Optimal. Leaf size=217 \[ -\frac {2 \left (77 b^2 c^2-22 a b c d+5 a^2 d^2\right ) \sqrt {c+d x^2}}{231 c^2 x^{3/2}}-\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{11 c x^{11/2}}-\frac {2 a (22 b c-5 a d) \left (c+d x^2\right )^{3/2}}{77 c^2 x^{7/2}}+\frac {2 d^{3/4} \left (77 b^2 c^2-22 a b c d+5 a^2 d^2\right ) \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{231 c^{9/4} \sqrt {c+d x^2}} \]
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Rubi [A]
time = 0.13, antiderivative size = 213, normalized size of antiderivative = 0.98, number of steps
used = 5, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {473, 464, 283,
335, 226} \begin {gather*} \frac {2 d^{3/4} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (5 a^2 d^2-22 a b c d+77 b^2 c^2\right ) F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{231 c^{9/4} \sqrt {c+d x^2}}-\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{11 c x^{11/2}}-\frac {2 \sqrt {c+d x^2} \left (77 b^2-\frac {a d (22 b c-5 a d)}{c^2}\right )}{231 x^{3/2}}-\frac {2 a \left (c+d x^2\right )^{3/2} (22 b c-5 a d)}{77 c^2 x^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 226
Rule 283
Rule 335
Rule 464
Rule 473
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^{13/2}} \, dx &=-\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{11 c x^{11/2}}+\frac {2 \int \frac {\left (\frac {1}{2} a (22 b c-5 a d)+\frac {11}{2} b^2 c x^2\right ) \sqrt {c+d x^2}}{x^{9/2}} \, dx}{11 c}\\ &=-\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{11 c x^{11/2}}-\frac {2 a (22 b c-5 a d) \left (c+d x^2\right )^{3/2}}{77 c^2 x^{7/2}}-\frac {1}{77} \left (-77 b^2+\frac {a d (22 b c-5 a d)}{c^2}\right ) \int \frac {\sqrt {c+d x^2}}{x^{5/2}} \, dx\\ &=-\frac {2 \left (77 b^2-\frac {a d (22 b c-5 a d)}{c^2}\right ) \sqrt {c+d x^2}}{231 x^{3/2}}-\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{11 c x^{11/2}}-\frac {2 a (22 b c-5 a d) \left (c+d x^2\right )^{3/2}}{77 c^2 x^{7/2}}+\frac {1}{231} \left (2 d \left (77 b^2-\frac {a d (22 b c-5 a d)}{c^2}\right )\right ) \int \frac {1}{\sqrt {x} \sqrt {c+d x^2}} \, dx\\ &=-\frac {2 \left (77 b^2-\frac {a d (22 b c-5 a d)}{c^2}\right ) \sqrt {c+d x^2}}{231 x^{3/2}}-\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{11 c x^{11/2}}-\frac {2 a (22 b c-5 a d) \left (c+d x^2\right )^{3/2}}{77 c^2 x^{7/2}}+\frac {1}{231} \left (4 d \left (77 b^2-\frac {a d (22 b c-5 a d)}{c^2}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c+d x^4}} \, dx,x,\sqrt {x}\right )\\ &=-\frac {2 \left (77 b^2-\frac {a d (22 b c-5 a d)}{c^2}\right ) \sqrt {c+d x^2}}{231 x^{3/2}}-\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{11 c x^{11/2}}-\frac {2 a (22 b c-5 a d) \left (c+d x^2\right )^{3/2}}{77 c^2 x^{7/2}}+\frac {2 d^{3/4} \left (77 b^2 c^2-22 a b c d+5 a^2 d^2\right ) \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{231 c^{9/4} \sqrt {c+d x^2}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 10.14, size = 187, normalized size = 0.86 \begin {gather*} -\frac {2 \sqrt {c+d x^2} \left (77 b^2 c^2 x^4+22 a b c x^2 \left (3 c+2 d x^2\right )+a^2 \left (21 c^2+6 c d x^2-10 d^2 x^4\right )\right )}{231 c^2 x^{11/2}}+\frac {4 i d \left (77 b^2 c^2-22 a b c d+5 a^2 d^2\right ) \sqrt {1+\frac {c}{d x^2}} x F\left (\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i \sqrt {c}}{\sqrt {d}}}}{\sqrt {x}}\right )\right |-1\right )}{231 c^2 \sqrt {\frac {i \sqrt {c}}{\sqrt {d}}} \sqrt {c+d x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 403, normalized size = 1.86
