Optimal. Leaf size=375 \[ -\frac {2 b (7 b c-18 a d) (e x)^{3/2} \sqrt {c+d x^2}}{45 d^2 e}+\frac {2 b^2 (e x)^{7/2} \sqrt {c+d x^2}}{9 d e^3}+\frac {2 \left (15 a^2 d^2+b c (7 b c-18 a d)\right ) \sqrt {e x} \sqrt {c+d x^2}}{15 d^{5/2} \left (\sqrt {c}+\sqrt {d} x\right )}-\frac {2 \sqrt [4]{c} \left (15 a^2 d^2+b c (7 b c-18 a d)\right ) \sqrt {e} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{15 d^{11/4} \sqrt {c+d x^2}}+\frac {\sqrt [4]{c} \left (15 a^2 d^2+b c (7 b c-18 a d)\right ) \sqrt {e} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{15 d^{11/4} \sqrt {c+d x^2}} \]
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Rubi [A]
time = 0.25, antiderivative size = 375, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {475, 470, 335,
311, 226, 1210} \begin {gather*} \frac {\sqrt [4]{c} \sqrt {e} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (15 a^2 d^2+b c (7 b c-18 a d)\right ) F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{15 d^{11/4} \sqrt {c+d x^2}}-\frac {2 \sqrt [4]{c} \sqrt {e} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (15 a^2 d^2+b c (7 b c-18 a d)\right ) E\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{15 d^{11/4} \sqrt {c+d x^2}}+\frac {2 \sqrt {e x} \sqrt {c+d x^2} \left (15 a^2 d^2+b c (7 b c-18 a d)\right )}{15 d^{5/2} \left (\sqrt {c}+\sqrt {d} x\right )}-\frac {2 b (e x)^{3/2} \sqrt {c+d x^2} (7 b c-18 a d)}{45 d^2 e}+\frac {2 b^2 (e x)^{7/2} \sqrt {c+d x^2}}{9 d e^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 226
Rule 311
Rule 335
Rule 470
Rule 475
Rule 1210
Rubi steps
\begin {align*} \int \frac {\sqrt {e x} \left (a+b x^2\right )^2}{\sqrt {c+d x^2}} \, dx &=\frac {2 b^2 (e x)^{7/2} \sqrt {c+d x^2}}{9 d e^3}+\frac {2 \int \frac {\sqrt {e x} \left (\frac {9 a^2 d}{2}-\frac {1}{2} b (7 b c-18 a d) x^2\right )}{\sqrt {c+d x^2}} \, dx}{9 d}\\ &=-\frac {2 b (7 b c-18 a d) (e x)^{3/2} \sqrt {c+d x^2}}{45 d^2 e}+\frac {2 b^2 (e x)^{7/2} \sqrt {c+d x^2}}{9 d e^3}+\frac {1}{15} \left (15 a^2+\frac {b c (7 b c-18 a d)}{d^2}\right ) \int \frac {\sqrt {e x}}{\sqrt {c+d x^2}} \, dx\\ &=-\frac {2 b (7 b c-18 a d) (e x)^{3/2} \sqrt {c+d x^2}}{45 d^2 e}+\frac {2 b^2 (e x)^{7/2} \sqrt {c+d x^2}}{9 d e^3}+\frac {\left (2 \left (15 a^2+\frac {b c (7 b c-18 a d)}{d^2}\right )\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{15 e}\\ &=-\frac {2 b (7 b c-18 a d) (e x)^{3/2} \sqrt {c+d x^2}}{45 d^2 e}+\frac {2 b^2 (e x)^{7/2} \sqrt {c+d x^2}}{9 d e^3}+\frac {\left (2 \sqrt {c} \left (15 a^2+\frac {b c (7 b c-18 a d)}{d^2}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{15 \sqrt {d}}-\frac {\left (2 \sqrt {c} \left (15 a^2+\frac {b c (7 b c-18 a d)}{d^2}\right )\right ) \text {Subst}\left (\int \frac {1-\frac {\sqrt {d} x^2}{\sqrt {c} e}}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{15 \sqrt {d}}\\ &=-\frac {2 b (7 b c-18 a d) (e x)^{3/2} \sqrt {c+d x^2}}{45 d^2 e}+\frac {2 b^2 (e x)^{7/2} \sqrt {c+d x^2}}{9 d e^3}+\frac {2 \left (15 a^2+\frac {b c (7 b c-18 a d)}{d^2}\right ) \sqrt {e x} \sqrt {c+d x^2}}{15 \sqrt {d} \left (\sqrt {c}+\sqrt {d} x\right )}-\frac {2 \sqrt [4]{c} \left (15 a^2+\frac {b c (7 b c-18 a d)}{d^2}\right ) \sqrt {e} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{15 d^{3/4} \sqrt {c+d x^2}}+\frac {\sqrt [4]{c} \left (15 a^2+\frac {b c (7 b c-18 a d)}{d^2}\right ) \sqrt {e} \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 )}{15 d^{3/4} \sqrt {c+d 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 = 20.11, size = 111, normalized size = 0.30 \begin {gather*} \frac {2 \sqrt {e x} \left (b x \left (c+d x^2\right ) \left (-7 b c+18 a d+5 b d x^2\right )+3 \left (7 b^2 c^2-18 a b c d+15 a^2 d^2\right ) \sqrt {1+\frac {c}{d x^2}} x \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};-\frac {c}{d x^2}\right )\right )}{45 d^2 \sqrt {c+d x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 604, normalized size = 1.61
