Optimal. Leaf size=449 \[ -\frac {2 b^{9/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (15 \sqrt {b} d-77 \sqrt {a} f\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{1155 a^{5/4} \sqrt {a+b x^4}}-\frac {4 b^{9/4} f \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 a^{3/4} \sqrt {a+b x^4}}+\frac {b^3 c \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{32 a^{3/2}}+\frac {4 b^{5/2} f x \sqrt {a+b x^4}}{15 a \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {b^2 c \sqrt {a+b x^4}}{32 a x^4}-\frac {4 b^2 d \sqrt {a+b x^4}}{77 a x^3}-\frac {b^2 e \sqrt {a+b x^4}}{10 a x^2}-\frac {4 b^2 f \sqrt {a+b x^4}}{15 a x}-\frac {\left (a+b x^4\right )^{3/2} \left (\frac {165 c}{x^{12}}+\frac {180 d}{x^{11}}+\frac {198 e}{x^{10}}+\frac {220 f}{x^9}\right )}{1980}-\frac {b \sqrt {a+b x^4} \left (\frac {1155 c}{x^8}+\frac {1440 d}{x^7}+\frac {1848 e}{x^6}+\frac {2464 f}{x^5}\right )}{18480} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.49, antiderivative size = 449, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 13, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.433, Rules used = {14, 1825, 1833, 1252, 835, 807, 266, 63, 208, 1282, 1198, 220, 1196} \[ \frac {b^3 c \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{32 a^{3/2}}-\frac {2 b^{9/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (15 \sqrt {b} d-77 \sqrt {a} f\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{1155 a^{5/4} \sqrt {a+b x^4}}-\frac {4 b^{9/4} f \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 a^{3/4} \sqrt {a+b x^4}}-\frac {b^2 c \sqrt {a+b x^4}}{32 a x^4}-\frac {4 b^2 d \sqrt {a+b x^4}}{77 a x^3}-\frac {b^2 e \sqrt {a+b x^4}}{10 a x^2}+\frac {4 b^{5/2} f x \sqrt {a+b x^4}}{15 a \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {4 b^2 f \sqrt {a+b x^4}}{15 a x}-\frac {b \sqrt {a+b x^4} \left (\frac {1155 c}{x^8}+\frac {1440 d}{x^7}+\frac {1848 e}{x^6}+\frac {2464 f}{x^5}\right )}{18480}-\frac {\left (a+b x^4\right )^{3/2} \left (\frac {165 c}{x^{12}}+\frac {180 d}{x^{11}}+\frac {198 e}{x^{10}}+\frac {220 f}{x^9}\right )}{1980} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 14
Rule 63
Rule 208
Rule 220
Rule 266
Rule 807
Rule 835
Rule 1196
Rule 1198
Rule 1252
Rule 1282
Rule 1825
Rule 1833
Rubi steps
\begin {align*} \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^{13}} \, dx &=-\frac {\left (\frac {165 c}{x^{12}}+\frac {180 d}{x^{11}}+\frac {198 e}{x^{10}}+\frac {220 f}{x^9}\right ) \left (a+b x^4\right )^{3/2}}{1980}-(6 b) \int \frac {\left (-\frac {c}{12}-\frac {d x}{11}-\frac {e x^2}{10}-\frac {f x^3}{9}\right ) \sqrt {a+b x^4}}{x^9} \, dx\\ &=-\frac {b \left (\frac {1155 c}{x^8}+\frac {1440 d}{x^7}+\frac {1848 e}{x^6}+\frac {2464 f}{x^5}\right ) \sqrt {a+b x^4}}{18480}-\frac {\left (\frac {165 c}{x^{12}}+\frac {180 d}{x^{11}}+\frac {198 e}{x^{10}}+\frac {220 f}{x^9}\right ) \left (a+b x^4\right )^{3/2}}{1980}+\left (12 b^2\right ) \int \frac {\frac {c}{96}+\frac {d x}{77}+\frac {e x^2}{60}+\frac {f x^3}{45}}{x^5 \sqrt {a+b x^4}} \, dx\\ &=-\frac {b \left (\frac {1155 c}{x^8}+\frac {1440 d}{x^7}+\frac {1848 e}{x^6}+\frac {2464 f}{x^5}\right ) \sqrt {a+b x^4}}{18480}-\frac {\left (\frac {165 c}{x^{12}}+\frac {180 d}{x^{11}}+\frac {198 e}{x^{10}}+\frac {220 f}{x^9}\right ) \left (a+b x^4\right )^{3/2}}{1980}+\left (12 b^2\right ) \int \left (\frac {\frac {c}{96}+\frac {e x^2}{60}}{x^5 \sqrt {a+b x^4}}+\frac {\frac {d}{77}+\frac {f x^2}{45}}{x^4 \sqrt {a+b x^4}}\right ) \, dx\\ &=-\frac {b \left (\frac {1155 c}{x^8}+\frac {1440 d}{x^7}+\frac {1848 e}{x^6}+\frac {2464 f}{x^5}\right ) \sqrt {a+b x^4}}{18480}-\frac {\left (\frac {165 c}{x^{12}}+\frac {180 d}{x^{11}}+\frac {198 e}{x^{10}}+\frac {220 f}{x^9}\right ) \left (a+b x^4\right )^{3/2}}{1980}+\left (12 b^2\right ) \int \frac {\frac {c}{96}+\frac {e x^2}{60}}{x^5 \sqrt {a+b x^4}} \, dx+\left (12 b^2\right ) \int \frac {\frac {d}{77}+\frac {f x^2}{45}}{x^4 \sqrt {a+b x^4}} \, dx\\ &=-\frac {b \left (\frac {1155 c}{x^8}+\frac {1440 d}{x^7}+\frac {1848 e}{x^6}+\frac {2464 f}{x^5}\right ) \sqrt {a+b x^4}}{18480}-\frac {4 b^2 d \sqrt {a+b x^4}}{77 a x^3}-\frac {\left (\frac {165 c}{x^{12}}+\frac {180 d}{x^{11}}+\frac {198 e}{x^{10}}+\frac {220 f}{x^9}\right ) \left (a+b x^4\right )^{3/2}}{1980}+\left (6 b^2\right ) \operatorname {Subst}\left (\int \frac {\frac {c}{96}+\frac {e x}{60}}{x^3 \sqrt {a+b x^2}} \, dx,x,x^2\right )-\frac {\left (4 b^2\right ) \int \frac {-\frac {a f}{15}+\frac {1}{77} b d x^2}{x^2 \sqrt {a+b x^4}} \, dx}{a}\\ &=-\frac {b \left (\frac {1155 c}{x^8}+\frac {1440 d}{x^7}+\frac {1848 e}{x^6}+\frac {2464 f}{x^5}\right ) \sqrt {a+b x^4}}{18480}-\frac {b^2 c \sqrt {a+b x^4}}{32 a x^4}-\frac {4 b^2 d \sqrt {a+b x^4}}{77 a x^3}-\frac {4 b^2 f \sqrt {a+b x^4}}{15 a x}-\frac {\left (\frac {165 c}{x^{12}}+\frac {180 d}{x^{11}}+\frac {198 e}{x^{10}}+\frac {220 f}{x^9}\right ) \left (a+b x^4\right )^{3/2}}{1980}+\frac {\left (4 b^2\right ) \int \frac {-\frac {1}{77} a b d+\frac {1}{15} a b f x^2}{\sqrt {a+b x^4}} \, dx}{a^2}-\frac {\left (3 b^2\right ) \operatorname {Subst}\left (\int \frac {-\frac {a e}{30}+\frac {b c x}{96}}{x^2 \sqrt {a+b x^2}} \, dx,x,x^2\right )}{a}\\ &=-\frac {b \left (\frac {1155 c}{x^8}+\frac {1440 d}{x^7}+\frac {1848 e}{x^6}+\frac {2464 f}{x^5}\right ) \sqrt {a+b x^4}}{18480}-\frac {b^2 c \sqrt {a+b x^4}}{32 a x^4}-\frac {4 b^2 d \sqrt {a+b x^4}}{77 a x^3}-\frac {b^2 e \sqrt {a+b x^4}}{10 a x^2}-\frac {4 b^2 f \sqrt {a+b x^4}}{15 a x}-\frac {\left (\frac {165 c}{x^{12}}+\frac {180 d}{x^{11}}+\frac {198 e}{x^{10}}+\frac {220 f}{x^9}\right ) \left (a+b x^4\right )^{3/2}}{1980}-\frac {\left (b^3 c\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x^2}} \, dx,x,x^2\right )}{32 a}-\frac {\left (4 b^{5/2} f\right ) \int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a}}}{\sqrt {a+b x^4}} \, dx}{15 \sqrt {a}}-\frac {\left (4 b^{5/2} \left (15 \sqrt {b} d-77 \sqrt {a} f\right )\right ) \int \frac {1}{\sqrt {a+b x^4}} \, dx}{1155 a}\\ &=-\frac {b \left (\frac {1155 c}{x^8}+\frac {1440 d}{x^7}+\frac {1848 e}{x^6}+\frac {2464 f}{x^5}\right ) \sqrt {a+b x^4}}{18480}-\frac {b^2 c \sqrt {a+b x^4}}{32 a x^4}-\frac {4 b^2 d \sqrt {a+b x^4}}{77 a x^3}-\frac {b^2 e \sqrt {a+b x^4}}{10 a x^2}-\frac {4 b^2 f \sqrt {a+b x^4}}{15 a x}+\frac {4 b^{5/2} f x \sqrt {a+b x^4}}{15 a \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {\left (\frac {165 c}{x^{12}}+\frac {180 d}{x^{11}}+\frac {198 e}{x^{10}}+\frac {220 f}{x^9}\right ) \left (a+b x^4\right )^{3/2}}{1980}-\frac {4 b^{9/4} f \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 a^{3/4} \sqrt {a+b x^4}}-\frac {2 b^{9/4} \left (15 \sqrt {b} d-77 \sqrt {a} f\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{1155 a^{5/4} \sqrt {a+b x^4}}-\frac {\left (b^3 c\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^4\right )}{64 a}\\ &=-\frac {b \left (\frac {1155 c}{x^8}+\frac {1440 d}{x^7}+\frac {1848 e}{x^6}+\frac {2464 f}{x^5}\right ) \sqrt {a+b x^4}}{18480}-\frac {b^2 c \sqrt {a+b x^4}}{32 a x^4}-\frac {4 b^2 d \sqrt {a+b x^4}}{77 a x^3}-\frac {b^2 e \sqrt {a+b x^4}}{10 a x^2}-\frac {4 b^2 f \sqrt {a+b x^4}}{15 a x}+\frac {4 b^{5/2} f x \sqrt {a+b x^4}}{15 a \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {\left (\frac {165 c}{x^{12}}+\frac {180 d}{x^{11}}+\frac {198 e}{x^{10}}+\frac {220 f}{x^9}\right ) \left (a+b x^4\right )^{3/2}}{1980}-\frac {4 b^{9/4} f \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 a^{3/4} \sqrt {a+b x^4}}-\frac {2 b^{9/4} \left (15 \sqrt {b} d-77 \sqrt {a} f\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{1155 a^{5/4} \sqrt {a+b x^4}}-\frac {\left (b^2 c\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^4}\right )}{32 a}\\ &=-\frac {b \left (\frac {1155 c}{x^8}+\frac {1440 d}{x^7}+\frac {1848 e}{x^6}+\frac {2464 f}{x^5}\right ) \sqrt {a+b x^4}}{18480}-\frac {b^2 c \sqrt {a+b x^4}}{32 a x^4}-\frac {4 b^2 d \sqrt {a+b x^4}}{77 a x^3}-\frac {b^2 e \sqrt {a+b x^4}}{10 a x^2}-\frac {4 b^2 f \sqrt {a+b x^4}}{15 a x}+\frac {4 b^{5/2} f x \sqrt {a+b x^4}}{15 a \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {\left (\frac {165 c}{x^{12}}+\frac {180 d}{x^{11}}+\frac {198 e}{x^{10}}+\frac {220 f}{x^9}\right ) \left (a+b x^4\right )^{3/2}}{1980}+\frac {b^3 c \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{32 a^{3/2}}-\frac {4 b^{9/4} f \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 a^{3/4} \sqrt {a+b x^4}}-\frac {2 b^{9/4} \left (15 \sqrt {b} d-77 \sqrt {a} f\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{1155 a^{5/4} \sqrt {a+b x^4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.21, size = 149, normalized size = 0.33 \[ -\frac {\sqrt {a+b x^4} \left (90 a^5 d \, _2F_1\left (-\frac {11}{4},-\frac {3}{2};-\frac {7}{4};-\frac {b x^4}{a}\right )+11 x \left (10 a^5 f x \, _2F_1\left (-\frac {9}{4},-\frac {3}{2};-\frac {5}{4};-\frac {b x^4}{a}\right )+9 \left (a+b x^4\right )^2 \sqrt {\frac {b x^4}{a}+1} \left (a^3 e-b^3 c x^{10} \, _2F_1\left (\frac {5}{2},4;\frac {7}{2};\frac {b x^4}{a}+1\right )\right )\right )\right )}{990 a^4 x^{11} \sqrt {\frac {b x^4}{a}+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b f x^{7} + b e x^{6} + b d x^{5} + b c x^{4} + a f x^{3} + a e x^{2} + a d x + a c\right )} \sqrt {b x^{4} + a}}{x^{13}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x^{4} + a\right )}^{\frac {3}{2}} {\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{13}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.20, size = 462, normalized size = 1.03 \[ -\frac {4 i \sqrt {-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, b^{\frac {5}{2}} f \EllipticE \left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, x , i\right )}{15 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, \sqrt {a}}+\frac {4 i \sqrt {-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, b^{\frac {5}{2}} f \EllipticF \left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, x , i\right )}{15 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, \sqrt {a}}-\frac {4 \sqrt {-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, b^{3} d \EllipticF \left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, x , i\right )}{77 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, a}+\frac {b^{3} c \ln \left (\frac {2 a +2 \sqrt {b \,x^{4}+a}\, \sqrt {a}}{x^{2}}\right )}{32 a^{\frac {3}{2}}}-\frac {4 \sqrt {b \,x^{4}+a}\, b^{2} f}{15 a x}-\frac {4 \sqrt {b \,x^{4}+a}\, b^{2} d}{77 a \,x^{3}}-\frac {\sqrt {b \,x^{4}+a}\, b^{2} c}{32 a \,x^{4}}-\frac {11 \sqrt {b \,x^{4}+a}\, b f}{45 x^{5}}-\frac {13 \sqrt {b \,x^{4}+a}\, b d}{77 x^{7}}-\frac {7 \sqrt {b \,x^{4}+a}\, b c}{48 x^{8}}-\frac {\sqrt {b \,x^{4}+a}\, a f}{9 x^{9}}-\frac {\sqrt {b \,x^{4}+a}\, a d}{11 x^{11}}-\frac {\sqrt {b \,x^{4}+a}\, a c}{12 x^{12}}-\frac {\sqrt {b \,x^{4}+a}\, \left (b^{2} x^{8}+2 a b \,x^{4}+a^{2}\right ) e}{10 a \,x^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{192} \, {\left (\frac {3 \, b^{3} \log \left (\frac {\sqrt {b x^{4} + a} - \sqrt {a}}{\sqrt {b x^{4} + a} + \sqrt {a}}\right )}{a^{\frac {3}{2}}} + \frac {2 \, {\left (3 \, {\left (b x^{4} + a\right )}^{\frac {5}{2}} b^{3} + 8 \, {\left (b x^{4} + a\right )}^{\frac {3}{2}} a b^{3} - 3 \, \sqrt {b x^{4} + a} a^{2} b^{3}\right )}}{{\left (b x^{4} + a\right )}^{3} a - 3 \, {\left (b x^{4} + a\right )}^{2} a^{2} + 3 \, {\left (b x^{4} + a\right )} a^{3} - a^{4}}\right )} c + \int \frac {{\left (b f x^{6} + b e x^{5} + b d x^{4} + a f x^{2} + a e x + a d\right )} \sqrt {b x^{4} + a}}{x^{12}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (b\,x^4+a\right )}^{3/2}\,\left (f\,x^3+e\,x^2+d\,x+c\right )}{x^{13}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 31.88, size = 403, normalized size = 0.90 \[ \frac {a^{\frac {3}{2}} d \Gamma \left (- \frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {11}{4}, - \frac {1}{2} \\ - \frac {7}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{11} \Gamma \left (- \frac {7}{4}\right )} + \frac {a^{\frac {3}{2}} f \Gamma \left (- \frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {9}{4}, - \frac {1}{2} \\ - \frac {5}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{9} \Gamma \left (- \frac {5}{4}\right )} + \frac {\sqrt {a} b d \Gamma \left (- \frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{4}, - \frac {1}{2} \\ - \frac {3}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{7} \Gamma \left (- \frac {3}{4}\right )} + \frac {\sqrt {a} b f \Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, - \frac {1}{2} \\ - \frac {1}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{5} \Gamma \left (- \frac {1}{4}\right )} - \frac {a^{2} c}{12 \sqrt {b} x^{14} \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {11 a \sqrt {b} c}{48 x^{10} \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {a \sqrt {b} e \sqrt {\frac {a}{b x^{4}} + 1}}{10 x^{8}} - \frac {17 b^{\frac {3}{2}} c}{96 x^{6} \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {b^{\frac {3}{2}} e \sqrt {\frac {a}{b x^{4}} + 1}}{5 x^{4}} - \frac {b^{\frac {5}{2}} c}{32 a x^{2} \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {b^{\frac {5}{2}} e \sqrt {\frac {a}{b x^{4}} + 1}}{10 a} + \frac {b^{3} c \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{2}} \right )}}{32 a^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________