Optimal. Leaf size=171 \[ -\frac {(4 b c-5 a d) e \sqrt {e x}}{2 b^2 \sqrt [4]{a+b x^2}}+\frac {d (e x)^{5/2}}{2 b e \sqrt [4]{a+b x^2}}+\frac {(4 b c-5 a d) e^{3/2} \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{4 b^{9/4}}+\frac {(4 b c-5 a d) e^{3/2} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{4 b^{9/4}} \]
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Rubi [A]
time = 0.07, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {470, 294, 335,
246, 218, 214, 211} \begin {gather*} \frac {e^{3/2} (4 b c-5 a d) \text {ArcTan}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{4 b^{9/4}}+\frac {e^{3/2} (4 b c-5 a d) \tanh ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{4 b^{9/4}}-\frac {e \sqrt {e x} (4 b c-5 a d)}{2 b^2 \sqrt [4]{a+b x^2}}+\frac {d (e x)^{5/2}}{2 b e \sqrt [4]{a+b x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 214
Rule 218
Rule 246
Rule 294
Rule 335
Rule 470
Rubi steps
\begin {align*} \int \frac {(e x)^{3/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{5/4}} \, dx &=\frac {d (e x)^{5/2}}{2 b e \sqrt [4]{a+b x^2}}-\frac {\left (-2 b c+\frac {5 a d}{2}\right ) \int \frac {(e x)^{3/2}}{\left (a+b x^2\right )^{5/4}} \, dx}{2 b}\\ &=-\frac {(4 b c-5 a d) e \sqrt {e x}}{2 b^2 \sqrt [4]{a+b x^2}}+\frac {d (e x)^{5/2}}{2 b e \sqrt [4]{a+b x^2}}+\frac {\left ((4 b c-5 a d) e^2\right ) \int \frac {1}{\sqrt {e x} \sqrt [4]{a+b x^2}} \, dx}{4 b^2}\\ &=-\frac {(4 b c-5 a d) e \sqrt {e x}}{2 b^2 \sqrt [4]{a+b x^2}}+\frac {d (e x)^{5/2}}{2 b e \sqrt [4]{a+b x^2}}+\frac {((4 b c-5 a d) e) \text {Subst}\left (\int \frac {1}{\sqrt [4]{a+\frac {b x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{2 b^2}\\ &=-\frac {(4 b c-5 a d) e \sqrt {e x}}{2 b^2 \sqrt [4]{a+b x^2}}+\frac {d (e x)^{5/2}}{2 b e \sqrt [4]{a+b x^2}}+\frac {((4 b c-5 a d) e) \text {Subst}\left (\int \frac {1}{1-\frac {b x^4}{e^2}} \, dx,x,\frac {\sqrt {e x}}{\sqrt [4]{a+b x^2}}\right )}{2 b^2}\\ &=-\frac {(4 b c-5 a d) e \sqrt {e x}}{2 b^2 \sqrt [4]{a+b x^2}}+\frac {d (e x)^{5/2}}{2 b e \sqrt [4]{a+b x^2}}+\frac {\left ((4 b c-5 a d) e^2\right ) \text {Subst}\left (\int \frac {1}{e-\sqrt {b} x^2} \, dx,x,\frac {\sqrt {e x}}{\sqrt [4]{a+b x^2}}\right )}{4 b^2}+\frac {\left ((4 b c-5 a d) e^2\right ) \text {Subst}\left (\int \frac {1}{e+\sqrt {b} x^2} \, dx,x,\frac {\sqrt {e x}}{\sqrt [4]{a+b x^2}}\right )}{4 b^2}\\ &=-\frac {(4 b c-5 a d) e \sqrt {e x}}{2 b^2 \sqrt [4]{a+b x^2}}+\frac {d (e x)^{5/2}}{2 b e \sqrt [4]{a+b x^2}}+\frac {(4 b c-5 a d) e^{3/2} \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{4 b^{9/4}}+\frac {(4 b c-5 a d) e^{3/2} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{4 b^{9/4}}\\ \end {align*}
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Mathematica [A]
