Optimal. Leaf size=530 \[ \frac {8 c^2 \left (51 a^2 d^2+b c (11 b c-42 a d)\right ) e (e x)^{3/2} \sqrt {c+d x^2}}{9945 d^3}+\frac {4 c \left (51 a^2 d^2+b c (11 b c-42 a d)\right ) (e x)^{7/2} \sqrt {c+d x^2}}{1989 d^2 e}-\frac {8 c^3 \left (51 a^2 d^2+b c (11 b c-42 a d)\right ) e^2 \sqrt {e x} \sqrt {c+d x^2}}{3315 d^{7/2} \left (\sqrt {c}+\sqrt {d} x\right )}+\frac {2 \left (51 a^2 d^2+b c (11 b c-42 a d)\right ) (e x)^{7/2} \left (c+d x^2\right )^{3/2}}{663 d^2 e}-\frac {2 b (11 b c-42 a d) (e x)^{7/2} \left (c+d x^2\right )^{5/2}}{357 d^2 e}+\frac {2 b^2 (e x)^{11/2} \left (c+d x^2\right )^{5/2}}{21 d e^3}+\frac {8 c^{13/4} \left (51 a^2 d^2+b c (11 b c-42 a d)\right ) e^{5/2} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{3315 d^{15/4} \sqrt {c+d x^2}}-\frac {4 c^{13/4} \left (51 a^2 d^2+b c (11 b c-42 a d)\right ) e^{5/2} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{3315 d^{15/4} \sqrt {c+d x^2}} \]
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Rubi [A]
time = 0.38, antiderivative size = 530, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 8, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {475, 470, 285,
327, 335, 311, 226, 1210} \begin {gather*} -\frac {4 c^{13/4} e^{5/2} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (51 a^2 d^2+b c (11 b c-42 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 )}{3315 d^{15/4} \sqrt {c+d x^2}}+\frac {8 c^{13/4} e^{5/2} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (51 a^2 d^2+b c (11 b c-42 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 )}{3315 d^{15/4} \sqrt {c+d x^2}}-\frac {8 c^3 e^2 \sqrt {e x} \sqrt {c+d x^2} \left (51 a^2 d^2+b c (11 b c-42 a d)\right )}{3315 d^{7/2} \left (\sqrt {c}+\sqrt {d} x\right )}+\frac {8 c^2 e (e x)^{3/2} \sqrt {c+d x^2} \left (51 a^2 d^2+b c (11 b c-42 a d)\right )}{9945 d^3}+\frac {2 (e x)^{7/2} \left (c+d x^2\right )^{3/2} \left (51 a^2 d^2+b c (11 b c-42 a d)\right )}{663 d^2 e}+\frac {4 c (e x)^{7/2} \sqrt {c+d x^2} \left (51 a^2 d^2+b c (11 b c-42 a d)\right )}{1989 d^2 e}-\frac {2 b (e x)^{7/2} \left (c+d x^2\right )^{5/2} (11 b c-42 a d)}{357 d^2 e}+\frac {2 b^2 (e x)^{11/2} \left (c+d x^2\right )^{5/2}}{21 d e^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 226
Rule 285
Rule 311
Rule 327
Rule 335
Rule 470
Rule 475
Rule 1210
Rubi steps
\begin {align*} \int (e x)^{5/2} \left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2} \, dx &=\frac {2 b^2 (e x)^{11/2} \left (c+d x^2\right )^{5/2}}{21 d e^3}+\frac {2 \int (e x)^{5/2} \left (c+d x^2\right )^{3/2} \left (\frac {21 a^2 d}{2}-\frac {1}{2} b (11 b c-42 a d) x^2\right ) \, dx}{21 d}\\ &=-\frac {2 b (11 b c-42 a d) (e x)^{7/2} \left (c+d x^2\right )^{5/2}}{357 d^2 e}+\frac {2 b^2 (e x)^{11/2} \left (c+d x^2\right )^{5/2}}{21 d e^3}+\frac {1}{51} \left (51 a^2+\frac {b c (11 b c-42 a d)}{d^2}\right ) \int (e x)^{5/2} \left (c+d x^2\right )^{3/2} \, dx\\ &=\frac {2 \left (51 a^2+\frac {b c (11 b c-42 a d)}{d^2}\right ) (e x)^{7/2} \left (c+d x^2\right )^{3/2}}{663 e}-\frac {2 b (11 b c-42 a d) (e x)^{7/2} \left (c+d x^2\right )^{5/2}}{357 d^2 e}+\frac {2 b^2 (e x)^{11/2} \left (c+d x^2\right )^{5/2}}{21 d e^3}+\frac {1}{221} \left (2 c \left (51 a^2+\frac {b c (11 b c-42 a d)}{d^2}\right )\right ) \int (e