Optimal. Leaf size=311 \[ \frac {2 d (2 b c-a d) x \sqrt {a+b x^2}}{3 c^2 \sqrt {c+d x^2}}-\frac {a \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 c x^3}-\frac {2 (2 b c-a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 c^2 x}-\frac {2 \sqrt {d} (2 b c-a d) \sqrt {a+b x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 c^{3/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {b (3 b c-a d) \sqrt {a+b x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 a \sqrt {c} \sqrt {d} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \]
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Rubi [A]
time = 0.20, antiderivative size = 311, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {485, 597, 545,
429, 506, 422} \begin {gather*} -\frac {2 \sqrt {d} \sqrt {a+b x^2} (2 b c-a d) E\left (\text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 c^{3/2} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {b \sqrt {a+b x^2} (3 b c-a d) F\left (\text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 a \sqrt {c} \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {2 \sqrt {a+b x^2} \sqrt {c+d x^2} (2 b c-a d)}{3 c^2 x}+\frac {2 d x \sqrt {a+b x^2} (2 b c-a d)}{3 c^2 \sqrt {c+d x^2}}-\frac {a \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 c x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 422
Rule 429
Rule 485
Rule 506
Rule 545
Rule 597
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^{3/2}}{x^4 \sqrt {c+d x^2}} \, dx &=-\frac {a \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 c x^3}+\frac {\int \frac {2 a (2 b c-a d)+b (3 b c-a d) x^2}{x^2 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{3 c}\\ &=-\frac {a \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 c x^3}-\frac {2 (2 b c-a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 c^2 x}-\frac {\int \frac {-a b c (3 b c-a d)-2 a b d (2 b c-a d) x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{3 a c^2}\\ &=-\frac {a \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 c x^3}-\frac {2 (2 b c-a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 c^2 x}+\frac {(2 b d (2 b c-a d)) \int \frac {x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{3 c^2}+\frac {(b (3 b c-a d)) \int \frac {1}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{3 c}\\ &=\frac {2 d (2 b c-a d) x \sqrt {a+b x^2}}{3 c^2 \sqrt {c+d x^2}}-\frac {a \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 c x^3}-\frac {2 (2 b c-a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 c^2 x}+\frac {b (3 b c-a d) \sqrt {a+b x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 a \sqrt {c} \sqrt {d} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {(2 d (2 b c-a d)) \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{3 c}\\ &=\frac {2 d (2 b c-a d) x \sqrt {a+b x^2}}{3 c^2 \sqrt {c+d x^2}}-\frac {a \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 c x^3}-\frac {2 (2 b c-a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 c^2 x}-\frac {2 \sqrt {d} (2 b c-a d) \sqrt {a+b x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 c^{3/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {b (3 b c-a d) \sqrt {a+b x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 a \sqrt {c} \sqrt {d} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 2.56, size = 227, normalized size = 0.73 \begin {gather*} \frac {\sqrt {\frac {b}{a}} \left (a+b x^2\right ) \left (c+d x^2\right ) \left (-a c-4 b c x^2+2 a d x^2\right )+2 i b c (-2 b c+a d) x^3 \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 )-i b c (-b c+a d) x^3 \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 )}{3 \sqrt {\frac {b}{a}} c^2 x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 433, normalized size = 1.39
method | result | size |
elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \left (-\frac {a \sqrt {b d \,x^{4}+a d \,x^{2}+c \,x^{2} b +a c}}{3 c \,x^{3}}+\frac {2 \left (a d -2 b c \right ) \sqrt {b d \,x^{4}+a d \,x^{2}+c \,x^{2} b +a c}}{3 c^{2} x}+\frac {\left (b^{2}-\frac {a b d}{3 c}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \EllipticF \left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c \,x^{2} b +a c}}+\frac {2 b \left (a d -2 b c \right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\EllipticF \left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\EllipticE \left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{3 c \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c \,x^{2} b +a c}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) | \(340\) |
risch | \(-\frac {\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}\, \left (-2 a d \,x^{2}+4 c \,x^{2} b +a c \right )}{3 c^{2} x^{3}}-\frac {b \left (-\frac {\left (2 a \,d^{2}-4 b c d \right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\EllipticF \left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\EllipticE \left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c \,x^{2} b +a c}\, d}+\frac {a c d \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \EllipticF \left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c \,x^{2} b +a c}}-\frac {3 b \,c^{2} \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \EllipticF \left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c \,x^{2} b +a c}}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}}{3 c^{2} \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) | \(403\) |
default | \(\frac {\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}\, \left (2 \sqrt {-\frac {b}{a}}\, a b \,d^{2} x^{6}-4 \sqrt {-\frac {b}{a}}\, b^{2} c d \,x^{6}+b d \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \EllipticF \left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) x^{3} a c -\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \EllipticF \left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{2} x^{3}-2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \EllipticE \left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b c d \,x^{3}+4 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \EllipticE \left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{2} x^{3}+2 \sqrt {-\frac {b}{a}}\, a^{2} d^{2} x^{4}-3 \sqrt {-\frac {b}{a}}\, a b c d \,x^{4}-4 \sqrt {-\frac {b}{a}}\, b^{2} c^{2} x^{4}+\sqrt {-\frac {b}{a}}\, a^{2} c d \,x^{2}-5 \sqrt {-\frac {b}{a}}\, a b \,c^{2} x^{2}-\sqrt {-\frac {b}{a}}\, a^{2} c^{2}\right )}{3 \left (b d \,x^{4}+a d \,x^{2}+c \,x^{2} b +a c \right ) c^{2} \sqrt {-\frac {b}{a}}\, x^{3}}\) | \(433\) |
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 {\left (a + b x^{2}\right )^{\frac {3}{2}}}{x^{4} \sqrt {c + d x^{2}}}\, 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 {{\left (b\,x^2+a\right )}^{3/2}}{x^4\,\sqrt {d\,x^2+c}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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