Optimal. Leaf size=403 \[ \frac {(b c-a d)^2 (e x)^{3/2}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}-\frac {(b c-a d) (3 b c+a d) (e x)^{3/2}}{2 c^2 d^2 e \sqrt {c+d x^2}}+\frac {\left (7 b^2 c^2-2 a b c d-a^2 d^2\right ) \sqrt {e x} \sqrt {c+d x^2}}{2 c^2 d^{5/2} \left (\sqrt {c}+\sqrt {d} x\right )}-\frac {\left (7 b^2 c^2-2 a b c d-a^2 d^2\right ) \sqrt {e} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{2 c^{7/4} d^{11/4} \sqrt {c+d x^2}}+\frac {\left (7 b^2 c^2-2 a b c d-a^2 d^2\right ) \sqrt {e} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{4 c^{7/4} d^{11/4} \sqrt {c+d x^2}} \]
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Rubi [A]
time = 0.23, antiderivative size = 403, 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 = {474, 468, 335,
311, 226, 1210} \begin {gather*} \frac {\sqrt {e} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (-a^2 d^2-2 a b c d+7 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 )}{4 c^{7/4} d^{11/4} \sqrt {c+d x^2}}-\frac {\sqrt {e} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (-a^2 d^2-2 a b c d+7 b^2 c^2\right ) E\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{2 c^{7/4} d^{11/4} \sqrt {c+d x^2}}+\frac {\sqrt {e x} \sqrt {c+d x^2} \left (-a^2 d^2-2 a b c d+7 b^2 c^2\right )}{2 c^2 d^{5/2} \left (\sqrt {c}+\sqrt {d} x\right )}-\frac {(e x)^{3/2} (a d+3 b c) (b c-a d)}{2 c^2 d^2 e \sqrt {c+d x^2}}+\frac {(e x)^{3/2} (b c-a d)^2}{3 c d^2 e \left (c+d x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 226
Rule 311
Rule 335
Rule 468
Rule 474
Rule 1210
Rubi steps
\begin {align*} \int \frac {\sqrt {e x} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx &=\frac {(b c-a d)^2 (e x)^{3/2}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}-\frac {\int \frac {\sqrt {e x} \left (-\frac {3}{2} \left (2 a^2 d^2-(b c-a d)^2\right )-3 b^2 c d x^2\right )}{\left (c+d x^2\right )^{3/2}} \, dx}{3 c d^2}\\ &=\frac {(b c-a d)^2 (e x)^{3/2}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}-\frac {(b c-a d) (3 b c+a d) (e x)^{3/2}}{2 c^2 d^2 e \sqrt {c+d x^2}}+\frac {\left (7 b^2 c^2-2 a b c d-a^2 d^2\right ) \int \frac {\sqrt {e x}}{\sqrt {c+d x^2}} \, dx}{4 c^2 d^2}\\ &=\frac {(b c-a d)^2 (e x)^{3/2}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}-\frac {(b c-a d) (3 b c+a d) (e x)^{3/2}}{2 c^2 d^2 e \sqrt {c+d x^2}}+\frac {\left (7 b^2 c^2-2 a b c d-a^2 d^2\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{2 c^2 d^2 e}\\ &=\frac {(b c-a d)^2 (e x)^{3/2}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}-\frac {(b c-a d) (3 b c+a d) (e x)^{3/2}}{2 c^2 d^2 e \sqrt {c+d x^2}}+\frac {\left (7 b^2 c^2-2 a b c d-a^2 d^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{2 c^{3/2} d^{5/2}}-\frac {\left (7 b^2 c^2-2 a b c d-a^2 d^2\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 )}{2 c^{3/2} d^{5/2}}\\ &=\frac {(b c-a d)^2 (e x)^{3/2}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}-\frac {(b c-a d) (3 b c+a d) (e x)^{3/2}}{2 c^2 d^2 e \sqrt {c+d x^2}}+\frac {\left (7 b^2 c^2-2 a b c d-a^2 d^2\right ) \sqrt {e x} \sqrt {c+d x^2}}{2 c^2 d^{5/2} \left (\sqrt {c}+\sqrt {d} x\right )}-\frac {\left (7 b^2 c^2-2 a b c d-a^2 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 )}{2 c^{7/4} d^{11/4} \sqrt {c+d x^2}}+\frac {\left (7 b^2 c^2-2 a b c d-a^2 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 )}{4 c^{7/4} d^{11/4} \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 = 20.16, size = 147, normalized size = 0.36 \begin {gather*} \frac {\sqrt {e x} \left (x \left (2 a b c d \left (c+3 d x^2\right )+a^2 d^2 \left (5 c+3 d x^2\right )-b^2 c^2 \left (7 c+9 d x^2\right )\right )+3 \left (7 b^2 c^2-2 a b c d-a^2 d^2\right ) \sqrt {1+\frac {c}{d x^2}} x \left (c+d x^2\right ) \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};-\frac {c}{d x^2}\right )\right )}{6 c^2 d^2 \left (c+d x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1175\) vs.
