Optimal. Leaf size=403 \[ \frac {(b c-4 a d) x \left (a+b x^2\right )}{15 b^2 d \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}+\frac {x^3 \left (a+b x^2\right )}{5 b \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}-\frac {\left (2 b^2 c^2+3 a b c d-8 a^2 d^2\right ) x \left (a+b x^2\right )}{15 b^3 d \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}+\frac {\sqrt {c} \left (2 b^2 c^2+3 a b c d-8 a^2 d^2\right ) \left (a+b x^2\right ) E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{15 b^3 d^{3/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}-\frac {c^{3/2} (b c-4 a d) \left (a+b x^2\right ) F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{15 b^2 d^{3/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )} \]
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Rubi [A]
time = 0.27, antiderivative size = 403, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {1986, 489, 596,
545, 429, 506, 422} \begin {gather*} \frac {\sqrt {c} \left (a+b x^2\right ) \left (-8 a^2 d^2+3 a b c d+2 b^2 c^2\right ) E\left (\text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{15 b^3 d^{3/2} \left (c+d x^2\right ) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}-\frac {x \left (a+b x^2\right ) \left (-8 a^2 d^2+3 a b c d+2 b^2 c^2\right )}{15 b^3 d \left (c+d x^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}-\frac {c^{3/2} \left (a+b x^2\right ) (b c-4 a d) F\left (\text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{15 b^2 d^{3/2} \left (c+d x^2\right ) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}+\frac {x \left (a+b x^2\right ) (b c-4 a d)}{15 b^2 d \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}+\frac {x^3 \left (a+b x^2\right )}{5 b \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 422
Rule 429
Rule 489
Rule 506
Rule 545
Rule 596
Rule 1986
Rubi steps
\begin {align*} \int \frac {x^4}{\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx &=\frac {\sqrt {a+b x^2} \int \frac {x^4 \sqrt {c+d x^2}}{\sqrt {a+b x^2}} \, dx}{\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}}\\ &=\frac {x^3 \left (a+b x^2\right )}{5 b \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}-\frac {\sqrt {a+b x^2} \int \frac {x^2 \left (3 a c+(-b c+4 a d) x^2\right )}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{5 b \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}}\\ &=\frac {(b c-4 a d) x \left (a+b x^2\right )}{15 b^2 d \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}+\frac {x^3 \left (a+b x^2\right )}{5 b \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}+\frac {\sqrt {a+b x^2} \int \frac {-a c (b c-4 a d)+\left (-2 b^2 c^2-3 a b c d+8 a^2 d^2\right ) x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{15 b^2 d \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}}\\ &=\frac {(b c-4 a d) x \left (a+b x^2\right )}{15 b^2 d \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}+\frac {x^3 \left (a+b x^2\right )}{5 b \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}-\frac {\left (a c (b c-4 a d) \sqrt {a+b x^2}\right ) \int \frac {1}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{15 b^2 d \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}}+\frac {\left (\left (-2 b^2 c^2-3 a b c d+8 a^2 d^2\right ) \sqrt {a+b x^2}\right ) \int \frac {x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{15 b^2 d \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}}\\ &=\frac {(b c-4 a d) x \left (a+b x^2\right )}{15 b^2 d \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}+\frac {x^3 \left (a+b x^2\right )}{5 b \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}-\frac {\left (2 b^2 c^2+3 a b c d-8 a^2 d^2\right ) x \left (a+b x^2\right )}{15 b^3 d \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}-\frac {c^{3/2} (b c-4 a d) \left (a+b x^2\right ) F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{15 b^2 d^{3/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}-\frac {\left (c \left (-2 b^2 c^2-3 a b c d+8 a^2 d^2\right ) \sqrt {a+b x^2}\right ) \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{15 b^3 d \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}}\\ &=\frac {(b c-4 a d) x \left (a+b x^2\right )}{15 b^2 d \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}+\frac {x^3 \left (a+b x^2\right )}{5 b \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}-\frac {\left (2 b^2 c^2+3 a b c d-8 a^2 d^2\right ) x \left (a+b x^2\right )}{15 b^3 d \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}+\frac {\sqrt {c} \left (2 b^2 c^2+3 a b c d-8 a^2 d^2\right ) \left (a+b x^2\right ) E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{15 b^3 d^{3/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}-\frac {c^{3/2} (b c-4 a d) \left (a+b x^2\right ) F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{15 b^2 d^{3/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 1.59, size = 258, normalized size = 0.64 \begin {gather*} \frac {-\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (c+d x^2\right ) \left (4 a d-b \left (c+3 d x^2\right )\right )-i c \left (-2 b^2 c^2-3 a b c d+8 a^2 d^2\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \sinh ^{-1}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+2 i c \left (-b^2 c^2-a b c d+2 a^2 d^2\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} F\left (i \sinh ^{-1}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )}{15 a^2 \left (\frac {b}{a}\right )^{5/2} d^2 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 554, normalized size = 1.37
method | result | size |
risch | \(-\frac {x \left (-3 b d \,x^{2}+4 a d -b c \right ) \left (b \,x^{2}+a \right )}{15 d \,b^{2} \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}}+\frac {\left (-\frac {2 \left (8 a^{2} d^{2}-3 a b c d -2 b^{2} c^{2}\right ) a c e \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\EllipticF \left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d e +b c e}{c e b}}\right )-\EllipticE \left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d e +b c e}{c e b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {d e b \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}\, \left (a d e +b c e +e \left (a d -b c \right )\right )}+\frac {4 a^{2} c d \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \EllipticF \left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d e +b c e}{c e b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {d e b \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}-\frac {a b \,c^{2} \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \EllipticF \left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d e +b c e}{c e b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {d e b \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}\right ) \sqrt {\left (d \,x^{2}+c \right ) e \left (b \,x^{2}+a \right )}}{15 d \,b^{2} \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right )}\) | \(492\) |
default | \(-\frac {\left (b \,x^{2}+a \right ) \left (-3 \sqrt {-\frac {b}{a}}\, b^{2} d^{3} x^{7}+\sqrt {-\frac {b}{a}}\, a b \,d^{3} x^{5}-4 \sqrt {-\frac {b}{a}}\, b^{2} c \,d^{2} x^{5}+4 \sqrt {-\frac {b}{a}}\, a^{2} d^{3} x^{3}-\sqrt {-\frac {b}{a}}\, b^{2} c^{2} d \,x^{3}+4 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \EllipticF \left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} c \,d^{2}-2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \EllipticF \left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b \,c^{2} d -2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \EllipticF \left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{3}-8 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \EllipticE \left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} c \,d^{2}+3 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \EllipticE \left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b \,c^{2} d +2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \EllipticE \left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{3}+4 \sqrt {-\frac {b}{a}}\, a^{2} c \,d^{2} x -\sqrt {-\frac {b}{a}}\, a b \,c^{2} d x \right )}{15 \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, b^{2} d^{2} \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}}\) | \(554\) |
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(-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 [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 {x^4}{\sqrt {\frac {e\,\left (b\,x^2+a\right )}{d\,x^2+c}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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