Optimal. Leaf size=530 \[ -\frac {4 c^{13/4} e^{5/2} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (51 a^2 d^2+b c (11 b c-42 a d)\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{3315 d^{15/4} \sqrt {c+d x^2}}+\frac {8 c^{13/4} e^{5/2} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (51 a^2 d^2+b c (11 b c-42 a d)\right ) E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{3315 d^{15/4} \sqrt {c+d x^2}}-\frac {8 c^3 e^2 \sqrt {e x} \sqrt {c+d x^2} \left (51 a^2 d^2+b c (11 b c-42 a d)\right )}{3315 d^{7/2} \left (\sqrt {c}+\sqrt {d} x\right )}+\frac {8 c^2 e (e x)^{3/2} \sqrt {c+d x^2} \left (51 a^2 d^2+b c (11 b c-42 a d)\right )}{9945 d^3}+\frac {2 (e x)^{7/2} \left (c+d x^2\right )^{3/2} \left (51 a^2 d^2+b c (11 b c-42 a d)\right )}{663 d^2 e}+\frac {4 c (e x)^{7/2} \sqrt {c+d x^2} \left (51 a^2 d^2+b c (11 b c-42 a d)\right )}{1989 d^2 e}-\frac {2 b (e x)^{7/2} \left (c+d x^2\right )^{5/2} (11 b c-42 a d)}{357 d^2 e}+\frac {2 b^2 (e x)^{11/2} \left (c+d x^2\right )^{5/2}}{21 d e^3} \]
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Rubi [A] time = 0.56, 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 = {464, 459, 279, 321, 329, 305, 220, 1196} \[ -\frac {8 c^3 e^2 \sqrt {e x} \sqrt {c+d x^2} \left (51 a^2 d^2+b c (11 b c-42 a d)\right )}{3315 d^{7/2} \left (\sqrt {c}+\sqrt {d} x\right )}-\frac {4 c^{13/4} e^{5/2} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (51 a^2 d^2+b c (11 b c-42 a d)\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{3315 d^{15/4} \sqrt {c+d x^2}}+\frac {8 c^{13/4} e^{5/2} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (51 a^2 d^2+b c (11 b c-42 a d)\right ) E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{3315 d^{15/4} \sqrt {c+d x^2}}+\frac {8 c^2 e (e x)^{3/2} \sqrt {c+d x^2} \left (51 a^2 d^2+b c (11 b c-42 a d)\right )}{9945 d^3}+\frac {2 (e x)^{7/2} \left (c+d x^2\right )^{3/2} \left (51 a^2 d^2+b c (11 b c-42 a d)\right )}{663 d^2 e}+\frac {4 c (e x)^{7/2} \sqrt {c+d x^2} \left (51 a^2 d^2+b c (11 b c-42 a d)\right )}{1989 d^2 e}-\frac {2 b (e x)^{7/2} \left (c+d x^2\right )^{5/2} (11 b c-42 a d)}{357 d^2 e}+\frac {2 b^2 (e x)^{11/2} \left (c+d x^2\right )^{5/2}}{21 d e^3} \]
Antiderivative was successfully verified.