method | result | size |
risch | \(\frac {2 b \,x^{2} \left (5 b d \,x^{2}+18 a d -7 b c \right ) \sqrt {d \,x^{2}+c}\, e}{45 d^{2} \sqrt {e x}}+\frac {\left (15 a^{2} d^{2}-18 a b c d +7 b^{2} c^{2}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \left (-\frac {2 \sqrt {-c d}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\sqrt {-c d}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}\right ) e \sqrt {e x \left (d \,x^{2}+c \right )}}{15 d^{3} \sqrt {d e \,x^{3}+c e x}\, \sqrt {e x}\, \sqrt {d \,x^{2}+c}}\) | \(252\) |
elliptic | \(\frac {\sqrt {e x \left (d \,x^{2}+c \right )}\, \sqrt {e x}\, \left (\frac {2 b^{2} x^{3} \sqrt {d e \,x^{3}+c e x}}{9 d}+\frac {2 \left (2 a b e -\frac {7 b^{2} c e}{9 d}\right ) x \sqrt {d e \,x^{3}+c e x}}{5 d e}+\frac {\left (a^{2} e -\frac {3 \left (2 a b e -\frac {7 b^{2} c e}{9 d}\right ) c}{5 d}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \left (-\frac {2 \sqrt {-c d}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\sqrt {-c d}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}\right )}{d \sqrt {d e \,x^{3}+c e x}}\right )}{e x \sqrt {d \,x^{2}+c}}\) | \(284\) |
default | \(\frac {\sqrt {e x}\, \left (10 b^{2} x^{6} d^{3}+90 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \EllipticE \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c \,d^{2}-108 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \EllipticE \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b \,c^{2} d +42 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \EllipticE \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{3}-45 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \EllipticF \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c \,d^{2}+54 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \EllipticF \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b \,c^{2} d -21 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \EllipticF \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{3}+36 a b \,d^{3} x^{4}-4 b^{2} c \,d^{2} x^{4}+36 a b c \,d^{2} x^{2}-14 b^{2} c^{2} d \,x^{2}\right )}{45 \sqrt {d \,x^{2}+c}\, d^{3} x}\) | \(604\) |
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 = 99, normalized size = 0.26 \begin {gather*} -\frac {2 \, {\left (3 \, {\left (7 \, b^{2} c^{2} - 18 \, a b c d + 15 \, a^{2} d^{2}\right )} \sqrt {d} e^{\frac {1}{2}} {\rm weierstrassZeta}\left (-\frac {4 \, c}{d}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right )\right ) - {\left (5 \, b^{2} d^{2} x^{3} - {\left (7 \, b^{2} c d - 18 \, a b d^{2}\right )} x\right )} \sqrt {d x^{2} + c} \sqrt {x} e^{\frac {1}{2}}\right )}}{45 \, d^{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 = 2.69, size = 143, normalized size = 0.38 \begin {gather*} \frac {a^{2} \left (e x\right )^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 \sqrt {c} e \Gamma \left (\frac {7}{4}\right )} + \frac {a b \left (e x\right )^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{\sqrt {c} e^{3} \Gamma \left (\frac {11}{4}\right )} + \frac {b^{2} \left (e x\right )^{\frac {11}{2}} \Gamma \left (\frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {11}{4} \\ \frac {15}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 \sqrt {c} e^{5} \Gamma \left (\frac {15}{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 {e\,x}\,{\left (b\,x^2+a\right )}^2}{\sqrt {d\,x^2+c}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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