{\sqrt {c+d x^4}}\\ &=-\frac {b (b c (5+m)-2 a d (7+m)) (e x)^{1+m} \sqrt {c+d x^4}}{d^2 e (3+m) (7+m)}+\frac {b^2 (e x)^{5+m} \sqrt {c+d x^4}}{d e^5 (7+m)}+\frac {\left (a^2+\frac {b c (1+m) (b c (5+m)-2 a d (7+m))}{d^2 (3+m) (7+m)}\right ) (e x)^{1+m} \sqrt {1+\frac {d x^4}{c}} \, _2F_1\left (\frac {1}{2},\frac {1+m}{4};\frac {5+m}{4};-\frac {d x^4}{c}\right )}{e (1+m) \sqrt {c+d x^4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.18, size = 164, normalized size = 0.82 \[ \frac {x \sqrt {\frac {d x^4}{c}+1} (e x)^m \left (a^2 \left (m^2+14 m+45\right ) \, _2F_1\left (\frac {1}{2},\frac {m+1}{4};\frac {m+5}{4};-\frac {d x^4}{c}\right )+b (m+1) x^4 \left (2 a (m+9) \, _2F_1\left (\frac {1}{2},\frac {m+5}{4};\frac {m+9}{4};-\frac {d x^4}{c}\right )+b (m+5) x^4 \, _2F_1\left (\frac {1}{2},\frac {m+9}{4};\frac {m+13}{4};-\frac {d x^4}{c}\right )\right )\right )}{(m+1) (m+5) (m+9) \sqrt {c+d x^4}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 1.08, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{2} x^{8} + 2 \, a b x^{4} + a^{2}\right )} \left (e x\right )^{m}}{\sqrt {d x^{4} + c}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x^{4} + a\right )}^{2} \left (e x\right )^{m}}{\sqrt {d x^{4} + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.56, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \,x^{4}+a \right )^{2} \left (e x \right )^{m}}{\sqrt {d \,x^{4}+c}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x^{4} + a\right )}^{2} \left (e x\right )^{m}}{\sqrt {d x^{4} + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (e\,x\right )}^m\,{\left (b\,x^4+a\right )}^2}{\sqrt {d\,x^4+c}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 14.27, size = 185, normalized size = 0.92 \[ \frac {a^{2} e^{m} x x^{m} \Gamma \left (\frac {m}{4} + \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {m}{4} + \frac {1}{4} \\ \frac {m}{4} + \frac {5}{4} \end {matrix}\middle | {\frac {d x^{4} e^{i \pi }}{c}} \right )}}{4 \sqrt {c} \Gamma \left (\frac {m}{4} + \frac {5}{4}\right )} + \frac {a b e^{m} x^{5} x^{m} \Gamma \left (\frac {m}{4} + \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {m}{4} + \frac {5}{4} \\ \frac {m}{4} + \frac {9}{4} \end {matrix}\middle | {\frac {d x^{4} e^{i \pi }}{c}} \right )}}{2 \sqrt {c} \Gamma \left (\frac {m}{4} + \frac {9}{4}\right )} + \frac {b^{2} e^{m} x^{9} x^{m} \Gamma \left (\frac {m}{4} + \frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {m}{4} + \frac {9}{4} \\ \frac {m}{4} + \frac {13}{4} \end {matrix}\middle | {\frac {d x^{4} e^{i \pi }}{c}} \right )}}{4 \sqrt {c} \Gamma \left (\frac {m}{4} + \frac {13}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________