Optimal. Leaf size=386 \[ \frac {12 a \sqrt {b} f x \sqrt {a+b x^4}}{5 \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {3}{4} b \left (c+e x^2\right ) \sqrt {a+b x^4}+\frac {2}{15} b x \left (5 d+9 f x^2\right ) \sqrt {a+b x^4}-\frac {1}{12} \left (\frac {3 c}{x^4}+\frac {4 d}{x^3}+\frac {6 e}{x^2}+\frac {12 f}{x}\right ) \left (a+b x^4\right )^{3/2}+\frac {3}{4} a \sqrt {b} e \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )-\frac {3}{4} \sqrt {a} b c \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )-\frac {12 a^{5/4} \sqrt [4]{b} f \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 \sqrt {a+b x^4}}+\frac {2 a^{3/4} \sqrt [4]{b} \left (5 \sqrt {b} d+9 \sqrt {a} f\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 \sqrt {a+b x^4}} \]
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Rubi [A]
time = 0.24, antiderivative size = 386, normalized size of antiderivative = 1.00, number
of steps used = 15, number of rules used = 15, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules
used = {14, 1839, 1847, 1266, 829, 858, 223, 212, 272, 65, 214, 1191, 1212, 226, 1210}
\begin {gather*} \frac {2 a^{3/4} \sqrt [4]{b} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (9 \sqrt {a} f+5 \sqrt {b} d\right ) F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 \sqrt {a+b x^4}}-\frac {12 a^{5/4} \sqrt [4]{b} 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 \text {ArcTan}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 \sqrt {a+b x^4}}-\frac {1}{12} \left (a+b x^4\right )^{3/2} \left (\frac {3 c}{x^4}+\frac {4 d}{x^3}+\frac {6 e}{x^2}+\frac {12 f}{x}\right )+\frac {3}{4} b \sqrt {a+b x^4} \left (c+e x^2\right )-\frac {3}{4} \sqrt {a} b c \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )+\frac {2}{15} b x \sqrt {a+b x^4} \left (5 d+9 f x^2\right )+\frac {3}{4} a \sqrt {b} e \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )+\frac {12 a \sqrt {b} f x \sqrt {a+b x^4}}{5 \left (\sqrt {a}+\sqrt {b} x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 65
Rule 212
Rule 214
Rule 223
Rule 226
Rule 272
Rule 829
Rule 858
Rule 1191
Rule 1210
Rule 1212
Rule 1266
Rule 1839
Rule 1847
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^5} \, dx &=-\frac {1}{12} \left (\frac {3 c}{x^4}+\frac {4 d}{x^3}+\frac {6 e}{x^2}+\frac {12 f}{x}\right ) \left (a+b x^4\right )^{3/2}-(6 b) \int \frac {\left (-\frac {c}{4}-\frac {d x}{3}-\frac {e x^2}{2}-f x^3\right ) \sqrt {a+b x^4}}{x} \, dx\\ &=-\frac {1}{12} \left (\frac {3 c}{x^4}+\frac {4 d}{x^3}+\frac {6 e}{x^2}+\frac {12 f}{x}\right ) \left (a+b x^4\right )^{3/2}-(6 b) \int \left (\frac {\left (-\frac {c}{4}-\frac {e x^2}{2}\right ) \sqrt {a+b x^4}}{x}+\left (-\frac {d}{3}-f x^2\right ) \sqrt {a+b x^4}\right ) \, dx\\ &=-\frac {1}{12} \left (\frac {3 c}{x^4}+\frac {4 d}{x^3}+\frac {6 e}{x^2}+\frac {12 f}{x}\right ) \left (a+b x^4\right )^{3/2}-(6 b) \int \frac {\left (-\frac {c}{4}-\frac {e x^2}{2}\right ) \sqrt {a+b x^4}}{x} \, dx-(6 b) \int \left (-\frac {d}{3}-f x^2\right ) \sqrt {a+b x^4} \, dx\\ &=\frac {2}{15} b x \left (5 d+9 f x^2\right ) \sqrt {a+b x^4}-\frac {1}{12} \left (\frac {3 c}{x^4}+\frac {4 d}{x^3}+\frac {6 e}{x^2}+\frac {12 f}{x}\right ) \left (a+b x^4\right )^{3/2}-\frac {1}{5} (2 b) \int \frac {-\frac {10 a d}{3}-6 a f x^2}{\sqrt {a+b x^4}} \, dx-(3 b) \text {Subst}\left (\int \frac {\left (-\frac {c}{4}-\frac {e x}{2}\right ) \sqrt {a+b x^2}}{x} \, dx,x,x^2\right )\\ &=\frac {3}{4} b \left (c+e x^2\right ) \sqrt {a+b x^4}+\frac {2}{15} b x \left (5 d+9 f x^2\right ) \sqrt {a+b x^4}-\frac {1}{12} \left (\frac {3 c}{x^4}+\frac {4 d}{x^3}+\frac {6 e}{x^2}+\frac {12 f}{x}\right ) \left (a+b x^4\right )^{3/2}-\frac {3}{2} \text {Subst}\left (\int \frac {-\frac {1}{2} a b c-\frac {1}{2} a b e x}{x \sqrt {a+b x^2}} \, dx,x,x^2\right )-\frac {1}{5} \left (12 a^{3/2} \sqrt {b} f\right ) \int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a}}}{\sqrt {a+b x^4}} \, dx+\frac {1}{15} \left (4 a b \left (5 d+\frac {9 \sqrt {a} f}{\sqrt {b}}\right )\right ) \int \frac {1}{\sqrt {a+b x^4}} \, dx\\ &=\frac {12 a \sqrt {b} f x \sqrt {a+b x^4}}{5 \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {3}{4} b \left (c+e x^2\right ) \sqrt {a+b x^4}+\frac {2}{15} b x \left (5 d+9 f x^2\right ) \sqrt {a+b x^4}-\frac {1}{12} \left (\frac {3 c}{x^4}+\frac {4 d}{x^3}+\frac {6 e}{x^2}+\frac {12 f}{x}\right ) \left (a+b x^4\right )^{3/2}-\frac {12 a^{5/4} \sqrt [4]{b} 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 )}{5 \sqrt {a+b x^4}}+\frac {2 a^{3/4} \sqrt [4]{b} \left (5 \sqrt {b} d+9 \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 )}{15 \sqrt {a+b x^4}}+\frac {1}{4} (3 a b c) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x^2}} \, dx,x,x^2\right )+\frac {1}{4} (3 a b e) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,x^2\right )\\ &=\frac {12 a \sqrt {b} f x \sqrt {a+b x^4}}{5 \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {3}{4} b \left (c+e x^2\right ) \sqrt {a+b x^4}+\frac {2}{15} b x \left (5 d+9 f x^2\right ) \sqrt {a+b x^4}-\frac {1}{12} \left (\frac {3 c}{x^4}+\frac {4 d}{x^3}+\frac {6 e}{x^2}+\frac {12 f}{x}\right ) \left (a+b x^4\right )^{3/2}-\frac {12 a^{5/4} \sqrt [4]{b} 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 )}{5 \sqrt {a+b x^4}}+\frac {2 a^{3/4} \sqrt [4]{b} \left (5 \sqrt {b} d+9 \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 )}{15 \sqrt {a+b x^4}}+\frac {1}{8} (3 a b c) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^4\right )+\frac {1}{4} (3 a b e) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x^2}{\sqrt {a+b x^4}}\right )\\ &=\frac {12 a \sqrt {b} f x \sqrt {a+b x^4}}{5 \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {3}{4} b \left (c+e x^2\right ) \sqrt {a+b x^4}+\frac {2}{15} b x \left (5 d+9 f x^2\right ) \sqrt {a+b x^4}-\frac {1}{12} \left (\frac {3 c}{x^4}+\frac {4 d}{x^3}+\frac {6 e}{x^2}+\frac {12 f}{x}\right ) \left (a+b x^4\right )^{3/2}+\frac {3}{4} a \sqrt {b} e \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )-\frac {12 a^{5/4} \sqrt [4]{b} 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 )}{5 \sqrt {a+b x^4}}+\frac {2 a^{3/4} \sqrt [4]{b} \left (5 \sqrt {b} d+9 \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 )}{15 \sqrt {a+b x^4}}+\frac {1}{4} (3 a c) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^4}\right )\\ &=\frac {12 a \sqrt {b} f x \sqrt {a+b x^4}}{5 \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {3}{4} b \left (c+e x^2\right ) \sqrt {a+b x^4}+\frac {2}{15} b x \left (5 d+9 f x^2\right ) \sqrt {a+b x^4}-\frac {1}{12} \left (\frac {3 c}{x^4}+\frac {4 d}{x^3}+\frac {6 e}{x^2}+\frac {12 f}{x}\right ) \left (a+b x^4\right )^{3/2}+\frac {3}{4} a \sqrt {b} e \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )-\frac {3}{4} \sqrt {a} b c \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )-\frac {12 a^{5/4} \sqrt [4]{b} 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 )}{5 \sqrt {a+b x^4}}+\frac {2 a^{3/4} \sqrt [4]{b} \left (5 \sqrt {b} d+9 \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 )}{15 \sqrt {a+b x^4}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 10.53, size = 329, normalized size = 0.85 \begin {gather*} \frac {\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} \left (-\left (\left (a+b x^4\right ) \left (5 a \left (3 c+4 d x+6 x^2 (e+2 f x)\right )-b x^4 (30 c+x (20 d+3 x (5 e+4 f x)))\right )\right )+45 a \sqrt {b} e x^4 \sqrt {a+b x^4} \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )-45 \sqrt {a} b c x^4 \sqrt {a+b x^4} \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )\right )+144 a^{3/2} \sqrt {b} f x^4 \sqrt {1+\frac {b x^4}{a}} E\left (\left .i \sinh ^{-1}\left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} x\right )\right |-1\right )-16 a \sqrt {b} \left (5 i \sqrt {b} d+9 \sqrt {a} f\right ) x^4 \sqrt {1+\frac {b x^4}{a}} F\left (\left .i \sinh ^{-1}\left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} x\right )\right |-1\right )}{60 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} x^4 \sqrt {a+b x^4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.43, size = 350, normalized size = 0.91
method | result | size |
