Optimal. Leaf size=174 \[ \frac {x (11 a d+4 b c)}{6 \sqrt {c+d x^2} (b c-a d)^3}+\frac {x (3 a d+2 b c)}{6 b \left (c+d x^2\right )^{3/2} (b c-a d)^2}+\frac {a x}{2 b \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} (b c-a d)}-\frac {\sqrt {a} (2 a d+3 b c) \tan ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 (b c-a d)^{7/2}} \]
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Rubi [A] time = 0.21, antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {470, 527, 12, 377, 205} \begin {gather*} \frac {x (11 a d+4 b c)}{6 \sqrt {c+d x^2} (b c-a d)^3}+\frac {x (3 a d+2 b c)}{6 b \left (c+d x^2\right )^{3/2} (b c-a d)^2}+\frac {a x}{2 b \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} (b c-a d)}-\frac {\sqrt {a} (2 a d+3 b c) \tan ^{-1}\left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 (b c-a d)^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 205
Rule 377
Rule 470
Rule 527
Rubi steps
\begin {align*} \int \frac {x^4}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx &=\frac {a x}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}-\frac {\int \frac {a c-2 (b c+a d) x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}} \, dx}{2 b (b c-a d)}\\ &=\frac {(2 b c+3 a d) x}{6 b (b c-a d)^2 \left (c+d x^2\right )^{3/2}}+\frac {a x}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}-\frac {\int \frac {5 a b c^2-2 b c (2 b c+3 a d) x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx}{6 b c (b c-a d)^2}\\ &=\frac {(2 b c+3 a d) x}{6 b (b c-a d)^2 \left (c+d x^2\right )^{3/2}}+\frac {a x}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}+\frac {(4 b c+11 a d) x}{6 (b c-a d)^3 \sqrt {c+d x^2}}-\frac {\int \frac {3 a b c^2 (3 b c+2 a d)}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{6 b c^2 (b c-a d)^3}\\ &=\frac {(2 b c+3 a d) x}{6 b (b c-a d)^2 \left (c+d x^2\right )^{3/2}}+\frac {a x}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}+\frac {(4 b c+11 a d) x}{6 (b c-a d)^3 \sqrt {c+d x^2}}-\frac {(a (3 b c+2 a d)) \int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{2 (b c-a d)^3}\\ &=\frac {(2 b c+3 a d) x}{6 b (b c-a d)^2 \left (c+d x^2\right )^{3/2}}+\frac {a x}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}+\frac {(4 b c+11 a d) x}{6 (b c-a d)^3 \sqrt {c+d x^2}}-\frac {(a (3 b c+2 a d)) \operatorname {Subst}\left (\int \frac {1}{a-(-b c+a d) x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{2 (b c-a d)^3}\\ &=\frac {(2 b c+3 a d) x}{6 b (b c-a d)^2 \left (c+d x^2\right )^{3/2}}+\frac {a x}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}+\frac {(4 b c+11 a d) x}{6 (b c-a d)^3 \sqrt {c+d x^2}}-\frac {\sqrt {a} (3 b c+2 a d) \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 (b c-a d)^{7/2}}\\ \end {align*}
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Mathematica [C] time = 1.17, size = 133, normalized size = 0.76 \begin {gather*} \frac {x^5 \left (\frac {8 x^2 \left (c+d x^2\right ) (b c-a d) \, _2F_1\left (2,3;\frac {11}{2};\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right )}{a+b x^2}+9 c \left (7 c+2 d x^2\right ) \, _2F_1\left (1,2;\frac {9}{2};\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right )\right )}{315 c^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 1.43, size = 189, normalized size = 1.09 \begin {gather*} \frac {\left (2 a^{3/2} d+3 \sqrt {a} b c\right ) \tan ^{-1}\left (\frac {a \sqrt {d}-b x \sqrt {c+d x^2}+b \sqrt {d} x^2}{\sqrt {a} \sqrt {b c-a d}}\right )}{2 (b c-a d)^{7/2}}-\frac {-6 a^2 c d x-8 a^2 d^2 x^3-9 a b c^2 x-16 a b c d x^3-11 a b d^2 x^5-6 b^2 c^2 x^3-4 b^2 c d x^5}{6 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} (b c-a d)^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 2.59, size = 1008, normalized size = 5.79 \begin {gather*} \left [-\frac {3 \, {\left ({\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{6} + 3 \, a b c^{3} + 2 \, a^{2} c^{2} d + {\left (6 \, b^{2} c^{2} d + 7 \, a b c d^{2} + 2 \, a^{2} d^{3}\right )} x^{4} + {\left (3 \, b^{2} c^{3} + 8 \, a b c^{2} d + 4 \, a^{2} c d^{2}\right )} x^{2}\right )} \sqrt {-\frac {a}{b c - a d}} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} + 4 \, {\left ({\left (b^{2} c^{2} - 3 \, a b c d + 2 \, a^{2} d^{2}\right )} x^{3} - {\left (a b c^{2} - a^{2} c d\right )} x\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {a}{b c - a d}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - 4 \, {\left ({\left (4 \, b^{2} c d + 11 \, a b d^{2}\right )} x^{5} + 2 \, {\left (3 \, b^{2} c^{2} + 8 \, a b c d + 4 \, a^{2} d^{2}\right )} x^{3} + 3 \, {\left (3 \, a b c^{2} + 2 \, a^{2} c d\right )} x\right )} \sqrt {d x^{2} + c}}{24 \, {\left (a b^{3} c^{5} - 3 \, a^{2} b^{2} c^{4} d + 3 \, a^{3} b c^{3} d^{2} - a^{4} c^{2} d^{3} + {\left (b^{4} c^{3} d^{2} - 3 \, a b^{3} c^{2} d^{3} + 3 \, a^{2} b^{2} c d^{4} - a^{3} b d^{5}\right )} x^{6} + {\left (2 \, b^{4} c^{4} d - 5 \, a b^{3} c^{3} d^{2} + 3 \, a^{2} b^{2} c^{2} d^{3} + a^{3} b c d^{4} - a^{4} d^{5}\right )} x^{4} + {\left (b^{4} c^{5} - a b^{3} c^{4} d - 3 \, a^{2} b^{2} c^{3} d^{2} + 5 \, a^{3} b c^{2} d^{3} - 2 \, a^{4} c d^{4}\right )} x^{2}\right )}}, \frac {3 \, {\left ({\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{6} + 3 \, a b c^{3} + 2 \, a^{2} c^{2} d + {\left (6 \, b^{2} c^{2} d + 7 \, a b c d^{2} + 2 \, a^{2} d^{3}\right )} x^{4} + {\left (3 \, b^{2} c^{3} + 8 \, a b c^{2} d + 4 \, a^{2} c d^{2}\right )} x^{2}\right )} \sqrt {\frac {a}{b c - a d}} \arctan \left (-\frac {{\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c} \sqrt {\frac {a}{b c - a d}}}{2 \, {\left (a d x^{3} + a c x\right )}}\right ) + 2 \, {\left ({\left (4 \, b^{2} c d + 11 \, a b d^{2}\right )} x^{5} + 2 \, {\left (3 \, b^{2} c^{2} + 8 \, a b c d + 4 \, a^{2} d^{2}\right )} x^{3} + 3 \, {\left (3 \, a b c^{2} + 2 \, a^{2} c d\right )} x\right )} \sqrt {d x^{2} + c}}{12 \, {\left (a b^{3} c^{5} - 3 \, a^{2} b^{2} c^{4} d + 3 \, a^{3} b c^{3} d^{2} - a^{4} c^{2} d^{3} + {\left (b^{4} c^{3} d^{2} - 3 \, a b^{3} c^{2} d^{3} + 3 \, a^{2} b^{2} c d^{4} - a^{3} b d^{5}\right )} x^{6} + {\left (2 \, b^{4} c^{4} d - 5 \, a b^{3} c^{3} d^{2} + 3 \, a^{2} b^{2} c^{2} d^{3} + a^{3} b c d^{4} - a^{4} d^{5}\right )} x^{4} + {\left (b^{4} c^{5} - a b^{3} c^{4} d - 3 \, a^{2} b^{2} c^{3} d^{2} + 5 \, a^{3} b c^{2} d^{3} - 2 \, a^{4} c d^{4}\right )} x^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 4.49, size = 594, normalized size = 3.41 \begin {gather*} \frac {{\left (\frac {2 \, {\left (b^{4} c^{5} d^{2} - a b^{3} c^{4} d^{3} - 3 \, a^{2} b^{2} c^{3} d^{4} + 5 \, a^{3} b c^{2} d^{5} - 2 \, a^{4} c d^{6}\right )} x^{2}}{b^{6} c^{7} d - 6 \, a b^{5} c^{6} d^{2} + 15 \, a^{2} b^{4} c^{5} d^{3} - 20 \, a^{3} b^{3} c^{4} d^{4} + 15 \, a^{4} b^{2} c^{3} d^{5} - 6 \, a^{5} b c^{2} d^{6} + a^{6} c d^{7}} + \frac {3 \, {\left (b^{4} c^{6} d - 2 \, a b^{3} c^{5} d^{2} + 2 \, a^{3} b c^{3} d^{4} - a^{4} c^{2} d^{5}\right )}}{b^{6} c^{7} d - 6 \, a b^{5} c^{6} d^{2} + 15 \, a^{2} b^{4} c^{5} d^{3} - 20 \, a^{3} b^{3} c^{4} d^{4} + 15 \, a^{4} b^{2} c^{3} d^{5} - 6 \, a^{5} b c^{2} d^{6} + a^{6} c d^{7}}\right )} x}{3 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}}} + \frac {{\left (3 \, a b c \sqrt {d} + 2 \, a^{2} d^{\frac {3}{2}}\right )} \arctan \left (\frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{2 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {a b c d - a^{2} d^{2}}} - \frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a b c \sqrt {d} - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a^{2} d^{\frac {3}{2}} - a b c^{2} \sqrt {d}}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} {\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c + 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a d + b c^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 2463, normalized size = 14.16
result 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*} \int \frac {x^{4}}{{\left (b x^{2} + a\right )}^{2} {\left (d x^{2} + c\right )}^{\frac {5}{2}}}\,{d x} \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 {x^4}{{\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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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4}}{\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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