Optimal. Leaf size=405 \[ -\frac {b \left (\frac {224 c}{x^5}+\frac {315 d}{x^4}+\frac {480 e}{x^3}+\frac {840 f}{x^2}\right ) \sqrt {a+b x^4}}{1680}-\frac {4 b^2 c \sqrt {a+b x^4}}{15 a x}+\frac {4 b^{5/2} c x \sqrt {a+b x^4}}{15 a \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {1}{504} \left (\frac {56 c}{x^9}+\frac {63 d}{x^8}+\frac {72 e}{x^7}+\frac {84 f}{x^6}\right ) \left (a+b x^4\right )^{3/2}+\frac {1}{2} b^{3/2} f \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )-\frac {3 b^2 d \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{16 \sqrt {a}}-\frac {4 b^{9/4} c \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 a^{3/4} \sqrt {a+b x^4}}+\frac {2 b^{7/4} \left (7 \sqrt {b} c+15 \sqrt {a} e\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{105 a^{3/4} \sqrt {a+b x^4}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.29, antiderivative size = 405, normalized size of antiderivative = 1.00, number of steps
used = 15, number of rules used = 14, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {14, 1839,
1847, 1296, 1212, 226, 1210, 1266, 858, 223, 212, 272, 65, 214} \begin {gather*} \frac {2 b^{7/4} \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 {a} e+7 \sqrt {b} c\right ) F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{105 a^{3/4} \sqrt {a+b x^4}}-\frac {4 b^{9/4} c \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 )}{15 a^{3/4} \sqrt {a+b x^4}}+\frac {4 b^{5/2} c x \sqrt {a+b x^4}}{15 a \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {1}{2} b^{3/2} f \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )-\frac {4 b^2 c \sqrt {a+b x^4}}{15 a x}-\frac {3 b^2 d \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{16 \sqrt {a}}-\frac {b \sqrt {a+b x^4} \left (\frac {224 c}{x^5}+\frac {315 d}{x^4}+\frac {480 e}{x^3}+\frac {840 f}{x^2}\right )}{1680}-\frac {1}{504} \left (a+b x^4\right )^{3/2} \left (\frac {56 c}{x^9}+\frac {63 d}{x^8}+\frac {72 e}{x^7}+\frac {84 f}{x^6}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 14
Rule 65
Rule 212
Rule 214
Rule 223
Rule 226
Rule 272
Rule 858
Rule 1210
Rule 1212
Rule 1266
Rule 1296
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^{10}} \, dx &=-\frac {1}{504} \left (\frac {56 c}{x^9}+\frac {63 d}{x^8}+\frac {72 e}{x^7}+\frac {84 f}{x^6}\right ) \left (a+b x^4\right )^{3/2}-(6 b) \int \frac {\left (-\frac {c}{9}-\frac {d x}{8}-\frac {e x^2}{7}-\frac {f x^3}{6}\right ) \sqrt {a+b x^4}}{x^6} \, dx\\ &=-\frac {b \left (\frac {224 c}{x^5}+\frac {315 d}{x^4}+\frac {480 e}{x^3}+\frac {840 f}{x^2}\right ) \sqrt {a+b x^4}}{1680}-\frac {1}{504} \left (\frac {56 c}{x^9}+\frac {63 d}{x^8}+\frac {72 e}{x^7}+\frac {84 f}{x^6}\right ) \left (a+b x^4\right )^{3/2}+\left (12 b^2\right ) \int \frac {\frac {c}{45}+\frac {d x}{32}+\frac {e x^2}{21}+\frac {f x^3}{12}}{x^2 \sqrt {a+b x^4}} \, dx\\ &=-\frac {b \left (\frac {224 c}{x^5}+\frac {315 d}{x^4}+\frac {480 e}{x^3}+\frac {840 f}{x^2}\right ) \sqrt {a+b x^4}}{1680}-\frac {1}{504} \left (\frac {56 c}{x^9}+\frac {63 d}{x^8}+\frac {72 e}{x^7}+\frac {84 f}{x^6}\right ) \left (a+b x^4\right )^{3/2}+\left (12 b^2\right ) \int \left (\frac {\frac {c}{45}+\frac {e x^2}{21}}{x^2 \sqrt {a+b x^4}}+\frac {\frac {d}{32}+\frac {f x^2}{12}}{x \sqrt {a+b x^4}}\right ) \, dx\\ &=-\frac {b \left (\frac {224 c}{x^5}+\frac {315 d}{x^4}+\frac {480 e}{x^3}+\frac {840 f}{x^2}\right ) \sqrt {a+b x^4}}{1680}-\frac {1}{504} \left (\frac {56 c}{x^9}+\frac {63 d}{x^8}+\frac {72 e}{x^7}+\frac {84 f}{x^6}\right ) \left (a+b x^4\right )^{3/2}+\left (12 b^2\right ) \int \frac {\frac {c}{45}+\frac {e x^2}{21}}{x^2 \sqrt {a+b x^4}} \, dx+\left (12 b^2\right ) \int \frac {\frac {d}{32}+\frac {f x^2}{12}}{x \sqrt {a+b x^4}} \, dx\\ &=-\frac {b \left (\frac {224 c}{x^5}+\frac {315 d}{x^4}+\frac {480 e}{x^3}+\frac {840 f}{x^2}\right ) \sqrt {a+b x^4}}{1680}-\frac {4 b^2 c \sqrt {a+b x^4}}{15 a x}-\frac {1}{504} \left (\frac {56 c}{x^9}+\frac {63 d}{x^8}+\frac {72 e}{x^7}+\frac {84 f}{x^6}\right ) \left (a+b x^4\right )^{3/2}+\left (6 b^2\right ) \text {Subst}\left (\int \frac {\frac {d}{32}+\frac {f x}{12}}{x \sqrt {a+b x^2}} \, dx,x,x^2\right )-\frac {\left (12 b^2\right ) \int \frac {-\frac {a e}{21}-\frac {1}{45} b c x^2}{\sqrt {a+b x^4}} \, dx}{a}\\ &=-\frac {b \left (\frac {224 c}{x^5}+\frac {315 d}{x^4}+\frac {480 e}{x^3}+\frac {840 f}{x^2}\right ) \sqrt {a+b x^4}}{1680}-\frac {4 b^2 c \sqrt {a+b x^4}}{15 a x}-\frac {1}{504} \left (\frac {56 c}{x^9}+\frac {63 d}{x^8}+\frac {72 e}{x^7}+\frac {84 f}{x^6}\right ) \left (a+b x^4\right )^{3/2}-\frac {\left (4 b^{5/2} c\right ) \int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a}}}{\sqrt {a+b x^4}} \, dx}{15 \sqrt {a}}+\frac {1}{16} \left (3 b^2 d\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x^2}} \, dx,x,x^2\right )+\frac {1}{105} \left (4 b^2 \left (\frac {7 \sqrt {b} c}{\sqrt {a}}+15 e\right )\right ) \int \frac {1}{\sqrt {a+b x^4}} \, dx+\frac {1}{2} \left (b^2 f\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,x^2\right )\\ &=-\frac {b \left (\frac {224 c}{x^5}+\frac {315 d}{x^4}+\frac {480 e}{x^3}+\frac {840 f}{x^2}\right ) \sqrt {a+b x^4}}{1680}-\frac {4 b^2 c \sqrt {a+b x^4}}{15 a x}+\frac {4 b^{5/2} c x \sqrt {a+b x^4}}{15 a \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {1}{504} \left (\frac {56 c}{x^9}+\frac {63 d}{x^8}+\frac {72 e}{x^7}+\frac {84 f}{x^6}\right ) \left (a+b x^4\right )^{3/2}-\frac {4 b^{9/4} c \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^{7/4} \left (7 \sqrt {b} c+15 \sqrt {a} e\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 )}{105 a^{3/4} \sqrt {a+b x^4}}+\frac {1}{32} \left (3 b^2 d\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^4\right )+\frac {1}{2} \left (b^2 f\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x^2}{\sqrt {a+b x^4}}\right )\\ &=-\frac {b \left (\frac {224 c}{x^5}+\frac {315 d}{x^4}+\frac {480 e}{x^3}+\frac {840 f}{x^2}\right ) \sqrt {a+b x^4}}{1680}-\frac {4 b^2 c \sqrt {a+b x^4}}{15 a x}+\frac {4 b^{5/2} c x \sqrt {a+b x^4}}{15 a \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {1}{504} \left (\frac {56 c}{x^9}+\frac {63 d}{x^8}+\frac {72 e}{x^7}+\frac {84 f}{x^6}\right ) \left (a+b x^4\right )^{3/2}+\frac {1}{2} b^{3/2} f \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )-\frac {4 b^{9/4} c \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^{7/4} \left (7 \sqrt {b} c+15 \sqrt {a} e\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 )}{105 a^{3/4} \sqrt {a+b x^4}}+\frac {1}{16} (3 b d) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^4}\right )\\ &=-\frac {b \left (\frac {224 c}{x^5}+\frac {315 d}{x^4}+\frac {480 e}{x^3}+\frac {840 f}{x^2}\right ) \sqrt {a+b x^4}}{1680}-\frac {4 b^2 c \sqrt {a+b x^4}}{15 a x}+\frac {4 b^{5/2} c x \sqrt {a+b x^4}}{15 a \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {1}{504} \left (\frac {56 c}{x^9}+\frac {63 d}{x^8}+\frac {72 e}{x^7}+\frac {84 f}{x^6}\right ) \left (a+b x^4\right )^{3/2}+\frac {1}{2} b^{3/2} f \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )-\frac {3 b^2 d \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{16 \sqrt {a}}-\frac {4 b^{9/4} c \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^{7/4} \left (7 \sqrt {b} c+15 \sqrt {a} e\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 )}{105 a^{3/4} \sqrt {a+b x^4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains complex when optimal does not.
time = 10.63, size = 351, normalized size = 0.87 \begin {gather*} \frac {-\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} \left (\left (a+b x^4\right ) \left (1344 b^2 c x^8+10 a^2 \left (56 c+63 d x+72 e x^2+84 f x^3\right )+a b x^4 (1232 c+15 x (105 d+16 x (9 e+14 f x)))\right )-2520 a b^{3/2} f x^9 \sqrt {a+b x^4} \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )+945 \sqrt {a} b^2 d x^9 \sqrt {a+b x^4} \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )\right )+1344 \sqrt {a} b^{5/2} c x^9 \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 )-192 i \sqrt {a} b^2 \left (-7 i \sqrt {b} c+15 \sqrt {a} e\right ) x^9 \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 )}{5040 a \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} x^9 \sqrt {a+b x^4}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains complex when optimal does not.
