Optimal. Leaf size=193 \[ -\frac {2 a^2 \sqrt {c+d x^2}}{7 c e (e x)^{7/2}}-\frac {2 a (14 b c-5 a d) \sqrt {c+d x^2}}{21 c^2 e^3 (e x)^{3/2}}+\frac {\left (21 b^2 c^2-14 a b c d+5 a^2 d^2\right ) \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{21 c^{9/4} \sqrt [4]{d} e^{9/2} \sqrt {c+d x^2}} \]
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Rubi [A]
time = 0.11, antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {473, 464, 335,
226} \begin {gather*} \frac {\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^2-14 a b c d+21 b^2 c^2\right ) F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{21 c^{9/4} \sqrt [4]{d} e^{9/2} \sqrt {c+d x^2}}-\frac {2 a^2 \sqrt {c+d x^2}}{7 c e (e x)^{7/2}}-\frac {2 a \sqrt {c+d x^2} (14 b c-5 a d)}{21 c^2 e^3 (e x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 226
Rule 335
Rule 464
Rule 473
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^2}{(e x)^{9/2} \sqrt {c+d x^2}} \, dx &=-\frac {2 a^2 \sqrt {c+d x^2}}{7 c e (e x)^{7/2}}+\frac {2 \int \frac {\frac {1}{2} a (14 b c-5 a d)+\frac {7}{2} b^2 c x^2}{(e x)^{5/2} \sqrt {c+d x^2}} \, dx}{7 c e^2}\\ &=-\frac {2 a^2 \sqrt {c+d x^2}}{7 c e (e x)^{7/2}}-\frac {2 a (14 b c-5 a d) \sqrt {c+d x^2}}{21 c^2 e^3 (e x)^{3/2}}-\frac {\left (4 \left (-\frac {21}{4} b^2 c^2+\frac {1}{4} a d (14 b c-5 a d)\right )\right ) \int \frac {1}{\sqrt {e x} \sqrt {c+d x^2}} \, dx}{21 c^2 e^4}\\ &=-\frac {2 a^2 \sqrt {c+d x^2}}{7 c e (e x)^{7/2}}-\frac {2 a (14 b c-5 a d) \sqrt {c+d x^2}}{21 c^2 e^3 (e x)^{3/2}}-\frac {\left (8 \left (-\frac {21}{4} b^2 c^2+\frac {1}{4} a d (14 b c-5 a d)\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{21 c^2 e^5}\\ &=-\frac {2 a^2 \sqrt {c+d x^2}}{7 c e (e x)^{7/2}}-\frac {2 a (14 b c-5 a d) \sqrt {c+d x^2}}{21 c^2 e^3 (e x)^{3/2}}+\frac {\left (21 b^2 c^2-a d (14 b c-5 a d)\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 {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{21 c^{9/4} \sqrt [4]{d} e^{9/2} \sqrt {c+d x^2}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 10.15, size = 159, normalized size = 0.82 \begin {gather*} \frac {x^{9/2} \left (\frac {2 a \left (c+d x^2\right ) \left (-3 a c-14 b c x^2+5 a d x^2\right )}{c^2 x^{7/2}}+\frac {2 i \left (21 b^2 c^2-14 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 )}{c^2 \sqrt {\frac {i \sqrt {c}}{\sqrt {d}}}}\right )}{21 (e x)^{9/2} \sqrt {c+d x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 370, normalized size = 1.92
method | result | size |
risch | \(-\frac {2 \sqrt {d \,x^{2}+c}\, a \left (-5 a d \,x^{2}+14 c \,x^{2} b +3 a c \right )}{21 c^{2} x^{3} e^{4} \sqrt {e x}}+\frac {\left (5 a^{2} d^{2}-14 a b c d +21 b^{2} c^{2}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {e x \left (d \,x^{2}+c \right )}}{21 c^{2} d \sqrt {d e \,x^{3}+c e x}\, e^{4} \sqrt {e x}\, \sqrt {d \,x^{2}+c}}\) | \(212\) |
elliptic | \(\frac {\sqrt {e x \left (d \,x^{2}+c \right )}\, \left (-\frac {2 a^{2} \sqrt {d e \,x^{3}+c e x}}{7 e^{5} c \,x^{4}}+\frac {2 a \left (5 a d -14 b c \right ) \sqrt {d e \,x^{3}+c e x}}{21 e^{5} c^{2} x^{2}}+\frac {\left (\frac {b^{2}}{e^{4}}+\frac {d a \left (5 a d -14 b c \right )}{21 c^{2} e^{4}}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d \sqrt {d e \,x^{3}+c e x}}\right )}{\sqrt {e x}\, \sqrt {d \,x^{2}+c}}\) | \(226\) |
default | \(\frac {5 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \EllipticF \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-c d}\, a^{2} d^{2} x^{3}-14 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \EllipticF \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-c d}\, a b c d \,x^{3}+21 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \EllipticF \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-c d}\, b^{2} c^{2} x^{3}+10 a^{2} d^{3} x^{4}-28 a b c \,d^{2} x^{4}+4 a^{2} c \,d^{2} x^{2}-28 a b \,c^{2} d \,x^{2}-6 a^{2} c^{2} d}{21 \sqrt {d \,x^{2}+c}\, x^{3} d \,c^{2} e^{4} \sqrt {e x}}\) | \(370\) |
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 = 0.26, size = 94, normalized size = 0.49 \begin {gather*} \frac {2 \, {\left ({\left (21 \, b^{2} c^{2} - 14 \, a b c d + 5 \, a^{2} d^{2}\right )} \sqrt {d} x^{4} {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right ) - {\left (3 \, a^{2} c d + {\left (14 \, a b c d - 5 \, a^{2} d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {x}\right )} e^{\left (-\frac {9}{2}\right )}}{21 \, c^{2} d x^{4}} \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 = 90.59, size = 155, normalized size = 0.80 \begin {gather*} \frac {a^{2} \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 )}}{2 \sqrt {c} e^{\frac {9}{2}} x^{\frac {7}{2}} \Gamma \left (- \frac {3}{4}\right )} + \frac {a b \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 )}}{\sqrt {c} e^{\frac {9}{2}} x^{\frac {3}{2}} \Gamma \left (\frac {1}{4}\right )} + \frac {b^{2} \sqrt {x} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 \sqrt {c} e^{\frac {9}{2}} \Gamma \left (\frac {5}{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 = -0.01 \begin {gather*} \int \frac {{\left (b\,x^2+a\right )}^2}{{\left (e\,x\right )}^{9/2}\,\sqrt {d\,x^2+c}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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