\(2(417)=834\).
time = 0.13, size = 1176, normalized size = 2.92
method | result | size |
elliptic | \(\frac {\sqrt {e x \left (d \,x^{2}+c \right )}\, \sqrt {e x}\, \left (\frac {x \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \sqrt {d e \,x^{3}+c e x}}{3 c \,d^{4} \left (x^{2}+\frac {c}{d}\right )^{2}}+\frac {e \,x^{2} \left (a^{2} d^{2}+2 a b c d -3 b^{2} c^{2}\right )}{2 d^{2} c^{2} \sqrt {\left (x^{2}+\frac {c}{d}\right ) d e x}}+\frac {\left (\frac {b^{2} e}{d^{2}}-\frac {e \left (a^{2} d^{2}+2 a b c d -3 b^{2} c^{2}\right )}{4 d^{2} c^{2}}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \left (-\frac {2 \sqrt {-c d}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\sqrt {-c d}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}\right )}{d \sqrt {d e \,x^{3}+c e x}}\right )}{e x \sqrt {d \,x^{2}+c}}\) | \(337\) |
default | \(-\frac {\left (6 \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^{3} x^{2}+12 \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^{2} x^{2}-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} d \,x^{2}-3 \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^{3} x^{2}-6 \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^{2} x^{2}+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} d \,x^{2}+6 \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^{2} d^{2}+12 \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^{3} 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^{4}-3 \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^{2} d^{2}-6 \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^{3} 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^{4}-6 a^{2} d^{4} x^{4}-12 a b c \,d^{3} x^{4}+18 b^{2} c^{2} d^{2} x^{4}-10 a^{2} c \,d^{3} x^{2}-4 a b \,c^{2} d^{2} x^{2}+14 b^{2} c^{3} d \,x^{2}\right ) \sqrt {e x}}{12 d^{3} c^{2} x \left (d \,x^{2}+c \right )^{\frac {3}{2}}}\) | \(1176\) |
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 = 0.25, size = 232, normalized size = 0.58 \begin {gather*} -\frac {3 \, {\left (7 \, b^{2} c^{4} - 2 \, a b c^{3} d - a^{2} c^{2} d^{2} + {\left (7 \, b^{2} c^{2} d^{2} - 2 \, a b c d^{3} - a^{2} d^{4}\right )} x^{4} + 2 \, {\left (7 \, b^{2} c^{3} d - 2 \, a b c^{2} d^{2} - a^{2} c d^{3}\right )} x^{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 (3 \, {\left (3 \, b^{2} c^{2} d^{2} - 2 \, a b c d^{3} - a^{2} d^{4}\right )} x^{3} + {\left (7 \, b^{2} c^{3} d - 2 \, a b c^{2} d^{2} - 5 \, a^{2} c d^{3}\right )} x\right )} \sqrt {d x^{2} + c} \sqrt {x} e^{\frac {1}{2}}}{6 \, {\left (c^{2} d^{5} x^{4} + 2 \, c^{3} d^{4} x^{2} + c^{4} d^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {e x} \left (a + b x^{2}\right )^{2}}{\left (c + d x^{2}\right )^{\frac {5}{2}}}\, dx \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}{{\left (d\,x^2+c\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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