Optimal. Leaf size=169 \[ -\frac {2 a^2 (b c-a d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{11/2}}+\frac {2 a^2 \sqrt {c+d x} (b c-a d)^2}{b^5}+\frac {2 a^2 (c+d x)^{3/2} (b c-a d)}{3 b^4}+\frac {2 a^2 (c+d x)^{5/2}}{5 b^3}-\frac {2 (c+d x)^{7/2} (a d+b c)}{7 b^2 d^2}+\frac {2 (c+d x)^{9/2}}{9 b d^2} \]
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Rubi [A] time = 0.11, antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {88, 50, 63, 208} \begin {gather*} \frac {2 a^2 (c+d x)^{5/2}}{5 b^3}+\frac {2 a^2 (c+d x)^{3/2} (b c-a d)}{3 b^4}+\frac {2 a^2 \sqrt {c+d x} (b c-a d)^2}{b^5}-\frac {2 a^2 (b c-a d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{11/2}}-\frac {2 (c+d x)^{7/2} (a d+b c)}{7 b^2 d^2}+\frac {2 (c+d x)^{9/2}}{9 b d^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 88
Rule 208
Rubi steps
\begin {align*} \int \frac {x^2 (c+d x)^{5/2}}{a+b x} \, dx &=\int \left (\frac {(-b c-a d) (c+d x)^{5/2}}{b^2 d}+\frac {a^2 (c+d x)^{5/2}}{b^2 (a+b x)}+\frac {(c+d x)^{7/2}}{b d}\right ) \, dx\\ &=-\frac {2 (b c+a d) (c+d x)^{7/2}}{7 b^2 d^2}+\frac {2 (c+d x)^{9/2}}{9 b d^2}+\frac {a^2 \int \frac {(c+d x)^{5/2}}{a+b x} \, dx}{b^2}\\ &=\frac {2 a^2 (c+d x)^{5/2}}{5 b^3}-\frac {2 (b c+a d) (c+d x)^{7/2}}{7 b^2 d^2}+\frac {2 (c+d x)^{9/2}}{9 b d^2}+\frac {\left (a^2 (b c-a d)\right ) \int \frac {(c+d x)^{3/2}}{a+b x} \, dx}{b^3}\\ &=\frac {2 a^2 (b c-a d) (c+d x)^{3/2}}{3 b^4}+\frac {2 a^2 (c+d x)^{5/2}}{5 b^3}-\frac {2 (b c+a d) (c+d x)^{7/2}}{7 b^2 d^2}+\frac {2 (c+d x)^{9/2}}{9 b d^2}+\frac {\left (a^2 (b c-a d)^2\right ) \int \frac {\sqrt {c+d x}}{a+b x} \, dx}{b^4}\\ &=\frac {2 a^2 (b c-a d)^2 \sqrt {c+d x}}{b^5}+\frac {2 a^2 (b c-a d) (c+d x)^{3/2}}{3 b^4}+\frac {2 a^2 (c+d x)^{5/2}}{5 b^3}-\frac {2 (b c+a d) (c+d x)^{7/2}}{7 b^2 d^2}+\frac {2 (c+d x)^{9/2}}{9 b d^2}+\frac {\left (a^2 (b c-a d)^3\right ) \int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx}{b^5}\\ &=\frac {2 a^2 (b c-a d)^2 \sqrt {c+d x}}{b^5}+\frac {2 a^2 (b c-a d) (c+d x)^{3/2}}{3 b^4}+\frac {2 a^2 (c+d x)^{5/2}}{5 b^3}-\frac {2 (b c+a d) (c+d x)^{7/2}}{7 b^2 d^2}+\frac {2 (c+d x)^{9/2}}{9 b d^2}+\frac {\left (2 a^2 (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^5 d}\\ &=\frac {2 a^2 (b c-a d)^2 \sqrt {c+d x}}{b^5}+\frac {2 a^2 (b c-a d) (c+d x)^{3/2}}{3 b^4}+\frac {2 a^2 (c+d x)^{5/2}}{5 b^3}-\frac {2 (b c+a d) (c+d x)^{7/2}}{7 b^2 d^2}+\frac {2 (c+d x)^{9/2}}{9 b d^2}-\frac {2 a^2 (b c-a d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{11/2}}\\ \end {align*}
