Optimal. Leaf size=204 \[ -\frac {a^2 (c+d x)^{7/2}}{b^2 (a+b x) (b c-a d)}+\frac {a (4 b c-9 a d) (b c-a d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{11/2}}-\frac {a \sqrt {c+d x} (4 b c-9 a d) (b c-a d)}{b^5}-\frac {a (c+d x)^{3/2} (4 b c-9 a d)}{3 b^4}-\frac {a (c+d x)^{5/2} (4 b c-9 a d)}{5 b^3 (b c-a d)}+\frac {2 (c+d x)^{7/2}}{7 b^2 d} \]
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Rubi [A] time = 0.22, antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {89, 80, 50, 63, 208} \[ -\frac {a^2 (c+d x)^{7/2}}{b^2 (a+b x) (b c-a d)}-\frac {a (c+d x)^{5/2} (4 b c-9 a d)}{5 b^3 (b c-a d)}-\frac {a (c+d x)^{3/2} (4 b c-9 a d)}{3 b^4}-\frac {a \sqrt {c+d x} (4 b c-9 a d) (b c-a d)}{b^5}+\frac {a (4 b c-9 a d) (b c-a d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{11/2}}+\frac {2 (c+d x)^{7/2}}{7 b^2 d} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 80
Rule 89
Rule 208
Rubi steps
\begin {align*} \int \frac {x^2 (c+d x)^{5/2}}{(a+b x)^2} \, dx &=-\frac {a^2 (c+d x)^{7/2}}{b^2 (b c-a d) (a+b x)}+\frac {\int \frac {(c+d x)^{5/2} \left (-\frac {1}{2} a (2 b c-7 a d)+b (b c-a d) x\right )}{a+b x} \, dx}{b^2 (b c-a d)}\\ &=\frac {2 (c+d x)^{7/2}}{7 b^2 d}-\frac {a^2 (c+d x)^{7/2}}{b^2 (b c-a d) (a+b x)}-\frac {(a (4 b c-9 a d)) \int \frac {(c+d x)^{5/2}}{a+b x} \, dx}{2 b^2 (b c-a d)}\\ &=-\frac {a (4 b c-9 a d) (c+d x)^{5/2}}{5 b^3 (b c-a d)}+\frac {2 (c+d x)^{7/2}}{7 b^2 d}-\frac {a^2 (c+d x)^{7/2}}{b^2 (b c-a d) (a+b x)}-\frac {(a (4 b c-9 a d)) \int \frac {(c+d x)^{3/2}}{a+b x} \, dx}{2 b^3}\\ &=-\frac {a (4 b c-9 a d) (c+d x)^{3/2}}{3 b^4}-\frac {a (4 b c-9 a d) (c+d x)^{5/2}}{5 b^3 (b c-a d)}+\frac {2 (c+d x)^{7/2}}{7 b^2 d}-\frac {a^2 (c+d x)^{7/2}}{b^2 (b c-a d) (a+b x)}-\frac {(a (4 b c-9 a d) (b c-a d)) \int \frac {\sqrt {c+d x}}{a+b x} \, dx}{2 b^4}\\ &=-\frac {a (4 b c-9 a d) (b c-a d) \sqrt {c+d x}}{b^5}-\frac {a (4 b c-9 a d) (c+d x)^{3/2}}{3 b^4}-\frac {a (4 b c-9 a d) (c+d x)^{5/2}}{5 b^3 (b c-a d)}+\frac {2 (c+d x)^{7/2}}{7 b^2 d}-\frac {a^2 (c+d x)^{7/2}}{b^2 (b c-a d) (a+b x)}-\frac {\left (a (4 b c-9 a d) (b c-a d)^2\right ) \int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx}{2 b^5}\\ &=-\frac {a (4 b c-9 a d) (b c-a d) \sqrt {c+d x}}{b^5}-\frac {a (4 b c-9 a d) (c+d x)^{3/2}}{3 b^4}-\frac {a (4 b c-9 a d) (c+d x)^{5/2}}{5 b^3 (b c-a d)}+\frac {2 (c+d x)^{7/2}}{7 b^2 d}-\frac {a^2 (c+d x)^{7/2}}{b^2 (b c-a d) (a+b x)}-\frac {\left (a (4 b c-9 a d) (b c-a d)^2\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 {a (4 b c-9 a d) (b c-a d) \sqrt {c+d x}}{b^5}-\frac {a (4 b c-9 a d) (c+d x)^{3/2}}{3 b^4}-\frac {a (4 b c-9 a d) (c+d x)^{5/2}}{5 b^3 (b c-a d)}+\frac {2 (c+d x)^{7/2}}{7 b^2 d}-\frac {a^2 (c+d x)^{7/2}}{b^2 (b c-a d) (a+b x)}+\frac {a (4 b c-9 a d) (b c-a d)^{3/2} \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 = 0.37, size = 181, normalized size = 0.89 \[ \frac {\frac {a (9 a d-4 b c) \left (\sqrt {b} \sqrt {c+d x} \left (15 a^2 d^2-5 a b d (7 c+d x)+b^2 \left (23 c^2+11 c d x+3 d^2 x^2\right )\right )-15 (b c-a d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )\right )}{15 b^{7/2}}-\frac {a^2 (c+d x)^{7/2}}{a+b x}+\frac {2 (c+d x)^{7/2} (b c-a d)}{7 d}}{b^2 (b c-a d)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 607, normalized size = 2.98 \[ \left [\frac {105 \, {\left (4 \, a^{2} b^{2} c^{2} d - 13 \, a^{3} b c d^{2} + 9 \, a^{4} d^{3} + {\left (4 \, a b^{3} c^{2} d - 13 \, a^{2} b^{2} c d^{2} + 9 \, a^{3} b d^{3}\right )} x\right )} \sqrt {\frac {b c - a d}{b}} \log \left (\frac {b d x + 2 \, b c - a d + 2 \, \sqrt {d x + c} b \sqrt {\frac {b c - a d}{b}}}{b x + a}\right ) + 2 \, {\left (30 \, b^{4} d^{3} x^{4} + 30 \, a b^{3} c^{3} - 749 \, a^{2} b^{2} c^{2} d + 1680 \, a^{3} b c d^{2} - 945 \, a^{4} d^{3} + 18 \, {\left (5 \, b^{4} c d^{2} - 3 \, a b^{3} d^{3}\right )} x^{3} + 2 \, {\left (45 \, b^{4} c^{2} d - 109 \, a b^{3} c d^{2} + 63 \, a^{2} b^{2} d^{3}\right )} x^{2} + 2 \, {\left (15 \, b^{4} c^{3} - 277 \, a b^{3} c^{2} d + 581 \, a^{2} b^{2} c d^{2} - 315 \, a^{3} b d^{3}\right )} x\right )} \sqrt {d x + c}}{210 \, {\left (b^{6} d x + a b^{5} d\right )}}, \frac {105 \, {\left (4 \, a^{2} b^{2} c^{2} d - 13 \, a^{3} b c d^{2} + 9 \, a^{4} d^{3} + {\left (4 \, a b^{3} c^{2} d - 13 \, a^{2} b^{2} c d^{2} + 9 \, a^{3} b d^{3}\right )} x\right )} \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {\sqrt {d x + c} b \sqrt {-\frac {b c - a d}{b}}}{b c - a d}\right ) + {\left (30 \, b^{4} d^{3} x^{4} + 30 \, a b^{3} c^{3} - 749 \, a^{2} b^{2} c^{2} d + 1680 \, a^{3} b c d^{2} - 945 \, a^{4} d^{3} + 18 \, {\left (5 \, b^{4} c d^{2} - 3 \, a b^{3} d^{3}\right )} x^{3} + 2 \, {\left (45 \, b^{4} c^{2} d - 109 \, a b^{3} c d^{2} + 63 \, a^{2} b^{2} d^{3}\right )} x^{2} + 2 \, {\left (15 \, b^{4} c^{3} - 277 \, a b^{3} c^{2} d + 581 \, a^{2} b^{2} c d^{2} - 315 \, a^{3} b d^{3}\right )} x\right )} \sqrt {d x + c}}{105 \, {\left (b^{6} d x + a b^{5} d\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.31, size = 285, normalized size = 1.40 \[ -\frac {{\left (4 \, a b^{3} c^{3} - 17 \, a^{2} b^{2} c^{2} d + 22 \, a^{3} b c d^{2} - 9 \, a^{4} d^{3}\right )} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{\sqrt {-b^{2} c + a b d} b^{5}} - \frac {\sqrt {d x + c} a^{2} b^{2} c^{2} d - 2 \, \sqrt {d x + c} a^{3} b c d^{2} + \sqrt {d x + c} a^{4} d^{3}}{{\left ({\left (d x + c\right )} b - b c + a d\right )} b^{5}} + \frac {2 \, {\left (15 \, {\left (d x + c\right )}^{\frac {7}{2}} b^{12} d^{6} - 42 \, {\left (d x + c\right )}^{\frac {5}{2}} a b^{11} d^{7} - 70 \, {\left (d x + c\right )}^{\frac {3}{2}} a b^{11} c d^{7} - 210 \, \sqrt {d x + c} a b^{11} c^{2} d^{7} + 105 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{2} b^{10} d^{8} + 630 \, \sqrt {d x + c} a^{2} b^{10} c d^{8} - 420 \, \sqrt {d x + c} a^{3} b^{9} d^{9}\right )}}{105 \, b^{14} d^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 377, normalized size = 1.85 \[ \frac {9 a^{4} d^{3} \arctan \left (\frac {\sqrt {d x +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )}{\sqrt {\left (a d -b c \right ) b}\, b^{5}}-\frac {22 a^{3} c \,d^{2} \arctan \left (\frac {\sqrt {d x +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )}{\sqrt {\left (a d -b c \right ) b}\, b^{4}}+\frac {17 a^{2} c^{2} d \arctan \left (\frac {\sqrt {d x +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )}{\sqrt {\left (a d -b c \right ) b}\, b^{3}}-\frac {4 a \,c^{3} \arctan \left (\frac {\sqrt {d x +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )}{\sqrt {\left (a d -b c \right ) b}\, b^{2}}-\frac {\sqrt {d x +c}\, a^{4} d^{3}}{\left (b d x +a d \right ) b^{5}}+\frac {2 \sqrt {d x +c}\, a^{3} c \,d^{2}}{\left (b d x +a d \right ) b^{4}}-\frac {\sqrt {d x +c}\, a^{2} c^{2} d}{\left (b d x +a d \right ) b^{3}}-\frac {8 \sqrt {d x +c}\, a^{3} d^{2}}{b^{5}}+\frac {12 \sqrt {d x +c}\, a^{2} c d}{b^{4}}-\frac {4 \sqrt {d x +c}\, a \,c^{2}}{b^{3}}+\frac {2 \left (d x +c \right )^{\frac {3}{2}} a^{2} d}{b^{4}}-\frac {4 \left (d x +c \right )^{\frac {3}{2}} a c}{3 b^{3}}-\frac {4 \left (d x +c \right )^{\frac {5}{2}} a}{5 b^{3}}+\frac {2 \left (d x +c \right )^{\frac {7}{2}}}{7 b^{2} d} \]
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 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.46, size = 419, normalized size = 2.05 \[ \left (\frac {\left (\frac {4\,c}{b^2\,d}+\frac {4\,\left (a\,d-b\,c\right )}{b^3\,d}\right )\,{\left (a\,d-b\,c\right )}^2}{b^2}-\frac {2\,\left (a\,d-b\,c\right )\,\left (\frac {2\,c^2}{b^2\,d}-\frac {2\,{\left (a\,d-b\,c\right )}^2}{b^4\,d}+\frac {2\,\left (\frac {4\,c}{b^2\,d}+\frac {4\,\left (a\,d-b\,c\right )}{b^3\,d}\right )\,\left (a\,d-b\,c\right )}{b}\right )}{b}\right )\,\sqrt {c+d\,x}+{\left (c+d\,x\right )}^{3/2}\,\left (\frac {2\,c^2}{3\,b^2\,d}-\frac {2\,{\left (a\,d-b\,c\right )}^2}{3\,b^4\,d}+\frac {2\,\left (\frac {4\,c}{b^2\,d}+\frac {4\,\left (a\,d-b\,c\right )}{b^3\,d}\right )\,\left (a\,d-b\,c\right )}{3\,b}\right )-\left (\frac {4\,c}{5\,b^2\,d}+\frac {4\,\left (a\,d-b\,c\right )}{5\,b^3\,d}\right )\,{\left (c+d\,x\right )}^{5/2}+\frac {2\,{\left (c+d\,x\right )}^{7/2}}{7\,b^2\,d}-\frac {\sqrt {c+d\,x}\,\left (a^4\,d^3-2\,a^3\,b\,c\,d^2+a^2\,b^2\,c^2\,d\right )}{b^6\,\left (c+d\,x\right )-b^6\,c+a\,b^5\,d}+\frac {a\,\mathrm {atan}\left (\frac {a\,\sqrt {b}\,{\left (a\,d-b\,c\right )}^{3/2}\,\left (9\,a\,d-4\,b\,c\right )\,\sqrt {c+d\,x}}{9\,a^4\,d^3-22\,a^3\,b\,c\,d^2+17\,a^2\,b^2\,c^2\,d-4\,a\,b^3\,c^3}\right )\,{\left (a\,d-b\,c\right )}^{3/2}\,\left (9\,a\,d-4\,b\,c\right )}{b^{11/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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