Optimal. Leaf size=240 \[ \frac {2 a \left (a^2-b^2 c\right )^3}{b^8 d^4 \left (a+b \sqrt {c+d x}\right )}+\frac {2 \left (7 a^2-b^2 c\right ) \left (a^2-b^2 c\right )^2 \log \left (a+b \sqrt {c+d x}\right )}{b^8 d^4}-\frac {12 a \left (a^2-b^2 c\right )^2 \sqrt {c+d x}}{b^7 d^4}-\frac {4 a \left (2 a^2-3 b^2 c\right ) (c+d x)^{3/2}}{3 b^5 d^4}+\frac {3 \left (a^2-b^2 c\right ) (c+d x)^2}{2 b^4 d^4}+\frac {x \left (5 a^4-9 a^2 b^2 c+3 b^4 c^2\right )}{b^6 d^3}-\frac {4 a (c+d x)^{5/2}}{5 b^3 d^4}+\frac {(c+d x)^3}{3 b^2 d^4} \]
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Rubi [A] time = 0.28, antiderivative size = 240, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {371, 1398, 772} \[ \frac {x \left (-9 a^2 b^2 c+5 a^4+3 b^4 c^2\right )}{b^6 d^3}+\frac {2 a \left (a^2-b^2 c\right )^3}{b^8 d^4 \left (a+b \sqrt {c+d x}\right )}-\frac {12 a \left (a^2-b^2 c\right )^2 \sqrt {c+d x}}{b^7 d^4}+\frac {3 \left (a^2-b^2 c\right ) (c+d x)^2}{2 b^4 d^4}-\frac {4 a \left (2 a^2-3 b^2 c\right ) (c+d x)^{3/2}}{3 b^5 d^4}+\frac {2 \left (7 a^2-b^2 c\right ) \left (a^2-b^2 c\right )^2 \log \left (a+b \sqrt {c+d x}\right )}{b^8 d^4}-\frac {4 a (c+d x)^{5/2}}{5 b^3 d^4}+\frac {(c+d x)^3}{3 b^2 d^4} \]
Antiderivative was successfully verified.
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Rule 371
Rule 772
Rule 1398
Rubi steps
\begin {align*} \int \frac {x^3}{\left (a+b \sqrt {c+d x}\right )^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(-c+x)^3}{\left (a+b \sqrt {x}\right )^2} \, dx,x,c+d x\right )}{d^4}\\ &=\frac {2 \operatorname {Subst}\left (\int \frac {x \left (-c+x^2\right )^3}{(a+b x)^2} \, dx,x,\sqrt {c+d x}\right )}{d^4}\\ &=\frac {2 \operatorname {Subst}\left (\int \left (-\frac {6 a \left (a^2-b^2 c\right )^2}{b^7}+\frac {\left (5 a^4-9 a^2 b^2 c+3 b^4 c^2\right ) x}{b^6}-\frac {2 a \left (2 a^2-3 b^2 c\right ) x^2}{b^5}-\frac {3 \left (-a^2+b^2 c\right ) x^3}{b^4}-\frac {2 a x^4}{b^3}+\frac {x^5}{b^2}-\frac {a \left (a^2-b^2 c\right )^3}{b^7 (a+b x)^2}-\frac {\left (-7 a^2+b^2 c\right ) \left (-a^2+b^2 c\right )^2}{b^7 (a+b x)}\right ) \, dx,x,\sqrt {c+d x}\right )}{d^4}\\ &=\frac {\left (5 a^4-9 a^2 b^2 c+3 b^4 c^2\right ) x}{b^6 d^3}-\frac {12 a \left (a^2-b^2 c\right )^2 \sqrt {c+d x}}{b^7 d^4}-\frac {4 a \left (2 a^2-3 b^2 c\right ) (c+d x)^{3/2}}{3 b^5 d^4}+\frac {3 \left (a^2-b^2 c\right ) (c+d x)^2}{2 b^4 d^4}-\frac {4 a (c+d x)^{5/2}}{5 b^3 d^4}+\frac {(c+d x)^3}{3 b^2 d^4}+\frac {2 a \left (a^2-b^2 c\right )^3}{b^8 d^4 \left (a+b \sqrt {c+d x}\right )}+\frac {2 \left (a^2-b^2 c\right )^2 \left (7 a^2-b^2 c\right ) \log \left (a+b \sqrt {c+d x}\right )}{b^8 d^4}\\ \end {align*}
