Optimal. Leaf size=303 \[ -\frac {2 \sqrt {a+b x^2}}{5 c (c x)^{5/2}}-\frac {4 b \sqrt {a+b x^2}}{5 a c^3 \sqrt {c x}}+\frac {4 b^{3/2} \sqrt {c x} \sqrt {a+b x^2}}{5 a c^4 \left (\sqrt {a}+\sqrt {b} x\right )}-\frac {4 b^{5/4} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )|\frac {1}{2}\right )}{5 a^{3/4} c^{7/2} \sqrt {a+b x^2}}+\frac {2 b^{5/4} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )|\frac {1}{2}\right )}{5 a^{3/4} c^{7/2} \sqrt {a+b x^2}} \]
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Rubi [A]
time = 0.16, antiderivative size = 303, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {283, 331, 335,
311, 226, 1210} \begin {gather*} \frac {2 b^{5/4} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )|\frac {1}{2}\right )}{5 a^{3/4} c^{7/2} \sqrt {a+b x^2}}-\frac {4 b^{5/4} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )|\frac {1}{2}\right )}{5 a^{3/4} c^{7/2} \sqrt {a+b x^2}}+\frac {4 b^{3/2} \sqrt {c x} \sqrt {a+b x^2}}{5 a c^4 \left (\sqrt {a}+\sqrt {b} x\right )}-\frac {4 b \sqrt {a+b x^2}}{5 a c^3 \sqrt {c x}}-\frac {2 \sqrt {a+b x^2}}{5 c (c x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 226
Rule 283
Rule 311
Rule 331
Rule 335
Rule 1210
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x^2}}{(c x)^{7/2}} \, dx &=-\frac {2 \sqrt {a+b x^2}}{5 c (c x)^{5/2}}+\frac {(2 b) \int \frac {1}{(c x)^{3/2} \sqrt {a+b x^2}} \, dx}{5 c^2}\\ &=-\frac {2 \sqrt {a+b x^2}}{5 c (c x)^{5/2}}-\frac {4 b \sqrt {a+b x^2}}{5 a c^3 \sqrt {c x}}+\frac {\left (2 b^2\right ) \int \frac {\sqrt {c x}}{\sqrt {a+b x^2}} \, dx}{5 a c^4}\\ &=-\frac {2 \sqrt {a+b x^2}}{5 c (c x)^{5/2}}-\frac {4 b \sqrt {a+b x^2}}{5 a c^3 \sqrt {c x}}+\frac {\left (4 b^2\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {a+\frac {b x^4}{c^2}}} \, dx,x,\sqrt {c x}\right )}{5 a c^5}\\ &=-\frac {2 \sqrt {a+b x^2}}{5 c (c x)^{5/2}}-\frac {4 b \sqrt {a+b x^2}}{5 a c^3 \sqrt {c x}}+\frac {\left (4 b^{3/2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+\frac {b x^4}{c^2}}} \, dx,x,\sqrt {c x}\right )}{5 \sqrt {a} c^4}-\frac {\left (4 b^{3/2}\right ) \text {Subst}\left (\int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a} c}}{\sqrt {a+\frac {b x^4}{c^2}}} \, dx,x,\sqrt {c x}\right )}{5 \sqrt {a} c^4}\\ &=-\frac {2 \sqrt {a+b x^2}}{5 c (c x)^{5/2}}-\frac {4 b \sqrt {a+b x^2}}{5 a c^3 \sqrt {c x}}+\frac {4 b^{3/2} \sqrt {c x} \sqrt {a+b x^2}}{5 a c^4 \left (\sqrt {a}+\sqrt {b} x\right )}-\frac {4 b^{5/4} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )|\frac {1}{2}\right )}{5 a^{3/4} c^{7/2} \sqrt {a+b x^2}}+\frac {2 b^{5/4} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )|\frac {1}{2}\right )}{5 a^{3/4} c^{7/2} \sqrt {a+b x^2}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.02, size = 56, normalized size = 0.18 \begin {gather*} -\frac {2 x \sqrt {a+b x^2} \, _2F_1\left (-\frac {5}{4},-\frac {1}{2};-\frac {1}{4};-\frac {b x^2}{a}\right )}{5 (c x)^{7/2} \sqrt {1+\frac {b x^2}{a}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 219, normalized size = 0.72
method | result | size |
default | \(\frac {\frac {4 \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \EllipticE \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a b \,x^{2}}{5}-\frac {2 \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a b \,x^{2}}{5}-\frac {4 b^{2} x^{4}}{5}-\frac {6 a b \,x^{2}}{5}-\frac {2 a^{2}}{5}}{x^{2} \sqrt {b \,x^{2}+a}\, c^{3} \sqrt {c x}\, a}\) | \(219\) |
risch | \(-\frac {2 \sqrt {b \,x^{2}+a}\, \left (2 b \,x^{2}+a \right )}{5 x^{2} a \,c^{3} \sqrt {c x}}+\frac {2 b \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right ) \sqrt {c x \left (b \,x^{2}+a \right )}}{5 a \sqrt {b c \,x^{3}+a c x}\, c^{3} \sqrt {c x}\, \sqrt {b \,x^{2}+a}}\) | \(225\) |
elliptic | \(\frac {\sqrt {c x \left (b \,x^{2}+a \right )}\, \left (-\frac {2 \sqrt {b c \,x^{3}+a c x}}{5 c^{4} x^{3}}-\frac {4 \left (c \,x^{2} b +a c \right ) b}{5 a \,c^{4} \sqrt {x \left (c \,x^{2} b +a c \right )}}+\frac {2 b \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{5 a \,c^{3} \sqrt {b c \,x^{3}+a c x}}\right )}{\sqrt {c x}\, \sqrt {b \,x^{2}+a}}\) | \(247\) |
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 [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.27, size = 63, normalized size = 0.21 \begin {gather*} -\frac {2 \, {\left (2 \, \sqrt {b c} b x^{3} {\rm weierstrassZeta}\left (-\frac {4 \, a}{b}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right )\right ) + {\left (2 \, b x^{2} + a\right )} \sqrt {b x^{2} + a} \sqrt {c x}\right )}}{5 \, a c^{4} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 5.36, size = 53, normalized size = 0.17 \begin {gather*} \frac {\sqrt {a} \Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, - \frac {1}{2} \\ - \frac {1}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 c^{\frac {7}{2}} x^{\frac {5}{2}} \Gamma \left (- \frac {1}{4}\right )} \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 {b\,x^2+a}}{{\left (c\,x\right )}^{7/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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