Optimal. Leaf size=466 \[ \frac {\left (8 b^2+13 a b c+3 a^2 c^2\right ) d^3 x \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{15 c^3 (b+a c)^2}-\frac {\left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{5 c x^5}+\frac {(4 b+3 a c) d \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{15 c^2 (b+a c) x^3}-\frac {\left (8 b^2+13 a b c+3 a^2 c^2\right ) d^2 \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{15 c^3 (b+a c)^2 x}-\frac {\left (8 b^2+13 a b c+3 a^2 c^2\right ) d^{5/2} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{15 c^{5/2} (b+a c)^2 \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}}+\frac {a (4 b+3 a c) d^{5/2} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{15 c^{3/2} (b+a c)^2 \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}} \]
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Rubi [A]
time = 0.41, antiderivative size = 466, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {1985, 1986,
486, 597, 545, 429, 506, 422} \begin {gather*} -\frac {d^{5/2} \left (3 a^2 c^2+13 a b c+8 b^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} E\left (\text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{15 c^{5/2} (a c+b)^2 \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}+\frac {d^3 x \left (3 a^2 c^2+13 a b c+8 b^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{15 c^3 (a c+b)^2}-\frac {d^2 \left (3 a^2 c^2+13 a b c+8 b^2\right ) \left (c+d x^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{15 c^3 x (a c+b)^2}+\frac {a d^{5/2} (3 a c+4 b) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} F\left (\text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{15 c^{3/2} (a c+b)^2 \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}+\frac {d (3 a c+4 b) \left (c+d x^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{15 c^2 x^3 (a c+b)}-\frac {\left (c+d x^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{5 c x^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 422
Rule 429
Rule 486
Rule 506
Rule 545
Rule 597
Rule 1985
Rule 1986
Rubi steps
\begin {align*} \int \frac {\sqrt {a+\frac {b}{c+d x^2}}}{x^6} \, dx &=\frac {\left (\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \int \frac {\sqrt {b+a \left (c+d x^2\right )}}{x^6 \sqrt {c+d x^2}} \, dx}{\sqrt {b+a \left (c+d x^2\right )}}\\ &=\frac {\left (\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \int \frac {\sqrt {b+a c+a d x^2}}{x^6 \sqrt {c+d x^2}} \, dx}{\sqrt {b+a \left (c+d x^2\right )}}\\ &=-\frac {\left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{5 c x^5 \sqrt {b+a \left (c+d x^2\right )}}+\frac {\left (\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \int \frac {-(4 b+3 a c) d-3 a d^2 x^2}{x^4 \sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{5 c \sqrt {b+a \left (c+d x^2\right )}}\\ &=-\frac {\left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{5 c x^5 \sqrt {b+a \left (c+d x^2\right )}}+\frac {(4 b+3 a c) d \left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{15 c^2 (b+a c) x^3 \sqrt {b+a \left (c+d x^2\right )}}-\frac {\left (\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \int \frac {-\left (8 b^2+13 a b c+3 a^2 c^2\right ) d^2-a (4 b+3 a c) d^3 x^2}{x^2 \sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{15 c^2 (b+a c) \sqrt {b+a \left (c+d x^2\right )}}\\ &=-\frac {\left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{5 c x^5 \sqrt {b+a \left (c+d x^2\right )}}+\frac {(4 b+3 a c) d \left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{15 c^2 (b+a c) x^3 \sqrt {b+a \left (c+d x^2\right )}}-\frac {\left (8 b^2+13 a b c+3 a^2 c^2\right ) d^2 \left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{15 c^3 (b+a c)^2 x \sqrt {b+a \left (c+d x^2\right )}}+\frac {\left (\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \int \frac {a c (b+a c) (4 b+3 a c) d^3+a \left (8 b^2+13 a b c+3 a^2 c^2\right ) d^4 x^2}{\sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{15 c^3 (b+a c)^2 \sqrt {b+a \left (c+d x^2\right )}}\\ &=-\frac {\left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{5 