Optimal. Leaf size=421 \[ -\frac {2 \left (b^2 c^2-3 a d (2 b c+5 a d)\right ) (e x)^{3/2} \sqrt {c+d x^2}}{15 c d e^3}-\frac {4 \left (b^2 c^2-3 a d (2 b c+5 a d)\right ) \sqrt {e x} \sqrt {c+d x^2}}{15 d^{3/2} e^2 \left (\sqrt {c}+\sqrt {d} x\right )}-\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{c e \sqrt {e x}}+\frac {2 b^2 (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{9 d e^3}+\frac {4 \sqrt [4]{c} \left (b^2 c^2-3 a d (2 b c+5 a d)\right ) \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{15 d^{7/4} e^{3/2} \sqrt {c+d x^2}}-\frac {2 \sqrt [4]{c} \left (b^2 c^2-3 a d (2 b c+5 a d)\right ) \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{15 d^{7/4} e^{3/2} \sqrt {c+d x^2}} \]
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Rubi [A]
time = 0.27, antiderivative size = 421, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {473, 470, 285,
335, 311, 226, 1210} \begin {gather*} -\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{c e \sqrt {e x}}-\frac {2 \sqrt [4]{c} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (b^2 c^2-3 a d (5 a d+2 b c)\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^{7/4} e^{3/2} \sqrt {c+d x^2}}+\frac {4 \sqrt [4]{c} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (b^2 c^2-3 a d (5 a d+2 b c)\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^{7/4} e^{3/2} \sqrt {c+d x^2}}-\frac {4 \sqrt {e x} \sqrt {c+d x^2} \left (b^2 c^2-3 a d (5 a d+2 b c)\right )}{15 d^{3/2} e^2 \left (\sqrt {c}+\sqrt {d} x\right )}-\frac {2 (e x)^{3/2} \sqrt {c+d x^2} \left (b^2 c^2-3 a d (5 a d+2 b c)\right )}{15 c d e^3}+\frac {2 b^2 (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{9 d e^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 226
Rule 285
Rule 311
Rule 335
Rule 470
Rule 473
Rule 1210
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{(e x)^{3/2}} \, dx &=-\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{c e \sqrt {e x}}+\frac {2 \int \sqrt {e x} \left (\frac {1}{2} a (2 b c+5 a d)+\frac {1}{2} b^2 c x^2\right ) \sqrt {c+d x^2} \, dx}{c e^2}\\ &=-\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{c e \sqrt {e x}}+\frac {2 b^2 (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{9 d e^3}-\frac {\left (b^2 c^2-3 a d (2 b c+5 a d)\right ) \int \sqrt {e x} \sqrt {c+d x^2} \, dx}{3 c d e^2}\\ &=-\frac {2 \left (b^2 c^2-3 a d (2 b c+5 a d)\right ) (e x)^{3/2} \sqrt {c+d x^2}}{15 c d e^3}-\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{c e \sqrt {e x}}+\frac {2 b^2 (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{9 d e^3}-\frac {\left (2 \left (b^2 c^2-3 a d (2 b c+5 a d)\right )\right ) \int \frac {\sqrt {e x}}{\sqrt {c+d x^2}} \, dx}{15 d e^2}\\ &=-\frac {2 \left (b^2 c^2-3 a d (2 b c+5 a d)\right ) (e x)^{3/2} \sqrt {c+d x^2}}{15 c d e^3}-\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{c e \sqrt {e x}}+\frac {2 b^2 (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{9 d e^3}-\frac {\left (4 \left (b^2 c^2-3 a d (2 b c+5 a d)\right )\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{15 d e^3}\\ &=-\frac {2 \left (b^2 c^2-3 a d (2 b c+5 a d)\right ) (e x)^{3/2} \sqrt {c+d x^2}}{15 c d e^3}-\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{c e \sqrt {e x}}+\frac {2 b^2 (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{9 d e^3}-\frac {\left (4 \sqrt {c} \left (b^2 c^2-3 a d (2 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 )}{15 d^{3/2} e^2}+\frac {\left (4 \sqrt {c} \left (b^2 c^2-3 a d (2 b c+5 a d)\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 d^{3/2} e^2}\\ &=-\frac {2 \left (b^2 c^2-3 a d (2 b c+5 a d)\right ) (e x)^{3/2} \sqrt {c+d x^2}}{15 c d e^3}-\frac {4 \left (b^2 c^2-3 a d (2 b c+5 a d)\right ) \sqrt {e x} \sqrt {c+d x^2}}{15 d^{3/2} e^2 \left (\sqrt {c}+\sqrt {d} x\right )}-\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{c e \sqrt {e x}}+\frac {2 b^2 (e x)^{3/2} \left (c+d x^2\right )^{3/2}}{9 d e^3}+\frac {4 \sqrt [4]{c} \left (b^2 c^2-3 a d (2 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}} 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^{7/4} e^{3/2} \sqrt {c+d x^2}}-\frac {2 \sqrt [4]{c} \left (b^2 c^2-3 a d (2 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 )}{15 d^{7/4} e^{3/2} \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 = 20.10, size = 129, normalized size = 0.31 \begin {gather*} \frac {x \left (2 \left (c+d x^2\right ) \left (-45 a^2 d+18 a b d x^2+b^2 x^2 \left (2 c+5 d x^2\right )\right )+12 \left (-b^2 c^2+6 a b c d+15 a^2 d^2\right ) \sqrt {1+\frac {c}{d x^2}} x^2 \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};-\frac {c}{d x^2}\right )\right )}{45 d (e x)^{3/2} \sqrt {c+d x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 624, normalized size = 1.48