elliptic | \(-\frac {a c \sqrt {b \,x^{4}+a}}{4 x^{4}}-\frac {a d \sqrt {b \,x^{4}+a}}{3 x^{3}}-\frac {a e \sqrt {b \,x^{4}+a}}{2 x^{2}}-\frac {a f \sqrt {b \,x^{4}+a}}{x}+\frac {b f \,x^{3} \sqrt {b \,x^{4}+a}}{5}+\frac {b e \,x^{2} \sqrt {b \,x^{4}+a}}{4}+\frac {b d x \sqrt {b \,x^{4}+a}}{3}+\frac {b c \sqrt {b \,x^{4}+a}}{2}+\frac {4 b d a \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{3 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {3 a \sqrt {b}\, e \ln \left (2 x^{2} \sqrt {b}+2 \sqrt {b \,x^{4}+a}\right )}{4}+\frac {12 i a^{\frac {3}{2}} \sqrt {b}\, f \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\EllipticE \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{5 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {3 \sqrt {a}\, b c \arctanh \left (\frac {\sqrt {a}}{\sqrt {b \,x^{4}+a}}\right )}{4}\) | \(344\) |
default | \(c \left (\frac {b \sqrt {b \,x^{4}+a}}{2}-\frac {3 \sqrt {a}\, b \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{4}+a}}{x^{2}}\right )}{4}-\frac {a \sqrt {b \,x^{4}+a}}{4 x^{4}}\right )+e \left (\frac {b \,x^{2} \sqrt {b \,x^{4}+a}}{4}+\frac {3 a \sqrt {b}\, \ln \left (x^{2} \sqrt {b}+\sqrt {b \,x^{4}+a}\right )}{4}-\frac {a \sqrt {b \,x^{4}+a}}{2 x^{2}}\right )+d \left (-\frac {a \sqrt {b \,x^{4}+a}}{3 x^{3}}+\frac {b x \sqrt {b \,x^{4}+a}}{3}+\frac {4 a b \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{3 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+f \left (-\frac {a \sqrt {b \,x^{4}+a}}{x}+\frac {\sqrt {b \,x^{4}+a}\, b \,x^{3}}{5}+\frac {12 i a^{\frac {3}{2}} \sqrt {b}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\EllipticE \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{5 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )\) | \(350\) |
risch | \(-\frac {a \sqrt {b \,x^{4}+a}\, \left (12 f \,x^{3}+6 e \,x^{2}+4 d x +3 c \right )}{12 x^{4}}+\frac {b f \,x^{3} \sqrt {b \,x^{4}+a}}{5}+\frac {12 i \sqrt {b}\, f \,a^{\frac {3}{2}} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{5 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {12 i \sqrt {b}\, f \,a^{\frac {3}{2}} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticE \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{5 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {b e \,x^{2} \sqrt {b \,x^{4}+a}}{4}+\frac {3 \sqrt {b}\, a e \ln \left (x^{2} \sqrt {b}+\sqrt {b \,x^{4}+a}\right )}{4}+\frac {b d x \sqrt {b \,x^{4}+a}}{3}+\frac {4 b d a \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{3 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {b c \sqrt {b \,x^{4}+a}}{2}-\frac {3 b \sqrt {a}\, c \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{4}+a}}{x^{2}}\right )}{4}\) | \(380\) |
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [F]
time = 0.27, size = 59, normalized size = 0.15 \begin {gather*} {\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^{5}}, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 5.74, size = 379, normalized size = 0.98 \begin {gather*} \frac {a^{\frac {3}{2}} d \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {1}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{3} \Gamma \left (\frac {1}{4}\right )} - \frac {a^{\frac {3}{2}} e}{2 x^{2} \sqrt {1 + \frac {b x^{4}}{a}}} + \frac {a^{\frac {3}{2}} f \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{4} \\ \frac {3}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x \Gamma \left (\frac {3}{4}\right )} - \frac {3 \sqrt {a} b c \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{2}} \right )}}{4} + \frac {\sqrt {a} b d x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} + \frac {\sqrt {a} b e x^{2} \sqrt {1 + \frac {b x^{4}}{a}}}{4} - \frac {\sqrt {a} b e x^{2}}{2 \sqrt {1 + \frac {b x^{4}}{a}}} + \frac {\sqrt {a} b f x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} - \frac {a \sqrt {b} c \sqrt {\frac {a}{b x^{4}} + 1}}{4 x^{2}} + \frac {a \sqrt {b} c}{2 x^{2} \sqrt {\frac {a}{b x^{4}} + 1}} + \frac {3 a \sqrt {b} e \operatorname {asinh}{\left (\frac {\sqrt {b} x^{2}}{\sqrt {a}} \right )}}{4} + \frac {b^{\frac {3}{2}} c x^{2}}{2 \sqrt {\frac {a}{b x^{4}} + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (b\,x^4+a\right )}^{3/2}\,\left (f\,x^3+e\,x^2+d\,x+c\right )}{x^5} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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