time = 0.42, size = 377, normalized size = 0.93
method | result | size |
elliptic | \(-\frac {a c \sqrt {b \,x^{4}+a}}{9 x^{9}}-\frac {a d \sqrt {b \,x^{4}+a}}{8 x^{8}}-\frac {a e \sqrt {b \,x^{4}+a}}{7 x^{7}}-\frac {a f \sqrt {b \,x^{4}+a}}{6 x^{6}}-\frac {11 b c \sqrt {b \,x^{4}+a}}{45 x^{5}}-\frac {5 b d \sqrt {b \,x^{4}+a}}{16 x^{4}}-\frac {3 b e \sqrt {b \,x^{4}+a}}{7 x^{3}}-\frac {2 b f \sqrt {b \,x^{4}+a}}{3 x^{2}}-\frac {4 b^{2} c \sqrt {b \,x^{4}+a}}{15 a x}+\frac {4 b^{2} e \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{7 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {b^{\frac {3}{2}} f \ln \left (2 x^{2} \sqrt {b}+2 \sqrt {b \,x^{4}+a}\right )}{2}+\frac {4 i b^{\frac {5}{2}} c \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\EllipticE \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{15 \sqrt {a}\, \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {3 b^{2} d \arctanh \left (\frac {\sqrt {a}}{\sqrt {b \,x^{4}+a}}\right )}{16 \sqrt {a}}\) | \(372\) |
default | \(c \left (-\frac {a \sqrt {b \,x^{4}+a}}{9 x^{9}}-\frac {11 b \sqrt {b \,x^{4}+a}}{45 x^{5}}-\frac {4 b^{2} \sqrt {b \,x^{4}+a}}{15 a x}+\frac {4 i b^{\frac {5}{2}} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\EllipticE \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{15 \sqrt {a}\, \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+f \left (\frac {b^{\frac {3}{2}} \ln \left (x^{2} \sqrt {b}+\sqrt {b \,x^{4}+a}\right )}{2}-\frac {a \sqrt {b \,x^{4}+a}}{6 x^{6}}-\frac {2 b \sqrt {b \,x^{4}+a}}{3 x^{2}}\right )+d \left (-\frac {a \sqrt {b \,x^{4}+a}}{8 x^{8}}-\frac {5 b \sqrt {b \,x^{4}+a}}{16 x^{4}}-\frac {3 b^{2} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{4}+a}}{x^{2}}\right )}{16 \sqrt {a}}\right )+e \left (-\frac {a \sqrt {b \,x^{4}+a}}{7 x^{7}}-\frac {3 b \sqrt {b \,x^{4}+a}}{7 x^{3}}+\frac {4 b^{2} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{7 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )\) | \(377\) |
risch | \(-\frac {\sqrt {b \,x^{4}+a}\, \left (1344 b^{2} c \,x^{8}+3360 a b f \,x^{7}+2160 a b e \,x^{6}+1575 a b d \,x^{5}+1232 a b c \,x^{4}+840 a^{2} f \,x^{3}+720 a^{2} e \,x^{2}+630 a^{2} d x +560 a^{2} c \right )}{5040 x^{9} a}+\frac {4 i b^{\frac {5}{2}} c \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{15 \sqrt {a}\, \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {4 i b^{\frac {5}{2}} c \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticE \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{15 \sqrt {a}\, \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {b^{\frac {3}{2}} f \ln \left (x^{2} \sqrt {b}+\sqrt {b \,x^{4}+a}\right )}{2}+\frac {4 b^{2} e \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{7 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {3 b^{2} d \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{4}+a}}{x^{2}}\right )}{16 \sqrt {a}}\) | \(378\) |
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 = 0.27, 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^{10}}, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] Result contains complex when optimal does not.
time = 7.39, size = 449, normalized size = 1.11 \begin {gather*} \frac {a^{\frac {3}{2}} c \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 {a^{\frac {3}{2}} e \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 c \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 {\sqrt {a} b e \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 {\sqrt {a} b f}{2 x^{2} \sqrt {1 + \frac {b x^{4}}{a}}} - \frac {a^{2} d}{8 \sqrt {b} x^{10} \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {3 a \sqrt {b} d}{16 x^{6} \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {a \sqrt {b} f \sqrt {\frac {a}{b x^{4}} + 1}}{6 x^{4}} - \frac {b^{\frac {3}{2}} d \sqrt {\frac {a}{b x^{4}} + 1}}{4 x^{2}} - \frac {b^{\frac {3}{2}} d}{16 x^{2} \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {b^{\frac {3}{2}} f \sqrt {\frac {a}{b x^{4}} + 1}}{6} + \frac {b^{\frac {3}{2}} f \operatorname {asinh}{\left (\frac {\sqrt {b} x^{2}}{\sqrt {a}} \right )}}{2} - \frac {3 b^{2} d \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{2}} \right )}}{16 \sqrt {a}} - \frac {b^{2} f x^{2}}{2 \sqrt {a} \sqrt {1 + \frac {b x^{4}}{a}}} \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 {{\left (b\,x^4+a\right )}^{3/2}\,\left (f\,x^3+e\,x^2+d\,x+c\right )}{x^{10}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________