Optimal. Leaf size=137 \[ -\frac {5}{2} a^{3/2} B c \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )-\frac {5 \left (a+c x^2\right )^{3/2} (a B-A c x)}{6 x^2}-\frac {5 a c \sqrt {a+c x^2} (A-B x)}{2 x}-\frac {\left (a+c x^2\right )^{5/2} (A-B x)}{3 x^3}+\frac {5}{2} a A c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \]
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Rubi [A] time = 0.11, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {813, 844, 217, 206, 266, 63, 208} \begin {gather*} -\frac {5}{2} a^{3/2} B c \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )-\frac {\left (a+c x^2\right )^{5/2} (A-B x)}{3 x^3}-\frac {5 \left (a+c x^2\right )^{3/2} (a B-A c x)}{6 x^2}-\frac {5 a c \sqrt {a+c x^2} (A-B x)}{2 x}+\frac {5}{2} a A c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 206
Rule 208
Rule 217
Rule 266
Rule 813
Rule 844
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a+c x^2\right )^{5/2}}{x^4} \, dx &=-\frac {(A-B x) \left (a+c x^2\right )^{5/2}}{3 x^3}-\frac {5}{18} \int \frac {(-6 a B-6 A c x) \left (a+c x^2\right )^{3/2}}{x^3} \, dx\\ &=-\frac {5 (a B-A c x) \left (a+c x^2\right )^{3/2}}{6 x^2}-\frac {(A-B x) \left (a+c x^2\right )^{5/2}}{3 x^3}+\frac {5}{48} \int \frac {(24 a A c+24 a B c x) \sqrt {a+c x^2}}{x^2} \, dx\\ &=-\frac {5 a c (A-B x) \sqrt {a+c x^2}}{2 x}-\frac {5 (a B-A c x) \left (a+c x^2\right )^{3/2}}{6 x^2}-\frac {(A-B x) \left (a+c x^2\right )^{5/2}}{3 x^3}-\frac {5}{96} \int \frac {-48 a^2 B c-48 a A c^2 x}{x \sqrt {a+c x^2}} \, dx\\ &=-\frac {5 a c (A-B x) \sqrt {a+c x^2}}{2 x}-\frac {5 (a B-A c x) \left (a+c x^2\right )^{3/2}}{6 x^2}-\frac {(A-B x) \left (a+c x^2\right )^{5/2}}{3 x^3}+\frac {1}{2} \left (5 a^2 B c\right ) \int \frac {1}{x \sqrt {a+c x^2}} \, dx+\frac {1}{2} \left (5 a A c^2\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx\\ &=-\frac {5 a c (A-B x) \sqrt {a+c x^2}}{2 x}-\frac {5 (a B-A c x) \left (a+c x^2\right )^{3/2}}{6 x^2}-\frac {(A-B x) \left (a+c x^2\right )^{5/2}}{3 x^3}+\frac {1}{4} \left (5 a^2 B c\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+c x}} \, dx,x,x^2\right )+\frac {1}{2} \left (5 a A c^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )\\ &=-\frac {5 a c (A-B x) \sqrt {a+c x^2}}{2 x}-\frac {5 (a B-A c x) \left (a+c x^2\right )^{3/2}}{6 x^2}-\frac {(A-B x) \left (a+c x^2\right )^{5/2}}{3 x^3}+\frac {5}{2} a A c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )+\frac {1}{2} \left (5 a^2 B\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{c}+\frac {x^2}{c}} \, dx,x,\sqrt {a+c x^2}\right )\\ &=-\frac {5 a c (A-B x) \sqrt {a+c x^2}}{2 x}-\frac {5 (a B-A c x) \left (a+c x^2\right )^{3/2}}{6 x^2}-\frac {(A-B x) \left (a+c x^2\right )^{5/2}}{3 x^3}+\frac {5}{2} a A c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )-\frac {5}{2} a^{3/2} B c \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )\\ \end {align*}
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Mathematica [C] time = 0.03, size = 94, normalized size = 0.69 \begin {gather*} \frac {B c \left (a+c x^2\right )^{7/2} \, _2F_1\left (2,\frac {7}{2};\frac {9}{2};\frac {c x^2}{a}+1\right )}{7 a^2}-\frac {a^2 A \sqrt {a+c x^2} \, _2F_1\left (-\frac {5}{2},-\frac {3}{2};-\frac {1}{2};-\frac {c x^2}{a}\right )}{3 x^3 \sqrt {\frac {c x^2}{a}+1}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.59, size = 140, normalized size = 1.02 \begin {gather*} 5 a^{3/2} B c \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}-\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )+\frac {\sqrt {a+c x^2} \left (-2 a^2 A-3 a^2 B x-14 a A c x^2+14 a B c x^3+3 A c^2 x^4+2 B c^2 x^5\right )}{6 x^3}-\frac {5}{2} a A c^{3/2} \log \left (\sqrt {a+c x^2}-\sqrt {c} x\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 529, normalized size = 3.86 \begin {gather*} \left [\frac {15 \, A a c^{\frac {3}{2}} x^{3} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 15 \, B a^{\frac {3}{2}} c x^{3} \log \left (-\frac {c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (2 \, B c^{2} x^{5} + 3 \, A c^{2} x^{4} + 14 \, B a c x^{3} - 14 \, A a c x^{2} - 3 \, B a^{2} x - 2 \, A a^{2}\right )} \sqrt {c x^{2} + a}}{12 \, x^{3}}, -\frac {30 \, A a \sqrt {-c} c x^{3} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - 15 \, B a^{\frac {3}{2}} c x^{3} \log \left (-\frac {c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (2 \, B c^{2} x^{5} + 3 \, A c^{2} x^{4} + 14 \, B a c x^{3} - 14 \, A a c x^{2} - 3 \, B a^{2} x - 2 \, A a^{2}\right )} \sqrt {c x^{2} + a}}{12 \, x^{3}}, \frac {30 \, B \sqrt {-a} a c x^{3} \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x^{2} + a}}\right ) + 15 \, A a c^{\frac {3}{2}} x^{3} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 2 \, {\left (2 \, B c^{2} x^{5} + 3 \, A c^{2} x^{4} + 14 \, B a c x^{3} - 14 \, A a c x^{2} - 3 \, B a^{2} x - 2 \, A a^{2}\right )} \sqrt {c x^{2} + a}}{12 \, x^{3}}, -\frac {15 \, A a \sqrt {-c} c x^{3} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - 15 \, B \sqrt {-a} a c x^{3} \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x^{2} + a}}\right ) - {\left (2 \, B c^{2} x^{5} + 3 \, A c^{2} x^{4} + 14 \, B a c x^{3} - 14 \, A a c x^{2} - 3 \, B a^{2} x - 2 \, A a^{2}\right )} \sqrt {c x^{2} + a}}{6 \, x^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.23, size = 239, normalized size = 1.74 \begin {gather*} \frac {5 \, B a^{2} c \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} - \frac {5}{2} \, A a c^{\frac {3}{2}} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right ) + \frac {1}{6} \, {\left (14 \, B a c + {\left (2 \, B c^{2} x + 3 \, A c^{2}\right )} x\right )} \sqrt {c x^{2} + a} + \frac {3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{5} B a^{2} c + 18 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} A a^{2} c^{\frac {3}{2}} - 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} A a^{3} c^{\frac {3}{2}} - 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} B a^{4} c + 14 \, A a^{4} c^{\frac {3}{2}}}{3 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} - a\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 207, normalized size = 1.51 \begin {gather*} \frac {5 A a \,c^{\frac {3}{2}} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{2}-\frac {5 B \,a^{\frac {3}{2}} c \ln \left (\frac {2 a +2 \sqrt {c \,x^{2}+a}\, \sqrt {a}}{x}\right )}{2}+\frac {5 \sqrt {c \,x^{2}+a}\, A \,c^{2} x}{2}+\frac {5 \left (c \,x^{2}+a \right )^{\frac {3}{2}} A \,c^{2} x}{3 a}+\frac {5 \sqrt {c \,x^{2}+a}\, B a c}{2}+\frac {4 \left (c \,x^{2}+a \right )^{\frac {5}{2}} A \,c^{2} x}{3 a^{2}}+\frac {5 \left (c \,x^{2}+a \right )^{\frac {3}{2}} B c}{6}+\frac {\left (c \,x^{2}+a \right )^{\frac {5}{2}} B c}{2 a}-\frac {4 \left (c \,x^{2}+a \right )^{\frac {7}{2}} A c}{3 a^{2} x}-\frac {\left (c \,x^{2}+a \right )^{\frac {7}{2}} B}{2 a \,x^{2}}-\frac {\left (c \,x^{2}+a \right )^{\frac {7}{2}} A}{3 a \,x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.66, size = 169, normalized size = 1.23 \begin {gather*} \frac {5}{2} \, \sqrt {c x^{2} + a} A c^{2} x + \frac {5 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} A c^{2} x}{3 \, a} + \frac {5}{2} \, A a c^{\frac {3}{2}} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right ) - \frac {5}{2} \, B a^{\frac {3}{2}} c \operatorname {arsinh}\left (\frac {a}{\sqrt {a c} {\left | x \right |}}\right ) + \frac {5}{6} \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} B c + \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} B c}{2 \, a} + \frac {5}{2} \, \sqrt {c x^{2} + a} B a c - \frac {4 \, {\left (c x^{2} + a\right )}^{\frac {5}{2}} A c}{3 \, a x} - \frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}} B}{2 \, a x^{2}} - \frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}} A}{3 \, a x^{3}} \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 (c\,x^2+a\right )}^{5/2}\,\left (A+B\,x\right )}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 9.68, size = 277, normalized size = 2.02 \begin {gather*} - \frac {2 A a^{\frac {3}{2}} c}{x \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {A \sqrt {a} c^{2} x \sqrt {1 + \frac {c x^{2}}{a}}}{2} - \frac {2 A \sqrt {a} c^{2} x}{\sqrt {1 + \frac {c x^{2}}{a}}} - \frac {A a^{2} \sqrt {c} \sqrt {\frac {a}{c x^{2}} + 1}}{3 x^{2}} - \frac {A a c^{\frac {3}{2}} \sqrt {\frac {a}{c x^{2}} + 1}}{3} + \frac {5 A a c^{\frac {3}{2}} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )}}{2} - \frac {5 B a^{\frac {3}{2}} c \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {c} x} \right )}}{2} - \frac {B a^{2} \sqrt {c} \sqrt {\frac {a}{c x^{2}} + 1}}{2 x} + \frac {2 B a^{2} \sqrt {c}}{x \sqrt {\frac {a}{c x^{2}} + 1}} + \frac {2 B a c^{\frac {3}{2}} x}{\sqrt {\frac {a}{c x^{2}} + 1}} + B c^{2} \left (\begin {cases} \frac {\sqrt {a} x^{2}}{2} & \text {for}\: c = 0 \\\frac {\left (a + c x^{2}\right )^{\frac {3}{2}}}{3 c} & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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