method | result | size |
risch | \(-\frac {2 \sqrt {d \,x^{2}+c}\, \left (-10 a^{2} d^{2} x^{4}+44 a b c d \,x^{4}+77 b^{2} c^{2} x^{4}+6 a^{2} c d \,x^{2}+66 a b \,c^{2} x^{2}+21 a^{2} c^{2}\right )}{231 x^{\frac {11}{2}} c^{2}}+\frac {2 \left (5 a^{2} d^{2}-22 a b c d +77 b^{2} c^{2}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {x \left (d \,x^{2}+c \right )}}{231 c^{2} \sqrt {d \,x^{3}+c x}\, \sqrt {x}\, \sqrt {d \,x^{2}+c}}\) | \(233\) |
elliptic | \(\frac {\sqrt {x \left (d \,x^{2}+c \right )}\, \left (-\frac {2 a^{2} \sqrt {d \,x^{3}+c x}}{11 x^{6}}-\frac {4 a \left (a d +11 b c \right ) \sqrt {d \,x^{3}+c x}}{77 c \,x^{4}}+\frac {2 \left (10 a^{2} d^{2}-44 a b c d -77 b^{2} c^{2}\right ) \sqrt {d \,x^{3}+c x}}{231 c^{2} x^{2}}+\frac {\left (b^{2} d +\frac {d \left (10 a^{2} d^{2}-44 a b c d -77 b^{2} c^{2}\right )}{231 c^{2}}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d \sqrt {d \,x^{3}+c x}}\right )}{\sqrt {x}\, \sqrt {d \,x^{2}+c}}\) | \(257\) |
default | \(\frac {\frac {10 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \EllipticF \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-c d}\, a^{2} d^{2} x^{5}}{231}-\frac {4 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \EllipticF \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-c d}\, a b c d \,x^{5}}{21}+\frac {2 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \EllipticF \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-c d}\, b^{2} c^{2} x^{5}}{3}+\frac {20 a^{2} d^{3} x^{6}}{231}-\frac {8 a b c \,d^{2} x^{6}}{21}-\frac {2 b^{2} c^{2} d \,x^{6}}{3}+\frac {8 a^{2} c \,d^{2} x^{4}}{231}-\frac {20 a b \,c^{2} d \,x^{4}}{21}-\frac {2 b^{2} c^{3} x^{4}}{3}-\frac {18 a^{2} c^{2} d \,x^{2}}{77}-\frac {4 a b \,c^{3} x^{2}}{7}-\frac {2 a^{2} c^{3}}{11}}{\sqrt {d \,x^{2}+c}\, x^{\frac {11}{2}} c^{2}}\) | \(403\) |
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 = 0.31, size = 118, normalized size = 0.54 \begin {gather*} \frac {2 \, {\left (2 \, {\left (77 \, b^{2} c^{2} - 22 \, a b c d + 5 \, a^{2} d^{2}\right )} \sqrt {d} x^{6} {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right ) - {\left ({\left (77 \, b^{2} c^{2} + 44 \, a b c d - 10 \, a^{2} d^{2}\right )} x^{4} + 21 \, a^{2} c^{2} + 6 \, {\left (11 \, a b c^{2} + a^{2} c d\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {x}\right )}}{231 \, c^{2} x^{6}} \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 = 92.68, size = 151, normalized size = 0.70 \begin {gather*} \frac {a^{2} \sqrt {c} \Gamma \left (- \frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {11}{4}, - \frac {1}{2} \\ - \frac {7}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 x^{\frac {11}{2}} \Gamma \left (- \frac {7}{4}\right )} + \frac {a b \sqrt {c} \Gamma \left (- \frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{4}, - \frac {1}{2} \\ - \frac {3}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{x^{\frac {7}{2}} \Gamma \left (- \frac {3}{4}\right )} + \frac {b^{2} \sqrt {c} \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {1}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 x^{\frac {3}{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 {{\left (b\,x^2+a\right )}^2\,\sqrt {d\,x^2+c}}{x^{13/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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