c x^5 \sqrt {b+a \left (c+d x^2\right )}}+\frac {(4 b+3 a c) d \left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{15 c^2 (b+a c) x^3 \sqrt {b+a \left (c+d x^2\right )}}-\frac {\left (8 b^2+13 a b c+3 a^2 c^2\right ) d^2 \left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{15 c^3 (b+a c)^2 x \sqrt {b+a \left (c+d x^2\right )}}+\frac {\left (a (4 b+3 a c) d^3 \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \int \frac {1}{\sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{15 c^2 (b+a c) \sqrt {b+a \left (c+d x^2\right )}}+\frac {\left (a \left (8 b^2+13 a b c+3 a^2 c^2\right ) d^4 \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \int \frac {x^2}{\sqrt {c+d x^2} \sqrt {b+a c+a d x^2}} \, dx}{15 c^3 (b+a c)^2 \sqrt {b+a \left (c+d x^2\right )}}\\ &=\frac {\left (8 b^2+13 a b c+3 a^2 c^2\right ) d^3 x \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{15 c^3 (b+a c)^2 \sqrt {b+a \left (c+d x^2\right )}}-\frac {\left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{5 c x^5 \sqrt {b+a \left (c+d x^2\right )}}+\frac {(4 b+3 a c) d \left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{15 c^2 (b+a c) x^3 \sqrt {b+a \left (c+d x^2\right )}}-\frac {\left (8 b^2+13 a b c+3 a^2 c^2\right ) d^2 \left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{15 c^3 (b+a c)^2 x \sqrt {b+a \left (c+d x^2\right )}}+\frac {a (4 b+3 a c) d^{5/2} \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{15 c^{3/2} (b+a c)^2 \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}} \sqrt {b+a \left (c+d x^2\right )}}-\frac {\left (\left (8 b^2+13 a b c+3 a^2 c^2\right ) d^3 \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \int \frac {\sqrt {b+a c+a d x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{15 c^2 (b+a c)^2 \sqrt {b+a \left (c+d x^2\right )}}\\ &=\frac {\left (8 b^2+13 a b c+3 a^2 c^2\right ) d^3 x \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{15 c^3 (b+a c)^2 \sqrt {b+a \left (c+d x^2\right )}}-\frac {\left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{5 c x^5 \sqrt {b+a \left (c+d x^2\right )}}+\frac {(4 b+3 a c) d \left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{15 c^2 (b+a c) x^3 \sqrt {b+a \left (c+d x^2\right )}}-\frac {\left (8 b^2+13 a b c+3 a^2 c^2\right ) d^2 \left (c+d x^2\right ) \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}}}{15 c^3 (b+a c)^2 x \sqrt {b+a \left (c+d x^2\right )}}-\frac {\left (8 b^2+13 a b c+3 a^2 c^2\right ) d^{5/2} \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{15 c^{5/2} (b+a c)^2 \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}} \sqrt {b+a \left (c+d x^2\right )}}+\frac {a (4 b+3 a c) d^{5/2} \sqrt {b+a c+a d x^2} \sqrt {a+\frac {b}{c+d x^2}} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{15 c^{3/2} (b+a c)^2 \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}} \sqrt {b+a \left (c+d x^2\right )}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 10.73, size = 402, normalized size = 0.86 \begin {gather*} -\frac {\sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \left (\sqrt {\frac {a d}{b+a c}} \left (c+d x^2\right ) \left (b^3 \left (3 c^2-4 c d x^2+8 d^2 x^4\right )+3 a^3 c^2 \left (c^3+d^3 x^6\right )+a b^2 \left (9 c^3-8 c^2 d x^2+17 c d^2 x^4+8 d^3 x^6\right )+a^2 b c \left (9 c^3-4 c^2 d x^2+9 c d^2 x^4+13 d^3 x^6\right )\right )+i a c \left (8 b^2+13 a b c+3 a^2 c^2\right ) d^3 x^5 \sqrt {\frac {b+a c+a d x^2}{b+a c}} \sqrt {1+\frac {d x^2}{c}} E\left (i \sinh ^{-1}\left (\sqrt {\frac {a d}{b+a c}} x\right )|1+\frac {b}{a c}\right )-2 i a b c (2 b+3 a c) d^3 x^5 \sqrt {\frac {b+a c+a d x^2}{b+a c}} \sqrt {1+\frac {d x^2}{c}} F\left (i \sinh ^{-1}\left (\sqrt {\frac {a d}{b+a c}} x\right )|1+\frac {b}{a c}\right )\right )}{15 c^3 (b+a c)^2 \sqrt {\frac {a d}{b+a c}} x^5 \left (b+a \left (c+d x^2\right )\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 955, normalized size = 2.05
method | result | size |