time = 0.77, size = 148, normalized size = 0.87 \begin {gather*} \frac {(e x)^{3/2} \left (2 \sqrt [4]{b} \sqrt {x} \left (-4 b c+5 a d+b d x^2\right )+(4 b c-5 a d) \sqrt [4]{a+b x^2} \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a+b x^2}}\right )+(4 b c-5 a d) \sqrt [4]{a+b x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a+b x^2}}\right )\right )}{4 b^{9/4} x^{3/2} \sqrt [4]{a+b x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (e x \right )^{\frac {3}{2}} \left (d \,x^{2}+c \right )}{\left (b \,x^{2}+a \right )^{\frac {5}{4}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 229, normalized size = 1.34 \begin {gather*} \frac {1}{8} \, {\left (d {\left (\frac {4 \, {\left (4 \, a b - \frac {5 \, {\left (b x^{2} + a\right )} a}{x^{2}}\right )}}{\frac {{\left (b x^{2} + a\right )}^{\frac {1}{4}} b^{3}}{\sqrt {x}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{4}} b^{2}}{x^{\frac {5}{2}}}} + \frac {5 \, a {\left (\frac {2 \, \arctan \left (\frac {{\left (b x^{2} + a\right )}^{\frac {1}{4}}}{b^{\frac {1}{4}} \sqrt {x}}\right )}{b^{\frac {1}{4}}} + \frac {\log \left (-\frac {b^{\frac {1}{4}} - \frac {{\left (b x^{2} + a\right )}^{\frac {1}{4}}}{\sqrt {x}}}{b^{\frac {1}{4}} + \frac {{\left (b x^{2} + a\right )}^{\frac {1}{4}}}{\sqrt {x}}}\right )}{b^{\frac {1}{4}}}\right )}}{b^{2}}\right )} - 4 \, c {\left (\frac {\frac {2 \, \arctan \left (\frac {{\left (b x^{2} + a\right )}^{\frac {1}{4}}}{b^{\frac {1}{4}} \sqrt {x}}\right )}{b^{\frac {1}{4}}} + \frac {\log \left (-\frac {b^{\frac {1}{4}} - \frac {{\left (b x^{2} + a\right )}^{\frac {1}{4}}}{\sqrt {x}}}{b^{\frac {1}{4}} + \frac {{\left (b x^{2} + a\right )}^{\frac {1}{4}}}{\sqrt {x}}}\right )}{b^{\frac {1}{4}}}}{b} + \frac {4 \, \sqrt {x}}{{\left (b x^{2} + a\right )}^{\frac {1}{4}} b}\right )}\right )} e^{\frac {3}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 891 vs.
\(2 (115) = 230\).
time = 1.68, size = 891, normalized size = 5.21 \begin {gather*} \frac {4 \, {\left (b d x^{2} - 4 \, b c + 5 \, a d\right )} {\left (b x^{2} + a\right )}^{\frac {3}{4}} \sqrt {x} e^{\frac {3}{2}} + 4 \, {\left (b^{3} x^{2} + a b^{2}\right )} \left (\frac {256 \, b^{4} c^{4} - 1280 \, a b^{3} c^{3} d + 2400 \, a^{2} b^{2} c^{2} d^{2} - 2000 \, a^{3} b c d^{3} + 625 \, a^{4} d^{4}}{b^{9}}\right )^{\frac {1}{4}} \arctan \left (\frac {{\left ({\left (4 \, b^{8} c - 5 \, a b^{7} d\right )} {\left (b x^{2} + a\right )}^{\frac {3}{4}} \sqrt {x} \left (\frac {256 \, b^{4} c^{4} - 1280 \, a b^{3} c^{3} d + 2400 \, a^{2} b^{2} c^{2} d^{2} - 2000 \, a^{3} b c d^{3} + 625 \, a^{4} d^{4}}{b^{9}}\right )^{\frac {3}{4}} e^{6} + {\left (b^{8} x^{2} + a b^{7}\right )} \sqrt {\frac {{\left (16 \, b^{2} c^{2} - 40 \, a b c d + 25 \, a^{2} d^{2}\right )} \sqrt {b x^{2} + a} x e^{3} + {\left (b^{5} x^{2} + a b^{4}\right )} \sqrt {\frac {256 \, b^{4} c^{4} - 1280 \, a b^{3} c^{3} d + 2400 \, a^{2} b^{2} c^{2} d^{2} - 2000 \, a^{3} b c d^{3} + 625 \, a^{4} d^{4}}{b^{9}}} e^{3}}{b x^{2} + a}} \left (\frac {256 \, b^{4} c^{4} - 1280 \, a b^{3} c^{3} d + 2400 \, a^{2} b^{2} c^{2} d^{2} - 2000 \, a^{3} b c d^{3} + 625 \, a^{4} d^{4}}{b^{9}}\right )^{\frac {3}{4}} e^{\frac {9}{2}}\right )} e^{\left (-6\right )}}{256 \, a b^{4} c^{4} - 1280 \, a^{2} b^{3} c^{3} d + 2400 \, a^{3} b^{2} c^{2} d^{2} - 2000 \, a^{4} b c d^{3} + 625 \, a^{5} d^{4} + {\left (256 \, b^{5} c^{4} - 1280 \, a b^{4} c^{3} d + 2400 \, a^{2} b^{3} c^{2} d^{2} - 2000 \, a^{3} b^{2} c d^{3} + 625 \, a^{4} b d^{4}\right )} x^{2}}\right ) e^{\frac {3}{2}} + {\left (b^{3} x^{2} + a b^{2}\right )} \left (\frac {256 \, b^{4} c^{4} - 1280 \, a b^{3} c^{3} d + 2400 \, a^{2} b^{2} c^{2} d^{2} - 2000 \, a^{3} b c d^{3} + 625 \, a^{4} d^{4}}{b^{9}}\right )^{\frac {1}{4}} e^{\frac {3}{2}} \log \left (-\frac {{\left (b x^{2} + a\right )}^{\frac {3}{4}} {\left (4 \, b c - 5 \, a d\right )} \sqrt {x} e^{\frac {3}{2}} + {\left (b^{3} x^{2} + a b^{2}\right )} \left (\frac {256 \, b^{4} c^{4} - 1280 \, a b^{3} c^{3} d + 2400 \, a^{2} b^{2} c^{2} d^{2} - 2000 \, a^{3} b c d^{3} + 625 \, a^{4} d^{4}}{b^{9}}\right )^{\frac {1}{4}} e^{\frac {3}{2}}}{b x^{2} + a}\right ) - {\left (b^{3} x^{2} + a b^{2}\right )} \left (\frac {256 \, b^{4} c^{4} - 1280 \, a b^{3} c^{3} d + 2400 \, a^{2} b^{2} c^{2} d^{2} - 2000 \, a^{3} b c d^{3} + 625 \, a^{4} d^{4}}{b^{9}}\right )^{\frac {1}{4}} e^{\frac {3}{2}} \log \left (-\frac {{\left (b x^{2} + a\right )}^{\frac {3}{4}} {\left (4 \, b c - 5 \, a d\right )} \sqrt {x} e^{\frac {3}{2}} - {\left (b^{3} x^{2} + a b^{2}\right )} \left (\frac {256 \, b^{4} c^{4} - 1280 \, a b^{3} c^{3} d + 2400 \, a^{2} b^{2} c^{2} d^{2} - 2000 \, a^{3} b c d^{3} + 625 \, a^{4} d^{4}}{b^{9}}\right )^{\frac {1}{4}} e^{\frac {3}{2}}}{b x^{2} + a}\right )}{8 \, {\left (b^{3} x^{2} + a b^{2}\right )}} \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 = 13.02, size = 94, normalized size = 0.55 \begin {gather*} \frac {c e^{\frac {3}{2}} x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{4}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {5}{4}} \Gamma \left (\frac {9}{4}\right )} + \frac {d e^{\frac {3}{2}} x^{\frac {9}{2}} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{4}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {5}{4}} \Gamma \left (\frac {13}{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 (e\,x\right )}^{3/2}\,\left (d\,x^2+c\right )}{{\left (b\,x^2+a\right )}^{5/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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