Optimal. Leaf size=135 \[ \frac {2 a (b c-a d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{9/2}}-\frac {2 a \sqrt {c+d x} (b c-a d)^2}{b^4}-\frac {2 a (c+d x)^{3/2} (b c-a d)}{3 b^3}-\frac {2 a (c+d x)^{5/2}}{5 b^2}+\frac {2 (c+d x)^{7/2}}{7 b d} \]
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Rubi [A] time = 0.07, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {80, 50, 63, 208} \begin {gather*} -\frac {2 a (c+d x)^{5/2}}{5 b^2}-\frac {2 a (c+d x)^{3/2} (b c-a d)}{3 b^3}-\frac {2 a \sqrt {c+d x} (b c-a d)^2}{b^4}+\frac {2 a (b c-a d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{9/2}}+\frac {2 (c+d x)^{7/2}}{7 b d} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 80
Rule 208
Rubi steps
\begin {align*} \int \frac {x (c+d x)^{5/2}}{a+b x} \, dx &=\frac {2 (c+d x)^{7/2}}{7 b d}-\frac {a \int \frac {(c+d x)^{5/2}}{a+b x} \, dx}{b}\\ &=-\frac {2 a (c+d x)^{5/2}}{5 b^2}+\frac {2 (c+d x)^{7/2}}{7 b d}-\frac {(a (b c-a d)) \int \frac {(c+d x)^{3/2}}{a+b x} \, dx}{b^2}\\ &=-\frac {2 a (b c-a d) (c+d x)^{3/2}}{3 b^3}-\frac {2 a (c+d x)^{5/2}}{5 b^2}+\frac {2 (c+d x)^{7/2}}{7 b d}-\frac {\left (a (b c-a d)^2\right ) \int \frac {\sqrt {c+d x}}{a+b x} \, dx}{b^3}\\ &=-\frac {2 a (b c-a d)^2 \sqrt {c+d x}}{b^4}-\frac {2 a (b c-a d) (c+d x)^{3/2}}{3 b^3}-\frac {2 a (c+d x)^{5/2}}{5 b^2}+\frac {2 (c+d x)^{7/2}}{7 b d}-\frac {\left (a (b c-a d)^3\right ) \int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx}{b^4}\\ &=-\frac {2 a (b c-a d)^2 \sqrt {c+d x}}{b^4}-\frac {2 a (b c-a d) (c+d x)^{3/2}}{3 b^3}-\frac {2 a (c+d x)^{5/2}}{5 b^2}+\frac {2 (c+d x)^{7/2}}{7 b d}-\frac {\left (2 a (b c-a d)^3\right ) \operatorname {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{b^4 d}\\ &=-\frac {2 a (b c-a d)^2 \sqrt {c+d x}}{b^4}-\frac {2 a (b c-a d) (c+d x)^{3/2}}{3 b^3}-\frac {2 a (c+d x)^{5/2}}{5 b^2}+\frac {2 (c+d x)^{7/2}}{7 b d}+\frac {2 a (b c-a d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.31, size = 126, normalized size = 0.93 \begin {gather*} -\frac {2 a (b c-a d) \left (\sqrt {b} \sqrt {c+d x} (-3 a d+4 b c+b d x)-3 (b c-a d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )\right )}{3 b^{9/2}}-\frac {2 a (c+d x)^{5/2}}{5 b^2}+\frac {2 (c+d x)^{7/2}}{7 b d} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.16, size = 160, normalized size = 1.19 \begin {gather*} \frac {2 \sqrt {c+d x} \left (-105 a^3 d^3+35 a^2 b d^2 (c+d x)+210 a^2 b c d^2-105 a b^2 c^2 d-21 a b^2 d (c+d x)^2-35 a b^2 c d (c+d x)+15 b^3 (c+d x)^3\right )}{105 b^4 d}-\frac {2 a (a d-b c)^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x} \sqrt {a d-b c}}{b c-a d}\right )}{b^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.33, size = 411, normalized size = 3.04 \begin {gather*} \left [\frac {105 \, {\left (a b^{2} c^{2} d - 2 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \sqrt {\frac {b c - a d}{b}} \log \left (\frac {b d x + 2 \, b c - a d + 2 \, \sqrt {d x + c} b \sqrt {\frac {b c - a d}{b}}}{b x + a}\right ) + 2 \, {\left (15 \, b^{3} d^{3} x^{3} + 15 \, b^{3} c^{3} - 161 \, a b^{2} c^{2} d + 245 \, a^{2} b c d^{2} - 105 \, a^{3} d^{3} + 3 \, {\left (15 \, b^{3} c d^{2} - 7 \, a b^{2} d^{3}\right )} x^{2} + {\left (45 \, b^{3} c^{2} d - 77 \, a b^{2} c d^{2} + 35 \, a^{2} b d^{3}\right )} x\right )} \sqrt {d x + c}}{105 \, b^{4} d}, \frac {2 \, {\left (105 \, {\left (a b^{2} c^{2} d - 2 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {\sqrt {d x + c} b \sqrt {-\frac {b c - a d}{b}}}{b c - a d}\right ) + {\left (15 \, b^{3} d^{3} x^{3} + 15 \, b^{3} c^{3} - 161 \, a b^{2} c^{2} d + 245 \, a^{2} b c d^{2} - 105 \, a^{3} d^{3} + 3 \, {\left (15 \, b^{3} c d^{2} - 7 \, a b^{2} d^{3}\right )} x^{2} + {\left (45 \, b^{3} c^{2} d - 77 \, a b^{2} c d^{2} + 35 \, a^{2} b d^{3}\right )} x\right )} \sqrt {d x + c}\right )}}{105 \, b^{4} d}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.31, size = 212, normalized size = 1.57 \begin {gather*} -\frac {2 \, {\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{\sqrt {-b^{2} c + a b d} b^{4}} + \frac {2 \, {\left (15 \, {\left (d x + c\right )}^{\frac {7}{2}} b^{6} d^{6} - 21 \, {\left (d x + c\right )}^{\frac {5}{2}} a b^{5} d^{7} - 35 \, {\left (d x + c\right )}^{\frac {3}{2}} a b^{5} c d^{7} - 105 \, \sqrt {d x + c} a b^{5} c^{2} d^{7} + 35 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{2} b^{4} d^{8} + 210 \, \sqrt {d x + c} a^{2} b^{4} c d^{8} - 105 \, \sqrt {d x + c} a^{3} b^{3} d^{9}\right )}}{105 \, b^{7} d^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 291, normalized size = 2.16 \begin {gather*} \frac {2 a^{4} d^{3} \arctan \left (\frac {\sqrt {d x +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )}{\sqrt {\left (a d -b c \right ) b}\, b^{4}}-\frac {6 a^{3} c \,d^{2} \arctan \left (\frac {\sqrt {d x +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )}{\sqrt {\left (a d -b c \right ) b}\, b^{3}}+\frac {6 a^{2} c^{2} d \arctan \left (\frac {\sqrt {d x +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )}{\sqrt {\left (a d -b c \right ) b}\, b^{2}}-\frac {2 a \,c^{3} \arctan \left (\frac {\sqrt {d x +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )}{\sqrt {\left (a d -b c \right ) b}\, b}-\frac {2 \sqrt {d x +c}\, a^{3} d^{2}}{b^{4}}+\frac {4 \sqrt {d x +c}\, a^{2} c d}{b^{3}}-\frac {2 \sqrt {d x +c}\, a \,c^{2}}{b^{2}}+\frac {2 \left (d x +c \right )^{\frac {3}{2}} a^{2} d}{3 b^{3}}-\frac {2 \left (d x +c \right )^{\frac {3}{2}} a c}{3 b^{2}}-\frac {2 \left (d x +c \right )^{\frac {5}{2}} a}{5 b^{2}}+\frac {2 \left (d x +c \right )^{\frac {7}{2}}}{7 b d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.40, size = 246, normalized size = 1.82 \begin {gather*} \frac {2\,{\left (c+d\,x\right )}^{7/2}}{7\,b\,d}-\left (\frac {2\,c}{5\,b\,d}+\frac {2\,\left (a\,d^2-b\,c\,d\right )}{5\,b^2\,d^2}\right )\,{\left (c+d\,x\right )}^{5/2}+\frac {2\,a\,\mathrm {atan}\left (\frac {a\,\sqrt {b}\,{\left (a\,d-b\,c\right )}^{5/2}\,\sqrt {c+d\,x}}{a^4\,d^3-3\,a^3\,b\,c\,d^2+3\,a^2\,b^2\,c^2\,d-a\,b^3\,c^3}\right )\,{\left (a\,d-b\,c\right )}^{5/2}}{b^{9/2}}-\frac {{\left (a\,d^2-b\,c\,d\right )}^2\,\left (\frac {2\,c}{b\,d}+\frac {2\,\left (a\,d^2-b\,c\,d\right )}{b^2\,d^2}\right )\,\sqrt {c+d\,x}}{b^2\,d^2}+\frac {\left (a\,d^2-b\,c\,d\right )\,\left (\frac {2\,c}{b\,d}+\frac {2\,\left (a\,d^2-b\,c\,d\right )}{b^2\,d^2}\right )\,{\left (c+d\,x\right )}^{3/2}}{3\,b\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 41.79, size = 148, normalized size = 1.10 \begin {gather*} - \frac {2 a \left (c + d x\right )^{\frac {5}{2}}}{5 b^{2}} + \frac {2 a \left (a d - b c\right )^{3} \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {\frac {a d - b c}{b}}} \right )}}{b^{5} \sqrt {\frac {a d - b c}{b}}} + \frac {2 \left (c + d x\right )^{\frac {7}{2}}}{7 b d} + \frac {\left (c + d x\right )^{\frac {3}{2}} \left (2 a^{2} d - 2 a b c\right )}{3 b^{3}} + \frac {\sqrt {c + d x} \left (- 2 a^{3} d^{2} + 4 a^{2} b c d - 2 a b^{2} c^{2}\right )}{b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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