risch | \(-\frac {\left (d \,x^{2}+c \right ) \left (3 a^{2} c^{2} d^{2} x^{4}+13 a c \,d^{2} b \,x^{4}-3 a^{2} c^{3} d \,x^{2}+8 b^{2} d^{2} x^{4}-7 a b \,c^{2} d \,x^{2}+3 a^{2} c^{4}-4 b^{2} c d \,x^{2}+6 a b \,c^{3}+3 b^{2} c^{2}\right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}{15 c^{3} x^{5} \left (a c +b \right )^{2}}+\frac {d^{3} a \left (-\frac {2 \left (3 a^{2} c^{2} d +13 a b c d +8 b^{2} d \right ) \left (c^{2} a +b c \right ) \sqrt {1+\frac {a d \,x^{2}}{a c +b}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\EllipticF \left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )-\EllipticE \left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )\right )}{\sqrt {-\frac {a d}{a c +b}}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}\, \left (2 a c d +2 b d \right )}+\frac {3 a^{2} c^{3} \sqrt {1+\frac {a d \,x^{2}}{a c +b}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \EllipticF \left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )}{\sqrt {-\frac {a d}{a c +b}}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}}+\frac {7 a b \,c^{2} \sqrt {1+\frac {a d \,x^{2}}{a c +b}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \EllipticF \left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )}{\sqrt {-\frac {a d}{a c +b}}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}}+\frac {4 b^{2} c \sqrt {1+\frac {a d \,x^{2}}{a c +b}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \EllipticF \left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )}{\sqrt {-\frac {a d}{a c +b}}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}}\right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}}{15 c^{3} \left (a c +b \right )^{2} \left (a d \,x^{2}+a c +b \right )}\) | \(778\) |
default | \(-\frac {\left (3 \sqrt {-\frac {a d}{a c +b}}\, a^{3} c^{2} d^{4} x^{8}+13 \sqrt {-\frac {a d}{a c +b}}\, a^{2} b c \,d^{4} x^{8}-3 \sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \EllipticE \left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right ) a^{3} c^{3} d^{3} x^{5}+3 \sqrt {-\frac {a d}{a c +b}}\, a^{3} c^{3} d^{3} x^{6}+8 \sqrt {-\frac {a d}{a c +b}}\, a \,b^{2} d^{4} x^{8}+6 \sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \EllipticF \left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right ) a^{2} b \,c^{2} d^{3} x^{5}-13 \sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \EllipticE \left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right ) a^{2} b \,c^{2} d^{3} x^{5}+22 \sqrt {-\frac {a d}{a c +b}}\, a^{2} b \,c^{2} d^{3} x^{6}+4 \sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \EllipticF \left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right ) a \,b^{2} c \,d^{3} x^{5}-8 \sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \EllipticE \left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right ) a \,b^{2} c \,d^{3} x^{5}+25 \sqrt {-\frac {a d}{a c +b}}\, a \,b^{2} c \,d^{3} x^{6}+5 \sqrt {-\frac {a d}{a c +b}}\, a^{2} b \,c^{3} d^{2} x^{4}+8 \sqrt {-\frac {a d}{a c +b}}\, b^{3} d^{3} x^{6}+3 \sqrt {-\frac {a d}{a c +b}}\, a^{3} c^{5} d \,x^{2}+9 \sqrt {-\frac {a d}{a c +b}}\, a \,b^{2} c^{2} d^{2} x^{4}+5 \sqrt {-\frac {a d}{a c +b}}\, a^{2} b \,c^{4} d \,x^{2}+4 \sqrt {-\frac {a d}{a c +b}}\, b^{3} c \,d^{2} x^{4}+3 \sqrt {-\frac {a d}{a c +b}}\, a^{3} c^{6}+\sqrt {-\frac {a d}{a c +b}}\, a \,b^{2} c^{3} d \,x^{2}+9 \sqrt {-\frac {a d}{a c +b}}\, a^{2} b \,c^{5}-\sqrt {-\frac {a d}{a c +b}}\, b^{3} c^{2} d \,x^{2}+9 \sqrt {-\frac {a d}{a c +b}}\, a \,b^{2} c^{4}+3 \sqrt {-\frac {a d}{a c +b}}\, b^{3} c^{3}\right ) \left (d \,x^{2}+c \right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}{15 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}\, \sqrt {-\frac {a d}{a c +b}}\, \left (a c +b \right )^{2} x^{5} c^{3} \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}}\) | \(955\) |
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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {\frac {a c + a d x^{2} + b}{c + d x^{2}}}}{x^{6}}\, dx \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 {\sqrt {a+\frac {b}{d\,x^2+c}}}{x^6} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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