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Mathematica [A] time = 0.27, size = 273, normalized size = 1.14 \[ \frac {96 a^7-324 a^6 b \sqrt {c+d x}-6 a^5 b^2 (102 c+35 d x)+2 a^4 b^3 \sqrt {c+d x} (284 c+35 d x)+a^3 b^4 \left (856 c^2+380 c d x-35 d^2 x^2\right )-3 a^2 b^5 \sqrt {c+d x} \left (76 c^2+36 c d x-7 d^2 x^2\right )+60 \left (a^2-b^2 c\right )^2 \left (7 a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right ) \log \left (a+b \sqrt {c+d x}\right )-a b^6 \left (324 c^3+162 c^2 d x-33 c d^2 x^2+14 d^3 x^3\right )+5 b^7 d x \sqrt {c+d x} \left (6 c^2-3 c d x+2 d^2 x^2\right )}{30 b^8 d^4 \left (a+b \sqrt {c+d x}\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 392, normalized size = 1.63 \[ \frac {10 \, b^{8} d^{4} x^{4} + 55 \, b^{8} c^{4} - 220 \, a^{2} b^{6} c^{3} + 195 \, a^{4} b^{4} c^{2} + 30 \, a^{6} b^{2} c - 60 \, a^{8} - 5 \, {\left (b^{8} c - 7 \, a^{2} b^{6}\right )} d^{3} x^{3} + 15 \, {\left (b^{8} c^{2} - 8 \, a^{2} b^{6} c + 7 \, a^{4} b^{4}\right )} d^{2} x^{2} + 5 \, {\left (17 \, b^{8} c^{3} - 87 \, a^{2} b^{6} c^{2} + 96 \, a^{4} b^{4} c - 30 \, a^{6} b^{2}\right )} d x - 60 \, {\left (b^{8} c^{4} - 10 \, a^{2} b^{6} c^{3} + 24 \, a^{4} b^{4} c^{2} - 22 \, a^{6} b^{2} c + 7 \, a^{8} + {\left (b^{8} c^{3} - 9 \, a^{2} b^{6} c^{2} + 15 \, a^{4} b^{4} c - 7 \, a^{6} b^{2}\right )} d x\right )} \log \left (\sqrt {d x + c} b + a\right ) - 4 \, {\left (6 \, a b^{7} d^{3} x^{3} + 81 \, a b^{7} c^{3} - 271 \, a^{3} b^{5} c^{2} + 295 \, a^{5} b^{3} c - 105 \, a^{7} b - 2 \, {\left (6 \, a b^{7} c - 7 \, a^{3} b^{5}\right )} d^{2} x^{2} + 2 \, {\left (24 \, a b^{7} c^{2} - 61 \, a^{3} b^{5} c + 35 \, a^{5} b^{3}\right )} d x\right )} \sqrt {d x + c}}{30 \, {\left (b^{10} d^{5} x + {\left (b^{10} c - a^{2} b^{8}\right )} d^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.48, size = 324, normalized size = 1.35 \[ -\frac {2 \, {\left (b^{6} c^{3} - 9 \, a^{2} b^{4} c^{2} + 15 \, a^{4} b^{2} c - 7 \, a^{6}\right )} \log \left ({\left | \sqrt {d x + c} b + a \right |}\right )}{b^{8} d^{4}} - \frac {2 \, {\left (a b^{6} c^{3} - 3 \, a^{3} b^{4} c^{2} + 3 \, a^{5} b^{2} c - a^{7}\right )}}{{\left (\sqrt {d x + c} b + a\right )} b^{8} d^{4}} + \frac {10 \, {\left (d x + c\right )}^{3} b^{10} d^{20} - 45 \, {\left (d x + c\right )}^{2} b^{10} c d^{20} + 90 \, {\left (d x + c\right )} b^{10} c^{2} d^{20} - 24 \, {\left (d x + c\right )}^{\frac {5}{2}} a b^{9} d^{20} + 120 \, {\left (d x + c\right )}^{\frac {3}{2}} a b^{9} c d^{20} - 360 \, \sqrt {d x + c} a b^{9} c^{2} d^{20} + 45 \, {\left (d x + c\right )}^{2} a^{2} b^{8} d^{20} - 270 \, {\left (d x + c\right )} a^{2} b^{8} c d^{20} - 80 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{3} b^{7} d^{20} + 720 \, \sqrt {d x + c} a^{3} b^{7} c d^{20} + 150 \, {\left (d x + c\right )} a^{4} b^{6} d^{20} - 360 \, \sqrt {d x + c} a^{5} b^{5} d^{20}}{30 \, b^{12} d^{24}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 416, normalized size = 1.73 \[ \frac {x^{3}}{3 b^{2} d}-\frac {c \,x^{2}}{2 b^{2} d^{2}}+\frac {3 a^{2} x^{2}}{2 b^{4} d^{2}}-\frac {2 a \,c^{3}}{\left (a +\sqrt {d x +c}\, b \right ) b^{2} d^{4}}-\frac {2 c^{3} \ln \left (a +\sqrt {d x +c}\, b \right )}{b^{2} d^{4}}+\frac {c^{2} x}{b^{2} d^{3}}+\frac {6 a^{3} c^{2}}{\left (a +\sqrt {d x +c}\, b \right ) b^{4} d^{4}}+\frac {18 a^{2} c^{2} \ln \left (a +\sqrt {d x +c}\, b \right )}{b^{4} d^{4}}-\frac {6 a^{2} c x}{b^{4} d^{3}}+\frac {11 c^{3}}{6 b^{2} d^{4}}-\frac {6 a^{5} c}{\left (a +\sqrt {d x +c}\, b \right ) b^{6} d^{4}}-\frac {30 a^{4} c \ln \left (a +\sqrt {d x +c}\, b \right )}{b^{6} d^{4}}+\frac {5 a^{4} x}{b^{6} d^{3}}-\frac {15 a^{2} c^{2}}{2 b^{4} d^{4}}-\frac {12 \sqrt {d x +c}\, a \,c^{2}}{b^{3} d^{4}}+\frac {2 a^{7}}{\left (a +\sqrt {d x +c}\, b \right ) b^{8} d^{4}}+\frac {14 a^{6} \ln \left (a +\sqrt {d x +c}\, b \right )}{b^{8} d^{4}}+\frac {5 a^{4} c}{b^{6} d^{4}}+\frac {24 \sqrt {d x +c}\, a^{3} c}{b^{5} d^{4}}+\frac {4 \left (d x +c \right )^{\frac {3}{2}} a c}{b^{3} d^{4}}-\frac {12 \sqrt {d x +c}\, a^{5}}{b^{7} d^{4}}-\frac {8 \left (d x +c \right )^{\frac {3}{2}} a^{3}}{3 b^{5} d^{4}}-\frac {4 \left (d x +c \right )^{\frac {5}{2}} a}{5 b^{3} d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.93, size = 251, normalized size = 1.05 \[ -\frac {\frac {60 \, {\left (a b^{6} c^{3} - 3 \, a^{3} b^{4} c^{2} + 3 \, a^{5} b^{2} c - a^{7}\right )}}{\sqrt {d x + c} b^{9} + a b^{8}} - \frac {10 \, {\left (d x + c\right )}^{3} b^{5} - 24 \, {\left (d x + c\right )}^{\frac {5}{2}} a b^{4} - 45 \, {\left (b^{5} c - a^{2} b^{3}\right )} {\left (d x + c\right )}^{2} + 40 \, {\left (3 \, a b^{4} c - 2 \, a^{3} b^{2}\right )} {\left (d x + c\right )}^{\frac {3}{2}} + 30 \, {\left (3 \, b^{5} c^{2} - 9 \, a^{2} b^{3} c + 5 \, a^{4} b\right )} {\left (d x + c\right )} - 360 \, {\left (a b^{4} c^{2} - 2 \, a^{3} b^{2} c + a^{5}\right )} \sqrt {d x + c}}{b^{7}} + \frac {60 \, {\left (b^{6} c^{3} - 9 \, a^{2} b^{4} c^{2} + 15 \, a^{4} b^{2} c - 7 \, a^{6}\right )} \log \left (\sqrt {d x + c} b + a\right )}{b^{8}}}{30 \, d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 461, normalized size = 1.92 \[ \left (\frac {4\,a^3}{3\,b^5\,d^4}+\frac {2\,a\,\left (\frac {6\,c}{b^2\,d^4}-\frac {6\,a^2}{b^4\,d^4}\right )}{3\,b}\right )\,{\left (c+d\,x\right )}^{3/2}-\left (\frac {3\,c}{2\,b^2\,d^4}-\frac {3\,a^2}{2\,b^4\,d^4}\right )\,{\left (c+d\,x\right )}^2-\left (\frac {2\,a\,\left (\frac {a^2\,\left (\frac {6\,c}{b^2\,d^4}-\frac {6\,a^2}{b^4\,d^4}\right )}{b^2}-\frac {2\,a\,\left (\frac {4\,a^3}{b^5\,d^4}+\frac {2\,a\,\left (\frac {6\,c}{b^2\,d^4}-\frac {6\,a^2}{b^4\,d^4}\right )}{b}\right )}{b}+\frac {6\,c^2}{b^2\,d^4}\right )}{b}+\frac {a^2\,\left (\frac {4\,a^3}{b^5\,d^4}+\frac {2\,a\,\left (\frac {6\,c}{b^2\,d^4}-\frac {6\,a^2}{b^4\,d^4}\right )}{b}\right )}{b^2}\right )\,\sqrt {c+d\,x}+\frac {{\left (c+d\,x\right )}^3}{3\,b^2\,d^4}+\frac {2\,\left (a^7-3\,a^5\,b^2\,c+3\,a^3\,b^4\,c^2-a\,b^6\,c^3\right )}{b\,\left (b^8\,d^4\,\sqrt {c+d\,x}+a\,b^7\,d^4\right )}+d\,x\,\left (\frac {a^2\,\left (\frac {6\,c}{b^2\,d^4}-\frac {6\,a^2}{b^4\,d^4}\right )}{2\,b^2}-\frac {a\,\left (\frac {4\,a^3}{b^5\,d^4}+\frac {2\,a\,\left (\frac {6\,c}{b^2\,d^4}-\frac {6\,a^2}{b^4\,d^4}\right )}{b}\right )}{b}+\frac {3\,c^2}{b^2\,d^4}\right )+\frac {\ln \left (a+b\,\sqrt {c+d\,x}\right )\,\left (14\,a^6-30\,a^4\,b^2\,c+18\,a^2\,b^4\,c^2-2\,b^6\,c^3\right )}{b^8\,d^4}-\frac {4\,a\,{\left (c+d\,x\right )}^{5/2}}{5\,b^3\,d^4} \]
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 \[ \int \frac {x^{3}}{\left (a + b \sqrt {c + d x}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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