Optimal. Leaf size=245 \[ -\frac {a \left (96 b^3 c-80 a b^2 d+70 a^2 b e-63 a^3 f\right ) x \sqrt {a+b x^2}}{256 b^5}+\frac {\left (96 b^3 c-80 a b^2 d+70 a^2 b e-63 a^3 f\right ) x^3 \sqrt {a+b x^2}}{384 b^4}+\frac {\left (80 b^2 d-70 a b e+63 a^2 f\right ) x^5 \sqrt {a+b x^2}}{480 b^3}+\frac {(10 b e-9 a f) x^7 \sqrt {a+b x^2}}{80 b^2}+\frac {f x^9 \sqrt {a+b x^2}}{10 b}+\frac {a^2 \left (96 b^3 c-80 a b^2 d+70 a^2 b e-63 a^3 f\right ) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{256 b^{11/2}} \]
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Rubi [A]
time = 0.18, antiderivative size = 245, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {1823, 1281,
470, 327, 223, 212} \begin {gather*} \frac {x^5 \sqrt {a+b x^2} \left (63 a^2 f-70 a b e+80 b^2 d\right )}{480 b^3}+\frac {a^2 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (-63 a^3 f+70 a^2 b e-80 a b^2 d+96 b^3 c\right )}{256 b^{11/2}}-\frac {a x \sqrt {a+b x^2} \left (-63 a^3 f+70 a^2 b e-80 a b^2 d+96 b^3 c\right )}{256 b^5}+\frac {x^3 \sqrt {a+b x^2} \left (-63 a^3 f+70 a^2 b e-80 a b^2 d+96 b^3 c\right )}{384 b^4}+\frac {x^7 \sqrt {a+b x^2} (10 b e-9 a f)}{80 b^2}+\frac {f x^9 \sqrt {a+b x^2}}{10 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 223
Rule 327
Rule 470
Rule 1281
Rule 1823
Rubi steps
\begin {align*} \int \frac {x^4 \left (c+d x^2+e x^4+f x^6\right )}{\sqrt {a+b x^2}} \, dx &=\frac {f x^9 \sqrt {a+b x^2}}{10 b}+\frac {\int \frac {x^4 \left (10 b c+10 b d x^2+(10 b e-9 a f) x^4\right )}{\sqrt {a+b x^2}} \, dx}{10 b}\\ &=\frac {(10 b e-9 a f) x^7 \sqrt {a+b x^2}}{80 b^2}+\frac {f x^9 \sqrt {a+b x^2}}{10 b}+\frac {\int \frac {x^4 \left (80 b^2 c+\left (80 b^2 d-70 a b e+63 a^2 f\right ) x^2\right )}{\sqrt {a+b x^2}} \, dx}{80 b^2}\\ &=\frac {\left (80 b^2 d-70 a b e+63 a^2 f\right ) x^5 \sqrt {a+b x^2}}{480 b^3}+\frac {(10 b e-9 a f) x^7 \sqrt {a+b x^2}}{80 b^2}+\frac {f x^9 \sqrt {a+b x^2}}{10 b}+\frac {\left (96 b^3 c-80 a b^2 d+70 a^2 b e-63 a^3 f\right ) \int \frac {x^4}{\sqrt {a+b x^2}} \, dx}{96 b^3}\\ &=\frac {\left (96 b^3 c-80 a b^2 d+70 a^2 b e-63 a^3 f\right ) x^3 \sqrt {a+b x^2}}{384 b^4}+\frac {\left (80 b^2 d-70 a b e+63 a^2 f\right ) x^5 \sqrt {a+b x^2}}{480 b^3}+\frac {(10 b e-9 a f) x^7 \sqrt {a+b x^2}}{80 b^2}+\frac {f x^9 \sqrt {a+b x^2}}{10 b}-\frac {\left (a \left (96 b^3 c-80 a b^2 d+70 a^2 b e-63 a^3 f\right )\right ) \int \frac {x^2}{\sqrt {a+b x^2}} \, dx}{128 b^4}\\ &=-\frac {a \left (96 b^3 c-80 a b^2 d+70 a^2 b e-63 a^3 f\right ) x \sqrt {a+b x^2}}{256 b^5}+\frac {\left (96 b^3 c-80 a b^2 d+70 a^2 b e-63 a^3 f\right ) x^3 \sqrt {a+b x^2}}{384 b^4}+\frac {\left (80 b^2 d-70 a b e+63 a^2 f\right ) x^5 \sqrt {a+b x^2}}{480 b^3}+\frac {(10 b e-9 a f) x^7 \sqrt {a+b x^2}}{80 b^2}+\frac {f x^9 \sqrt {a+b x^2}}{10 b}+\frac {\left (a^2 \left (96 b^3 c-80 a b^2 d+70 a^2 b e-63 a^3 f\right )\right ) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{256 b^5}\\ &=-\frac {a \left (96 b^3 c-80 a b^2 d+70 a^2 b e-63 a^3 f\right ) x \sqrt {a+b x^2}}{256 b^5}+\frac {\left (96 b^3 c-80 a b^2 d+70 a^2 b e-63 a^3 f\right ) x^3 \sqrt {a+b x^2}}{384 b^4}+\frac {\left (80 b^2 d-70 a b e+63 a^2 f\right ) x^5 \sqrt {a+b x^2}}{480 b^3}+\frac {(10 b e-9 a f) x^7 \sqrt {a+b x^2}}{80 b^2}+\frac {f x^9 \sqrt {a+b x^2}}{10 b}+\frac {\left (a^2 \left (96 b^3 c-80 a b^2 d+70 a^2 b e-63 a^3 f\right )\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{256 b^5}\\ &=-\frac {a \left (96 b^3 c-80 a b^2 d+70 a^2 b e-63 a^3 f\right ) x \sqrt {a+b x^2}}{256 b^5}+\frac {\left (96 b^3 c-80 a b^2 d+70 a^2 b e-63 a^3 f\right ) x^3 \sqrt {a+b x^2}}{384 b^4}+\frac {\left (80 b^2 d-70 a b e+63 a^2 f\right ) x^5 \sqrt {a+b x^2}}{480 b^3}+\frac {(10 b e-9 a f) x^7 \sqrt {a+b x^2}}{80 b^2}+\frac {f x^9 \sqrt {a+b x^2}}{10 b}+\frac {a^2 \left (96 b^3 c-80 a b^2 d+70 a^2 b e-63 a^3 f\right ) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{256 b^{11/2}}\\ \end {align*}
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Mathematica [A]
time = 0.34, size = 186, normalized size = 0.76 \begin {gather*} \frac {\sqrt {b} x \sqrt {a+b x^2} \left (945 a^4 f-210 a^3 b \left (5 e+3 f x^2\right )+4 a^2 b^2 \left (300 d+175 e x^2+126 f x^4\right )+32 b^4 x^2 \left (30 c+20 d x^2+15 e x^4+12 f x^6\right )-16 a b^3 \left (90 c+50 d x^2+35 e x^4+27 f x^6\right )\right )+15 a^2 \left (-96 b^3 c+80 a b^2 d-70 a^2 b e+63 a^3 f\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{3840 b^{11/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 402, normalized size = 1.64
method | result | size |
risch | \(\frac {x \left (384 f \,x^{8} b^{4}-432 a \,b^{3} f \,x^{6}+480 b^{4} e \,x^{6}+504 a^{2} b^{2} f \,x^{4}-560 a \,b^{3} e \,x^{4}+640 b^{4} d \,x^{4}-630 a^{3} b f \,x^{2}+700 a^{2} b^{2} e \,x^{2}-800 a \,b^{3} d \,x^{2}+960 b^{4} c \,x^{2}+945 a^{4} f -1050 a^{3} b e +1200 a^{2} b^{2} d -1440 a \,b^{3} c \right ) \sqrt {b \,x^{2}+a}}{3840 b^{5}}-\frac {63 a^{5} \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right ) f}{256 b^{\frac {11}{2}}}+\frac {35 a^{4} \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right ) e}{128 b^{\frac {9}{2}}}-\frac {5 a^{3} \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right ) d}{16 b^{\frac {7}{2}}}+\frac {3 a^{2} \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right ) c}{8 b^{\frac {5}{2}}}\) | \(247\) |
default | \(f \left (\frac {x^{9} \sqrt {b \,x^{2}+a}}{10 b}-\frac {9 a \left (\frac {x^{7} \sqrt {b \,x^{2}+a}}{8 b}-\frac {7 a \left (\frac {x^{5} \sqrt {b \,x^{2}+a}}{6 b}-\frac {5 a \left (\frac {x^{3} \sqrt {b \,x^{2}+a}}{4 b}-\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2 b}-\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}\right )}{4 b}\right )}{6 b}\right )}{8 b}\right )}{10 b}\right )+e \left (\frac {x^{7} \sqrt {b \,x^{2}+a}}{8 b}-\frac {7 a \left (\frac {x^{5} \sqrt {b \,x^{2}+a}}{6 b}-\frac {5 a \left (\frac {x^{3} \sqrt {b \,x^{2}+a}}{4 b}-\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2 b}-\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}\right )}{4 b}\right )}{6 b}\right )}{8 b}\right )+d \left (\frac {x^{5} \sqrt {b \,x^{2}+a}}{6 b}-\frac {5 a \left (\frac {x^{3} \sqrt {b \,x^{2}+a}}{4 b}-\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2 b}-\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}\right )}{4 b}\right )}{6 b}\right )+c \left (\frac {x^{3} \sqrt {b \,x^{2}+a}}{4 b}-\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2 b}-\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}\right )}{4 b}\right )\) | \(402\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 344, normalized size = 1.40 \begin {gather*} \frac {\sqrt {b x^{2} + a} f x^{9}}{10 \, b} - \frac {9 \, \sqrt {b x^{2} + a} a f x^{7}}{80 \, b^{2}} + \frac {\sqrt {b x^{2} + a} x^{7} e}{8 \, b} + \frac {\sqrt {b x^{2} + a} d x^{5}}{6 \, b} + \frac {21 \, \sqrt {b x^{2} + a} a^{2} f x^{5}}{160 \, b^{3}} - \frac {7 \, \sqrt {b x^{2} + a} a x^{5} e}{48 \, b^{2}} + \frac {\sqrt {b x^{2} + a} c x^{3}}{4 \, b} - \frac {5 \, \sqrt {b x^{2} + a} a d x^{3}}{24 \, b^{2}} - \frac {21 \, \sqrt {b x^{2} + a} a^{3} f x^{3}}{128 \, b^{4}} + \frac {35 \, \sqrt {b x^{2} + a} a^{2} x^{3} e}{192 \, b^{3}} - \frac {3 \, \sqrt {b x^{2} + a} a c x}{8 \, b^{2}} + \frac {5 \, \sqrt {b x^{2} + a} a^{2} d x}{16 \, b^{3}} + \frac {63 \, \sqrt {b x^{2} + a} a^{4} f x}{256 \, b^{5}} + \frac {3 \, a^{2} c \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{8 \, b^{\frac {5}{2}}} - \frac {5 \, a^{3} d \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{16 \, b^{\frac {7}{2}}} - \frac {63 \, a^{5} f \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{256 \, b^{\frac {11}{2}}} - \frac {35 \, \sqrt {b x^{2} + a} a^{3} x e}{128 \, b^{4}} + \frac {35 \, a^{4} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right ) e}{128 \, b^{\frac {9}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.20, size = 432, normalized size = 1.76 \begin {gather*} \left [\frac {15 \, {\left (96 \, a^{2} b^{3} c - 80 \, a^{3} b^{2} d - 63 \, a^{5} f + 70 \, a^{4} b e\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (384 \, b^{5} f x^{9} - 432 \, a b^{4} f x^{7} + 8 \, {\left (80 \, b^{5} d + 63 \, a^{2} b^{3} f\right )} x^{5} + 10 \, {\left (96 \, b^{5} c - 80 \, a b^{4} d - 63 \, a^{3} b^{2} f\right )} x^{3} - 15 \, {\left (96 \, a b^{4} c - 80 \, a^{2} b^{3} d - 63 \, a^{4} b f\right )} x + 10 \, {\left (48 \, b^{5} x^{7} - 56 \, a b^{4} x^{5} + 70 \, a^{2} b^{3} x^{3} - 105 \, a^{3} b^{2} x\right )} e\right )} \sqrt {b x^{2} + a}}{7680 \, b^{6}}, -\frac {15 \, {\left (96 \, a^{2} b^{3} c - 80 \, a^{3} b^{2} d - 63 \, a^{5} f + 70 \, a^{4} b e\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (384 \, b^{5} f x^{9} - 432 \, a b^{4} f x^{7} + 8 \, {\left (80 \, b^{5} d + 63 \, a^{2} b^{3} f\right )} x^{5} + 10 \, {\left (96 \, b^{5} c - 80 \, a b^{4} d - 63 \, a^{3} b^{2} f\right )} x^{3} - 15 \, {\left (96 \, a b^{4} c - 80 \, a^{2} b^{3} d - 63 \, a^{4} b f\right )} x + 10 \, {\left (48 \, b^{5} x^{7} - 56 \, a b^{4} x^{5} + 70 \, a^{2} b^{3} x^{3} - 105 \, a^{3} b^{2} x\right )} e\right )} \sqrt {b x^{2} + a}}{3840 \, b^{6}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.87, size = 224, normalized size = 0.91 \begin {gather*} \frac {1}{3840} \, {\left (2 \, {\left (4 \, {\left (6 \, {\left (\frac {8 \, f x^{2}}{b} - \frac {9 \, a b^{7} f - 10 \, b^{8} e}{b^{9}}\right )} x^{2} + \frac {80 \, b^{8} d + 63 \, a^{2} b^{6} f - 70 \, a b^{7} e}{b^{9}}\right )} x^{2} + \frac {5 \, {\left (96 \, b^{8} c - 80 \, a b^{7} d - 63 \, a^{3} b^{5} f + 70 \, a^{2} b^{6} e\right )}}{b^{9}}\right )} x^{2} - \frac {15 \, {\left (96 \, a b^{7} c - 80 \, a^{2} b^{6} d - 63 \, a^{4} b^{4} f + 70 \, a^{3} b^{5} e\right )}}{b^{9}}\right )} \sqrt {b x^{2} + a} x - \frac {{\left (96 \, a^{2} b^{3} c - 80 \, a^{3} b^{2} d - 63 \, a^{5} f + 70 \, a^{4} b e\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{256 \, b^{\frac {11}{2}}} \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 {x^4\,\left (f\,x^6+e\,x^4+d\,x^2+c\right )}{\sqrt {b\,x^2+a}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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