Optimal. Leaf size=436 \[ \frac {(b c-a d)^2 (e x)^{7/2}}{c d^2 e \sqrt {c+d x^2}}-\frac {\left (77 b^2 c^2-126 a b c d+45 a^2 d^2\right ) e (e x)^{3/2} \sqrt {c+d x^2}}{45 c d^3}+\frac {2 b^2 (e x)^{7/2} \sqrt {c+d x^2}}{9 d^2 e}+\frac {\left (77 b^2 c^2-126 a b c d+45 a^2 d^2\right ) e^2 \sqrt {e x} \sqrt {c+d x^2}}{15 d^{7/2} \left (\sqrt {c}+\sqrt {d} x\right )}-\frac {\sqrt [4]{c} \left (77 b^2 c^2-126 a b c d+45 a^2 d^2\right ) e^{5/2} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{15 d^{15/4} \sqrt {c+d x^2}}+\frac {\sqrt [4]{c} \left (77 b^2 c^2-126 a b c d+45 a^2 d^2\right ) e^{5/2} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{30 d^{15/4} \sqrt {c+d x^2}} \]
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Rubi [A]
time = 0.27, antiderivative size = 436, 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 = {474, 470, 327,
335, 311, 226, 1210} \begin {gather*} \frac {\sqrt [4]{c} 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 (45 a^2 d^2-126 a b c d+77 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 )}{30 d^{15/4} \sqrt {c+d x^2}}-\frac {\sqrt [4]{c} 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 (45 a^2 d^2-126 a b c d+77 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 )}{15 d^{15/4} \sqrt {c+d x^2}}+\frac {e^2 \sqrt {e x} \sqrt {c+d x^2} \left (45 a^2 d^2-126 a b c d+77 b^2 c^2\right )}{15 d^{7/2} \left (\sqrt {c}+\sqrt {d} x\right )}-\frac {e (e x)^{3/2} \sqrt {c+d x^2} \left (45 a^2 d^2-126 a b c d+77 b^2 c^2\right )}{45 c d^3}+\frac {(e x)^{7/2} (b c-a d)^2}{c d^2 e \sqrt {c+d x^2}}+\frac {2 b^2 (e x)^{7/2} \sqrt {c+d x^2}}{9 d^2 e} \end {gather*}
Antiderivative was successfully verified.
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Rule 226
Rule 311
Rule 327
Rule 335
Rule 470
Rule 474
Rule 1210
Rubi steps
\begin {align*} \int \frac {(e x)^{5/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx &=\frac {(b c-a d)^2 (e x)^{7/2}}{c d^2 e \sqrt {c+d x^2}}-\frac {\int \frac {(e x)^{5/2} \left (\frac {1}{2} \left (-2 a^2 d^2+7 (b c-a d)^2\right )-b^2 c d x^2\right )}{\sqrt {c+d x^2}} \, dx}{c d^2}\\ &=\frac {(b c-a d)^2 (e x)^{7/2}}{c d^2 e \sqrt {c+d x^2}}+\frac {2 b^2 (e x)^{7/2} \sqrt {c+d x^2}}{9 d^2 e}-\frac {\left (77 b^2 c^2-126 a b c d+45 a^2 d^2\right ) \int \frac {(e x)^{5/2}}{\sqrt {c+d x^2}} \, dx}{18 c d^2}\\ &=\frac {(b c-a d)^2 (e x)^{7/2}}{c d^2 e \sqrt {c+d x^2}}-\frac {\left (77 b^2 c^2-126 a b c d+45 a^2 d^2\right ) e (e x)^{3/2} \sqrt {c+d x^2}}{45 c d^3}+\frac {2 b^2 (e x)^{7/2} \sqrt {c+d x^2}}{9 d^2 e}+\frac {\left (\left (77 b^2 c^2-126 a b c d+45 a^2 d^2\right ) e^2\right ) \int \frac {\sqrt {e x}}{\sqrt {c+d x^2}} \, dx}{30 d^3}\\ &=\frac {(b c-a d)^2 (e x)^{7/2}}{c d^2 e \sqrt {c+d x^2}}-\frac {\left (77 b^2 c^2-126 a b c d+45 a^2 d^2\right ) e (e x)^{3/2} \sqrt {c+d x^2}}{45 c d^3}+\frac {2 b^2 (e x)^{7/2} \sqrt {c+d x^2}}{9 d^2 e}+\frac {\left (\left (77 b^2 c^2-126 a b c d+45 a^2 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 )}{15 d^3}\\ &=\frac {(b c-a d)^2 (e x)^{7/2}}{c d^2 e \sqrt {c+d x^2}}-\frac {\left (77 b^2 c^2-126 a b c d+45 a^2 d^2\right ) e (e x)^{3/2} \sqrt {c+d x^2}}{45 c d^3}+\frac {2 b^2 (e x)^{7/2} \sqrt {c+d x^2}}{9 d^2 e}+\frac {\left (\sqrt {c} \left (77 b^2 c^2-126 a b c d+45 a^2 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 )}{15 d^{7/2}}-\frac {\left (\sqrt {c} \left (77 b^2 c^2-126 a b c d+45 a^2 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 )}{15 d^{7/2}}\\ &=\frac {(b c-a d)^2 (e x)^{7/2}}{c d^2 e \sqrt {c+d x^2}}-\frac {\left (77 b^2 c^2-126 a b c d+45 a^2 d^2\right ) e (e x)^{3/2} \sqrt {c+d x^2}}{45 c d^3}+\frac {2 b^2 (e x)^{7/2} \sqrt {c+d x^2}}{9 d^2 e}+\frac {\left (77 b^2 c^2-126 a b c d+45 a^2 d^2\right ) e^2 \sqrt {e x} \sqrt {c+d x^2}}{15 d^{7/2} \left (\sqrt {c}+\sqrt {d} x\right )}-\frac {\sqrt [4]{c} \left (77 b^2 c^2-126 a b c d+45 a^2 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 )}{15 d^{15/4} \sqrt {c+d x^2}}+\frac {\sqrt [4]{c} \left (77 b^2 c^2-126 a b c d+45 a^2 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 )}{30 d^{15/4} \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 = 20.13, size = 133, normalized size = 0.31 \begin {gather*} \frac {e (e x)^{3/2} \left (-45 a^2 d^2+18 a b d \left (7 c+2 d x^2\right )+b^2 \left (-77 c^2-22 c d x^2+10 d^2 x^4\right )+3 \left (77 b^2 c^2-126 a b c d+45 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 )}{45 d^3 \sqrt {c+d x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 618, normalized size = 1.42