x)^{5/2} \sqrt {c+d x^2} \, dx\\ &=\frac {4 c \left (51 a^2+\frac {b c (11 b c-42 a d)}{d^2}\right ) (e x)^{7/2} \sqrt {c+d x^2}}{1989 e}+\frac {2 \left (51 a^2+\frac {b c (11 b c-42 a d)}{d^2}\right ) (e x)^{7/2} \left (c+d x^2\right )^{3/2}}{663 e}-\frac {2 b (11 b c-42 a d) (e x)^{7/2} \left (c+d x^2\right )^{5/2}}{357 d^2 e}+\frac {2 b^2 (e x)^{11/2} \left (c+d x^2\right )^{5/2}}{21 d e^3}+\frac {\left (4 c^2 \left (51 a^2+\frac {b c (11 b c-42 a d)}{d^2}\right )\right ) \int \frac {(e x)^{5/2}}{\sqrt {c+d x^2}} \, dx}{1989}\\ &=\frac {8 c^2 \left (51 a^2+\frac {b c (11 b c-42 a d)}{d^2}\right ) e (e x)^{3/2} \sqrt {c+d x^2}}{9945 d}+\frac {4 c \left (51 a^2+\frac {b c (11 b c-42 a d)}{d^2}\right ) (e x)^{7/2} \sqrt {c+d x^2}}{1989 e}+\frac {2 \left (51 a^2+\frac {b c (11 b c-42 a d)}{d^2}\right ) (e x)^{7/2} \left (c+d x^2\right )^{3/2}}{663 e}-\frac {2 b (11 b c-42 a d) (e x)^{7/2} \left (c+d x^2\right )^{5/2}}{357 d^2 e}+\frac {2 b^2 (e x)^{11/2} \left (c+d x^2\right )^{5/2}}{21 d e^3}-\frac {\left (4 c^3 \left (51 a^2+\frac {b c (11 b c-42 a d)}{d^2}\right ) e^2\right ) \int \frac {\sqrt {e x}}{\sqrt {c+d x^2}} \, dx}{3315 d}\\ &=\frac {8 c^2 \left (51 a^2+\frac {b c (11 b c-42 a d)}{d^2}\right ) e (e x)^{3/2} \sqrt {c+d x^2}}{9945 d}+\frac {4 c \left (51 a^2+\frac {b c (11 b c-42 a d)}{d^2}\right ) (e x)^{7/2} \sqrt {c+d x^2}}{1989 e}+\frac {2 \left (51 a^2+\frac {b c (11 b c-42 a d)}{d^2}\right ) (e x)^{7/2} \left (c+d x^2\right )^{3/2}}{663 e}-\frac {2 b (11 b c-42 a d) (e x)^{7/2} \left (c+d x^2\right )^{5/2}}{357 d^2 e}+\frac {2 b^2 (e x)^{11/2} \left (c+d x^2\right )^{5/2}}{21 d e^3}-\frac {\left (8 c^3 \left (51 a^2+\frac {b c (11 b c-42 a d)}{d^2}\right ) e\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{3315 d}\\ &=\frac {8 c^2 \left (51 a^2+\frac {b c (11 b c-42 a d)}{d^2}\right ) e (e x)^{3/2} \sqrt {c+d x^2}}{9945 d}+\frac {4 c \left (51 a^2+\frac {b c (11 b c-42 a d)}{d^2}\right ) (e x)^{7/2} \sqrt {c+d x^2}}{1989 e}+\frac {2 \left (51 a^2+\frac {b c (11 b c-42 a d)}{d^2}\right ) (e x)^{7/2} \left (c+d x^2\right )^{3/2}}{663 e}-\frac {2 b (11 b c-42 a d) (e x)^{7/2} \left (c+d x^2\right )^{5/2}}{357 d^2 e}+\frac {2 b^2 (e x)^{11/2} \left (c+d x^2\right )^{5/2}}{21 d e^3}-\frac {\left (8 c^{7/2} \left (51 a^2+\frac {b c (11 b c-42 a d)}{d^2}\right ) e^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{3315 d^{3/2}}+\frac {\left (8 c^{7/2} \left (51 a^2+\frac {b c (11 b c-42 a d)}{d^2}\right ) e^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 )}{3315 d^{3/2}}\\ &=\frac {8 c^2 \left (51 a^2+\frac {b c (11 b c-42 a d)}{d^2}\right ) e (e x)^{3/2} \sqrt {c+d x^2}}{9945 d}+\frac {4 c \left (51 a^2+\frac {b c (11 b c-42 a d)}{d^2}\right ) (e x)^{7/2} \sqrt {c+d x^2}}{1989 e}-\frac {8 c^3 \left (51 a^2+\frac {b c (11 b c-42 a d)}{d^2}\right ) e^2 \sqrt {e x} \sqrt {c+d x^2}}{3315 d^{3/2} \left (\sqrt {c}+\sqrt {d} x\right )}+\frac {2 \left (51 a^2+\frac {b c (11 b c-42 a d)}{d^2}\right ) (e x)^{7/2} \left (c+d x^2\right )^{3/2}}{663 e}-\frac {2 b (11 b c-42 a d) (e x)^{7/2} \left (c+d x^2\right )^{5/2}}{357 d^2 e}+\frac {2 b^2 (e x)^{11/2} \left (c+d x^2\right )^{5/2}}{21 d e^3}+\frac {8 c^{13/4} \left (51 a^2+\frac {b c (11 b c-42 a d)}{d^2}\right ) e^{5/2} \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 )}{3315 d^{7/4} \sqrt {c+d x^2}}-\frac {4 c^{13/4} \left (51 a^2+\frac {b c (11 b c-42 a d)}{d^2}\right ) e^{5/2} \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 )}{3315 d^{7/4} \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 = 210, normalized size = 0.40 \begin {gather*} \frac {2 e (e x)^{3/2} \left (\left (c+d x^2\right ) \left (357 a^2 d^2 \left (4 c^2+25 c d x^2+15 d^2 x^4\right )+42 a b d \left (-28 c^3+20 c^2 d x^2+285 c d^2 x^4+195 d^3 x^6\right )+b^2 \left (308 c^4-220 c^3 d x^2+180 c^2 d^2 x^4+4485 c d^3 x^6+3315 d^4 x^8\right )\right )-84 c^3 \left (11 b^2 c^2-42 a b c d+51 a^2 d^2\right ) \sqrt {1+\frac {c}{d x^2}} \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};-\frac {c}{d x^2}\right )\right )}{69615 d^3 \sqrt {c+d x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 743, normalized size = 1.40 Too large to display
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 = 0.48, size = 210, normalized size = 0.40 \begin {gather*} \frac {2 \, {\left (84 \, {\left (11 \, b^{2} c^{5} - 42 \, a b c^{4} d + 51 \, a^{2} c^{3} d^{2}\right )} \sqrt {d} e^{\frac {5}{2}} {\rm weierstrassZeta}\left (-\frac {4 \, c}{d}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right )\right ) + {\left (3315 \, b^{2} d^{5} x^{9} + 195 \, {\left (23 \, b^{2} c d^{4} + 42 \, a b d^{5}\right )} x^{7} + 45 \, {\left (4 \, b^{2} c^{2} d^{3} + 266 \, a b c d^{4} + 119 \, a^{2} d^{5}\right )} x^{5} - 5 \, {\left (44 \, b^{2} c^{3} d^{2} - 168 \, a b c^{2} d^{3} - 1785 \, a^{2} c d^{4}\right )} x^{3} + 28 \, {\left (11 \, b^{2} c^{4} d - 42 \, a b c^{3} d^{2} + 51 \, a^{2} c^{2} d^{3}\right )} x\right )} \sqrt {d x^{2} + c} \sqrt {x} e^{\frac {5}{2}}\right )}}{69615 \, d^{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 = 99.08, size = 306, normalized size = 0.58 \begin {gather*} \frac {a^{2} c^{\frac {3}{2}} e^{\frac {5}{2}} 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 \Gamma \left (\frac {11}{4}\right )} + \frac {a^{2} \sqrt {c} d e^{\frac {5}{2}} x^{\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 \Gamma \left (\frac {15}{4}\right )} + \frac {a b c^{\frac {3}{2}} e^{\frac {5}{2}} x^{\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 )}}{\Gamma \left (\frac {15}{4}\right )} + \frac {a b \sqrt {c} d e^{\frac {5}{2}} x^{\frac {15}{2}} \Gamma \left (\frac {15}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {15}{4} \\ \frac {19}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{\Gamma \left (\frac {19}{4}\right )} + \frac {b^{2} c^{\frac {3}{2}} e^{\frac {5}{2}} x^{\frac {15}{2}} \Gamma \left (\frac {15}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {15}{4} \\ \frac {19}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 \Gamma \left (\frac {19}{4}\right )} + \frac {b^{2} \sqrt {c} d e^{\frac {5}{2}} x^{\frac {19}{2}} \Gamma \left (\frac {19}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {19}{4} \\ \frac {23}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 \Gamma \left (\frac {23}{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 {\left (e\,x\right )}^{5/2}\,{\left (b\,x^2+a\right )}^2\,{\left (d\,x^2+c\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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