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Mathematica [A] time = 0.30, size = 151, normalized size = 0.89 \begin {gather*} \frac {2 \left (\frac {105 a^2 (a d-b c) \left (3 (b c-a d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )-\sqrt {b} \sqrt {c+d x} (-3 a d+4 b c+b d x)\right )}{b^{5/2}}+63 a^2 (c+d x)^{5/2}-\frac {45 b (c+d x)^{7/2} (a d+b c)}{d^2}+\frac {35 b^2 (c+d x)^{9/2}}{d^2}\right )}{315 b^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.22, size = 201, normalized size = 1.19 \begin {gather*} \frac {2 a^2 (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^{11/2}}+\frac {2 \sqrt {c+d x} \left (315 a^4 d^4-105 a^3 b d^3 (c+d x)-630 a^3 b c d^3+315 a^2 b^2 c^2 d^2+63 a^2 b^2 d^2 (c+d x)^2+105 a^2 b^2 c d^2 (c+d x)-45 a b^3 d (c+d x)^3+35 b^4 (c+d x)^4-45 b^4 c (c+d x)^3\right )}{315 b^5 d^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.21, size = 552, normalized size = 3.27 \begin {gather*} \left [\frac {315 \, {\left (a^{2} b^{2} c^{2} d^{2} - 2 \, a^{3} b c d^{3} + a^{4} d^{4}\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 (35 \, b^{4} d^{4} x^{4} - 10 \, b^{4} c^{4} - 45 \, a b^{3} c^{3} d + 483 \, a^{2} b^{2} c^{2} d^{2} - 735 \, a^{3} b c d^{3} + 315 \, a^{4} d^{4} + 5 \, {\left (19 \, b^{4} c d^{3} - 9 \, a b^{3} d^{4}\right )} x^{3} + 3 \, {\left (25 \, b^{4} c^{2} d^{2} - 45 \, a b^{3} c d^{3} + 21 \, a^{2} b^{2} d^{4}\right )} x^{2} + {\left (5 \, b^{4} c^{3} d - 135 \, a b^{3} c^{2} d^{2} + 231 \, a^{2} b^{2} c d^{3} - 105 \, a^{3} b d^{4}\right )} x\right )} \sqrt {d x + c}}{315 \, b^{5} d^{2}}, -\frac {2 \, {\left (315 \, {\left (a^{2} b^{2} c^{2} d^{2} - 2 \, a^{3} b c d^{3} + a^{4} d^{4}\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 (35 \, b^{4} d^{4} x^{4} - 10 \, b^{4} c^{4} - 45 \, a b^{3} c^{3} d + 483 \, a^{2} b^{2} c^{2} d^{2} - 735 \, a^{3} b c d^{3} + 315 \, a^{4} d^{4} + 5 \, {\left (19 \, b^{4} c d^{3} - 9 \, a b^{3} d^{4}\right )} x^{3} + 3 \, {\left (25 \, b^{4} c^{2} d^{2} - 45 \, a b^{3} c d^{3} + 21 \, a^{2} b^{2} d^{4}\right )} x^{2} + {\left (5 \, b^{4} c^{3} d - 135 \, a b^{3} c^{2} d^{2} + 231 \, a^{2} b^{2} c d^{3} - 105 \, a^{3} b d^{4}\right )} x\right )} \sqrt {d x + c}\right )}}{315 \, b^{5} d^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.32, size = 252, normalized size = 1.49 \begin {gather*} \frac {2 \, {\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} 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^{5}} + \frac {2 \, {\left (35 \, {\left (d x + c\right )}^{\frac {9}{2}} b^{8} d^{16} - 45 \, {\left (d x + c\right )}^{\frac {7}{2}} b^{8} c d^{16} - 45 \, {\left (d x + c\right )}^{\frac {7}{2}} a b^{7} d^{17} + 63 \, {\left (d x + c\right )}^{\frac {5}{2}} a^{2} b^{6} d^{18} + 105 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{2} b^{6} c d^{18} + 315 \, \sqrt {d x + c} a^{2} b^{6} c^{2} d^{18} - 105 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{3} b^{5} d^{19} - 630 \, \sqrt {d x + c} a^{3} b^{5} c d^{19} + 315 \, \sqrt {d x + c} a^{4} b^{4} d^{20}\right )}}{315 \, b^{9} d^{18}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 331, normalized size = 1.96 \begin {gather*} -\frac {2 a^{5} 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^{5}}+\frac {6 a^{4} 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^{4}}-\frac {6 a^{3} 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^{3}}+\frac {2 a^{2} 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^{2}}+\frac {2 \sqrt {d x +c}\, a^{4} d^{2}}{b^{5}}-\frac {4 \sqrt {d x +c}\, a^{3} c d}{b^{4}}+\frac {2 \sqrt {d x +c}\, a^{2} c^{2}}{b^{3}}-\frac {2 \left (d x +c \right )^{\frac {3}{2}} a^{3} d}{3 b^{4}}+\frac {2 \left (d x +c \right )^{\frac {3}{2}} a^{2} c}{3 b^{3}}+\frac {2 \left (d x +c \right )^{\frac {5}{2}} a^{2}}{5 b^{3}}-\frac {2 \left (d x +c \right )^{\frac {7}{2}} a}{7 b^{2} d}-\frac {2 \left (d x +c \right )^{\frac {7}{2}} c}{7 b \,d^{2}}+\frac {2 \left (d x +c \right )^{\frac {9}{2}}}{9 b \,d^{2}} \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 = 397, normalized size = 2.35 \begin {gather*} \left (\frac {2\,c^2}{5\,b\,d^2}+\frac {\left (\frac {4\,c}{b\,d^2}+\frac {2\,\left (a\,d^3-b\,c\,d^2\right )}{b^2\,d^4}\right )\,\left (a\,d^3-b\,c\,d^2\right )}{5\,b\,d^2}\right )\,{\left (c+d\,x\right )}^{5/2}-\left (\frac {4\,c}{7\,b\,d^2}+\frac {2\,\left (a\,d^3-b\,c\,d^2\right )}{7\,b^2\,d^4}\right )\,{\left (c+d\,x\right )}^{7/2}+\frac {2\,{\left (c+d\,x\right )}^{9/2}}{9\,b\,d^2}-\frac {2\,a^2\,\mathrm {atan}\left (\frac {a^2\,\sqrt {b}\,{\left (a\,d-b\,c\right )}^{5/2}\,\sqrt {c+d\,x}}{a^5\,d^3-3\,a^4\,b\,c\,d^2+3\,a^3\,b^2\,c^2\,d-a^2\,b^3\,c^3}\right )\,{\left (a\,d-b\,c\right )}^{5/2}}{b^{11/2}}-\frac {\left (\frac {2\,c^2}{b\,d^2}+\frac {\left (\frac {4\,c}{b\,d^2}+\frac {2\,\left (a\,d^3-b\,c\,d^2\right )}{b^2\,d^4}\right )\,\left (a\,d^3-b\,c\,d^2\right )}{b\,d^2}\right )\,\left (a\,d^3-b\,c\,d^2\right )\,{\left (c+d\,x\right )}^{3/2}}{3\,b\,d^2}+\frac {\left (\frac {2\,c^2}{b\,d^2}+\frac {\left (\frac {4\,c}{b\,d^2}+\frac {2\,\left (a\,d^3-b\,c\,d^2\right )}{b^2\,d^4}\right )\,\left (a\,d^3-b\,c\,d^2\right )}{b\,d^2}\right )\,{\left (a\,d^3-b\,c\,d^2\right )}^2\,\sqrt {c+d\,x}}{b^2\,d^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 57.68, size = 185, normalized size = 1.09 \begin {gather*} \frac {2 a^{2} \left (c + d x\right )^{\frac {5}{2}}}{5 b^{3}} - \frac {2 a^{2} \left (a d - b c\right )^{3} \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {\frac {a d - b c}{b}}} \right )}}{b^{6} \sqrt {\frac {a d - b c}{b}}} + \frac {2 \left (c + d x\right )^{\frac {9}{2}}}{9 b d^{2}} + \frac {\left (c + d x\right )^{\frac {7}{2}} \left (- 2 a d - 2 b c\right )}{7 b^{2} d^{2}} + \frac {\left (c + d x\right )^{\frac {3}{2}} \left (- 2 a^{3} d + 2 a^{2} b c\right )}{3 b^{4}} + \frac {\sqrt {c + d x} \left (2 a^{4} d^{2} - 4 a^{3} b c d + 2 a^{2} b^{2} c^{2}\right )}{b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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