method | result | size |
elliptic | \(\frac {\sqrt {e x \left (d \,x^{2}+c \right )}\, \sqrt {e x}\, \left (-\frac {e^{3} x^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{d^{3} \sqrt {\left (x^{2}+\frac {c}{d}\right ) d e x}}+\frac {2 b^{2} e^{2} x^{3} \sqrt {d e \,x^{3}+c e x}}{9 d^{2}}+\frac {2 \left (\frac {b \left (2 a d -b c \right ) e^{3}}{d^{2}}-\frac {7 b^{2} e^{3} c}{9 d^{2}}\right ) x \sqrt {d e \,x^{3}+c e x}}{5 d e}+\frac {\left (\frac {3 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) e^{3}}{2 d^{3}}-\frac {3 \left (\frac {b \left (2 a d -b c \right ) e^{3}}{d^{2}}-\frac {7 b^{2} e^{3} c}{9 d^{2}}\right ) c}{5 d}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \left (-\frac {2 \sqrt {-c d}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\sqrt {-c d}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}\right )}{d \sqrt {d e \,x^{3}+c e x}}\right )}{e x \sqrt {d \,x^{2}+c}}\) | \(386\) |
risch | \(\frac {2 b \,x^{2} \left (5 b d \,x^{2}+18 a d -16 b c \right ) \sqrt {d \,x^{2}+c}\, e^{3}}{45 d^{3} \sqrt {e x}}+\frac {\left (\frac {\left (15 a^{2} d^{2}-48 a b c d +31 b^{2} c^{2}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \left (-\frac {2 \sqrt {-c d}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\sqrt {-c d}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}\right )}{d \sqrt {d e \,x^{3}+c e x}}-15 c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {x^{2}}{c \sqrt {\left (x^{2}+\frac {c}{d}\right ) d e x}}-\frac {\sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \left (-\frac {2 \sqrt {-c d}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\sqrt {-c d}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}\right )}{2 c d \sqrt {d e \,x^{3}+c e x}}\right )\right ) e^{3} \sqrt {e x \left (d \,x^{2}+c \right )}}{15 d^{3} \sqrt {e x}\, \sqrt {d \,x^{2}+c}}\) | \(471\) |
default | \(\frac {e^{2} \sqrt {e x}\, \left (20 b^{2} x^{6} d^{3}+270 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \EllipticE \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c \,d^{2}-756 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \EllipticE \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b \,c^{2} d +462 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \EllipticE \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{3}-135 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \EllipticF \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c \,d^{2}+378 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \EllipticF \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b \,c^{2} d -231 \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \EllipticF \left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{3}+72 a b \,d^{3} x^{4}-44 b^{2} c \,d^{2} x^{4}-90 a^{2} d^{3} x^{2}+252 a b c \,d^{2} x^{2}-154 b^{2} c^{2} d \,x^{2}\right )}{90 x \sqrt {d \,x^{2}+c}\, d^{4}}\) | \(618\) |
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 = 0.29, size = 177, normalized size = 0.41 \begin {gather*} -\frac {3 \, {\left (77 \, b^{2} c^{3} - 126 \, a b c^{2} d + 45 \, a^{2} c d^{2} + {\left (77 \, b^{2} c^{2} d - 126 \, a b c d^{2} + 45 \, a^{2} d^{3}\right )} x^{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 (10 \, b^{2} d^{3} x^{5} - 2 \, {\left (11 \, b^{2} c d^{2} - 18 \, a b d^{3}\right )} x^{3} - {\left (77 \, b^{2} c^{2} d - 126 \, a b c d^{2} + 45 \, a^{2} d^{3}\right )} x\right )} \sqrt {d x^{2} + c} \sqrt {x} e^{\frac {5}{2}}}{45 \, {\left (d^{5} x^{2} + c d^{4}\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 {\left (e x\right )^{\frac {5}{2}} \left (a + b x^{2}\right )^{2}}{\left (c + d x^{2}\right )^{\frac {3}{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 {{\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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