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Rule 220
Rule 279
Rule 305
Rule 321
Rule 329
Rule 459
Rule 464
Rule 1196
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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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] time = 0.18, size = 210, normalized size = 0.40 \[ \frac {2 e (e x)^{3/2} \left (\left (c+d x^2\right ) \left (357 a^2 d^2 \left (4 c^2+25 c d x^2+15 d^2 x^4\right )+42 a b d \left (-28 c^3+20 c^2 d x^2+285 c d^2 x^4+195 d^3 x^6\right )+b^2 \left (308 c^4-220 c^3 d x^2+180 c^2 d^2 x^4+4485 c d^3 x^6+3315 d^4 x^8\right )\right )-84 c^3 \sqrt {\frac {c}{d x^2}+1} \left (51 a^2 d^2-42 a b c d+11 b^2 c^2\right ) \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};-\frac {c}{d x^2}\right )\right )}{69615 d^3 \sqrt {c+d x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{2} d e^{2} x^{8} + {\left (b^{2} c + 2 \, a b d\right )} e^{2} x^{6} + a^{2} c e^{2} x^{2} + {\left (2 \, a b c + a^{2} d\right )} e^{2} x^{4}\right )} \sqrt {d x^{2} + c} \sqrt {e x}, x\right ) \]
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 \[ \int {\left (b x^{2} + a\right )}^{2} {\left (d x^{2} + c\right )}^{\frac {3}{2}} \left (e x\right )^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 743, normalized size = 1.40 \[ -\frac {2 \sqrt {e x}\, \left (-3315 b^{2} d^{6} x^{12}-8190 a b \,d^{6} x^{10}-7800 b^{2} c \,d^{5} x^{10}-5355 a^{2} d^{6} x^{8}-20160 a b c \,d^{5} x^{8}-4665 b^{2} c^{2} d^{4} x^{8}-14280 a^{2} c \,d^{5} x^{6}-12810 a b \,c^{2} d^{4} x^{6}+40 b^{2} c^{3} d^{3} x^{6}-10353 a^{2} c^{2} d^{4} x^{4}+336 a b \,c^{3} d^{3} x^{4}-88 b^{2} c^{4} d^{2} x^{4}-1428 a^{2} c^{3} d^{3} x^{2}+1176 a b \,c^{4} d^{2} x^{2}-308 b^{2} c^{5} d \,x^{2}+4284 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, a^{2} c^{4} d^{2} \EllipticE \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )-2142 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, a^{2} c^{4} d^{2} \EllipticF \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )-3528 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, a b \,c^{5} d \EllipticE \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )+1764 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, a b \,c^{5} d \EllipticF \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )+924 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, b^{2} c^{6} \EllipticE \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )-462 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {d x}{\sqrt {-c d}}}\, b^{2} c^{6} \EllipticF \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )\right ) e^{2}}{69615 \sqrt {d \,x^{2}+c}\, d^{4} x} \]
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 \[ \int {\left (b x^{2} + a\right )}^{2} {\left (d x^{2} + c\right )}^{\frac {3}{2}} \left (e x\right )^{\frac {5}{2}}\,{d x} \]
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 \[ \int {\left (e\,x\right )}^{5/2}\,{\left (b\,x^2+a\right )}^2\,{\left (d\,x^2+c\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 148.82, size = 306, normalized size = 0.58 \[ \frac {a^{2} c^{\frac {3}{2}} e^{\frac {5}{2}} x^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 \Gamma \left (\frac {11}{4}\right )} + \frac {a^{2} \sqrt {c} d e^{\frac {5}{2}} x^{\frac {11}{2}} \Gamma \left (\frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {11}{4} \\ \frac {15}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 \Gamma \left (\frac {15}{4}\right )} + \frac {a b c^{\frac {3}{2}} e^{\frac {5}{2}} x^{\frac {11}{2}} \Gamma \left (\frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {11}{4} \\ \frac {15}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{\Gamma \left (\frac {15}{4}\right )} + \frac {a b \sqrt {c} d e^{\frac {5}{2}} x^{\frac {15}{2}} \Gamma \left (\frac {15}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {15}{4} \\ \frac {19}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{\Gamma \left (\frac {19}{4}\right )} + \frac {b^{2} c^{\frac {3}{2}} e^{\frac {5}{2}} x^{\frac {15}{2}} \Gamma \left (\frac {15}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {15}{4} \\ \frac {19}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 \Gamma \left (\frac {19}{4}\right )} + \frac {b^{2} \sqrt {c} d e^{\frac {5}{2}} x^{\frac {19}{2}} \Gamma \left (\frac {19}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {19}{4} \\ \frac {23}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 \Gamma \left (\frac {23}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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