method | result | size |
risch | \(-\frac {2 \sqrt {d \,x^{2}+c}\, \left (-5 b^{2} d \,x^{4}-18 a b d \,x^{2}-2 b^{2} c \,x^{2}+45 a^{2} d \right )}{45 d e \sqrt {e x}}+\frac {2 \left (15 a^{2} d^{2}+6 a b c d -b^{2} c^{2}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \left (-\frac {2 \sqrt {-c d}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\sqrt {-c d}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}\right ) \sqrt {e x \left (d \,x^{2}+c \right )}}{15 d^{2} \sqrt {d e \,x^{3}+c e x}\, e \sqrt {e x}\, \sqrt {d \,x^{2}+c}}\) | \(269\) |
elliptic | \(\frac {\sqrt {e x \left (d \,x^{2}+c \right )}\, \left (-\frac {2 \left (d e \,x^{2}+c e \right ) a^{2}}{e^{2} \sqrt {x \left (d e \,x^{2}+c e \right )}}+\frac {2 b^{2} x^{3} \sqrt {d e \,x^{3}+c e x}}{9 e^{2}}+\frac {2 \left (\frac {b \left (2 a d +b c \right )}{e}-\frac {7 b^{2} c}{9 e}\right ) x \sqrt {d e \,x^{3}+c e x}}{5 d e}+\frac {\left (\frac {a \left (a d +2 b c \right )}{e}+\frac {d \,a^{2}}{e}-\frac {3 \left (\frac {b \left (2 a d +b c \right )}{e}-\frac {7 b^{2} c}{9 e}\right ) c}{5 d}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \left (-\frac {2 \sqrt {-c d}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\sqrt {-c d}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}\right )}{d \sqrt {d e \,x^{3}+c e x}}\right )}{\sqrt {e x}\, \sqrt {d \,x^{2}+c}}\) | \(340\) |
default | \(\frac {\frac {2 b^{2} x^{6} d^{3}}{9}+4 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \EllipticE \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c \,d^{2}+\frac {8 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \EllipticE \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b \,c^{2} d}{5}-\frac {4 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \EllipticE \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{3}}{15}-2 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \EllipticF \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c \,d^{2}-\frac {4 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \EllipticF \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b \,c^{2} d}{5}+\frac {2 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \EllipticF \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{3}}{15}+\frac {4 a b \,d^{3} x^{4}}{5}+\frac {14 b^{2} c \,d^{2} x^{4}}{45}-2 a^{2} d^{3} x^{2}+\frac {4 a b c \,d^{2} x^{2}}{5}+\frac {4 b^{2} c^{2} d \,x^{2}}{45}-2 a^{2} c \,d^{2}}{\sqrt {d \,x^{2}+c}\, d^{2} e \sqrt {e x}}\) | \(624\) |
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 = 108, normalized size = 0.26 \begin {gather*} \frac {2 \, {\left (6 \, {\left (b^{2} c^{2} - 6 \, a b c d - 15 \, a^{2} d^{2}\right )} \sqrt {d} x {\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^{4} - 45 \, a^{2} d^{2} + 2 \, {\left (b^{2} c d + 9 \, a b d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {x}\right )} e^{\left (-\frac {3}{2}\right )}}{45 \, d^{2} x} \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 = 3.68, size = 153, normalized size = 0.36 \begin {gather*} \frac {a^{2} \sqrt {c} \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{4} \\ \frac {3}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 e^{\frac {3}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} + \frac {a b \sqrt {c} x^{\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 )}}{e^{\frac {3}{2}} \Gamma \left (\frac {7}{4}\right )} + \frac {b^{2} \sqrt {c} x^{\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 )}}{2 e^{\frac {3}{2}} \Gamma \left (\frac {11}{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}}{{\left (e\,x\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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