Optimal. Leaf size=229 \[ \frac {(b c-a d) x \sqrt {c+d x^2}}{3 a b \left (a+b x^2\right )^{3/2}}+\frac {2 (b c+a d) \sqrt {c+d x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{3 a^{3/2} b^{3/2} \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {c^{3/2} \sqrt {d} \sqrt {a+b x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 a^2 b \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \]
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Rubi [A]
time = 0.08, antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {424, 539, 429,
422} \begin {gather*} \frac {2 \sqrt {c+d x^2} (a d+b c) E\left (\text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{3 a^{3/2} b^{3/2} \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {c^{3/2} \sqrt {d} \sqrt {a+b x^2} F\left (\text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 a^2 b \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {x \sqrt {c+d x^2} (b c-a d)}{3 a b \left (a+b x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 422
Rule 424
Rule 429
Rule 539
Rubi steps
\begin {align*} \int \frac {\left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^{5/2}} \, dx &=\frac {(b c-a d) x \sqrt {c+d x^2}}{3 a b \left (a+b x^2\right )^{3/2}}+\frac {\int \frac {c (2 b c+a d)+d (b c+2 a d) x^2}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx}{3 a b}\\ &=\frac {(b c-a d) x \sqrt {c+d x^2}}{3 a b \left (a+b x^2\right )^{3/2}}-\frac {(c d) \int \frac {1}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{3 a b}+\frac {(2 (b c+a d)) \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^{3/2}} \, dx}{3 a b}\\ &=\frac {(b c-a d) x \sqrt {c+d x^2}}{3 a b \left (a+b x^2\right )^{3/2}}+\frac {2 (b c+a d) \sqrt {c+d x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{3 a^{3/2} b^{3/2} \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {c^{3/2} \sqrt {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^2 b \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 = 4.73, size = 232, normalized size = 1.01 \begin {gather*} \frac {\sqrt {\frac {b}{a}} x \left (c+d x^2\right ) \left (a^2 d+2 b^2 c x^2+a b \left (3 c+2 d x^2\right )\right )+2 i c (b c+a d) \left (a+b x^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 )-i c (2 b c+a d) \left (a+b x^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 )}{3 a^3 \left (\frac {b}{a}\right )^{3/2} \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(606\) vs.
\(2(269)=538\).
time = 0.09, size = 607, normalized size = 2.65
method | result | size |
elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \left (-\frac {\left (a d -b c \right ) x \sqrt {b d \,x^{4}+a d \,x^{2}+c \,x^{2} b +a c}}{3 a \,b^{3} \left (x^{2}+\frac {a}{b}\right )^{2}}+\frac {2 \left (b d \,x^{2}+b c \right ) x \left (a d +b c \right )}{3 b^{2} a^{2} \sqrt {\left (x^{2}+\frac {a}{b}\right ) \left (b d \,x^{2}+b c \right )}}+\frac {\left (\frac {d^{2}}{b^{2}}-\frac {d \left (a d -b c \right )}{3 b^{2} a}-\frac {2 \left (a d +b c \right ) \left (a d -b c \right )}{3 b^{2} a^{2}}-\frac {2 c \left (a d +b c \right )}{3 b \,a^{2}}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \EllipticF \left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c \,x^{2} b +a c}}+\frac {2 \left (a d +b c \right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\EllipticF \left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\EllipticE \left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{3 b \,a^{2} \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c \,x^{2} b +a c}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) | \(421\) |
default | \(\frac {2 \sqrt {-\frac {b}{a}}\, a b \,d^{2} x^{5}+2 \sqrt {-\frac {b}{a}}\, b^{2} c d \,x^{5}+\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \EllipticF \left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b c d \,x^{2}+2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \EllipticF \left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{2} x^{2}-2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \EllipticE \left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b c d \,x^{2}-2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \EllipticE \left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{2} x^{2}+\sqrt {-\frac {b}{a}}\, a^{2} d^{2} x^{3}+5 \sqrt {-\frac {b}{a}}\, a b c d \,x^{3}+2 \sqrt {-\frac {b}{a}}\, b^{2} c^{2} x^{3}+\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \EllipticF \left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} c d +2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \EllipticF \left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b \,c^{2}-2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \EllipticE \left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} c d -2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \EllipticE \left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b \,c^{2}+\sqrt {-\frac {b}{a}}\, a^{2} c d x +3 \sqrt {-\frac {b}{a}}\, a b \,c^{2} x}{3 \sqrt {d \,x^{2}+c}\, a^{2} \sqrt {-\frac {b}{a}}\, \left (b \,x^{2}+a \right )^{\frac {3}{2}} b}\) | \(607\) |
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 (c + d x^{2}\right )^{\frac {3}{2}}}{\left (a + b x^{2}\right )^{\frac {5}{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 (d\,x^2+c\right )}^{3/2}}{{\left (b\,x^2+a\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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