Optimal. Leaf size=230 \[ -\frac {a \left (a^4-3 a^2 b^2 c+3 b^4 c^2\right ) x}{b^6 d^3}+\frac {2 \left (a^2-b^2 c\right )^3 \sqrt {c+d x}}{b^7 d^4}+\frac {2 \left (a^4-3 a^2 b^2 c+3 b^4 c^2\right ) (c+d x)^{3/2}}{3 b^5 d^4}-\frac {a \left (a^2-3 b^2 c\right ) (c+d x)^2}{2 b^4 d^4}+\frac {2 \left (a^2-3 b^2 c\right ) (c+d x)^{5/2}}{5 b^3 d^4}-\frac {a (c+d x)^3}{3 b^2 d^4}+\frac {2 (c+d x)^{7/2}}{7 b d^4}-\frac {2 a \left (a^2-b^2 c\right )^3 \log \left (a+b \sqrt {c+d x}\right )}{b^8 d^4} \]
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Rubi [A]
time = 0.19, antiderivative size = 230, 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 = {378, 1412, 786}
\begin {gather*} -\frac {2 a \left (a^2-b^2 c\right )^3 \log \left (a+b \sqrt {c+d x}\right )}{b^8 d^4}+\frac {2 \left (a^2-b^2 c\right )^3 \sqrt {c+d x}}{b^7 d^4}-\frac {a \left (a^2-3 b^2 c\right ) (c+d x)^2}{2 b^4 d^4}+\frac {2 \left (a^2-3 b^2 c\right ) (c+d x)^{5/2}}{5 b^3 d^4}-\frac {a x \left (a^4-3 a^2 b^2 c+3 b^4 c^2\right )}{b^6 d^3}+\frac {2 \left (a^4-3 a^2 b^2 c+3 b^4 c^2\right ) (c+d x)^{3/2}}{3 b^5 d^4}-\frac {a (c+d x)^3}{3 b^2 d^4}+\frac {2 (c+d x)^{7/2}}{7 b d^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 378
Rule 786
Rule 1412
Rubi steps
\begin {align*} \int \frac {x^3}{a+b \sqrt {c+d x}} \, dx &=\frac {\text {Subst}\left (\int \frac {(-c+x)^3}{a+b \sqrt {x}} \, dx,x,c+d x\right )}{d^4}\\ &=\frac {2 \text {Subst}\left (\int \frac {x \left (-c+x^2\right )^3}{a+b x} \, dx,x,\sqrt {c+d x}\right )}{d^4}\\ &=\frac {2 \text {Subst}\left (\int \left (-\frac {\left (-a^2+b^2 c\right )^3}{b^7}-\frac {a \left (a^4-3 a^2 b^2 c+3 b^4 c^2\right ) x}{b^6}+\frac {\left (a^4-3 a^2 b^2 c+3 b^4 c^2\right ) x^2}{b^5}-\frac {a \left (a^2-3 b^2 c\right ) x^3}{b^4}-\frac {\left (-a^2+3 b^2 c\right ) x^4}{b^3}-\frac {a x^5}{b^2}+\frac {x^6}{b}-\frac {a \left (a^2-b^2 c\right )^3}{b^7 (a+b x)}\right ) \, dx,x,\sqrt {c+d x}\right )}{d^4}\\ &=-\frac {a \left (a^4-3 a^2 b^2 c+3 b^4 c^2\right ) x}{b^6 d^3}+\frac {2 \left (a^2-b^2 c\right )^3 \sqrt {c+d x}}{b^7 d^4}+\frac {2 \left (a^4-3 a^2 b^2 c+3 b^4 c^2\right ) (c+d x)^{3/2}}{3 b^5 d^4}-\frac {a \left (a^2-3 b^2 c\right ) (c+d x)^2}{2 b^4 d^4}+\frac {2 \left (a^2-3 b^2 c\right ) (c+d x)^{5/2}}{5 b^3 d^4}-\frac {a (c+d x)^3}{3 b^2 d^4}+\frac {2 (c+d x)^{7/2}}{7 b d^4}-\frac {2 a \left (a^2-b^2 c\right )^3 \log \left (a+b \sqrt {c+d x}\right )}{b^8 d^4}\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 235, normalized size = 1.02 \begin {gather*} \frac {b \left (420 a^6 \sqrt {c+d x}-140 a^4 b^2 (8 c-d x) \sqrt {c+d x}-210 a^5 b (c+d x)+105 a^3 b^3 \left (5 c^2+4 c d x-d^2 x^2\right )+84 a^2 b^4 \sqrt {c+d x} \left (11 c^2-3 c d x+d^2 x^2\right )-35 a b^5 \left (11 c^3+6 c^2 d x-3 c d^2 x^2+2 d^3 x^3\right )+12 b^6 \sqrt {c+d x} \left (-16 c^3+8 c^2 d x-6 c d^2 x^2+5 d^3 x^3\right )\right )-420 a \left (a^2-b^2 c\right )^3 \log \left (a+b \sqrt {c+d x}\right )}{210 b^8 d^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 288, normalized size = 1.25
method | result | size |
derivativedivides | \(\frac {\frac {2 \left (\frac {\left (d x +c \right )^{\frac {7}{2}} b^{6}}{7}-\frac {a \left (d x +c \right )^{3} b^{5}}{6}-\frac {3 b^{6} c \left (d x +c \right )^{\frac {5}{2}}}{5}+\frac {a^{2} b^{4} \left (d x +c \right )^{\frac {5}{2}}}{5}+\frac {3 a \,b^{5} c \left (d x +c \right )^{2}}{4}+b^{6} c^{2} \left (d x +c \right )^{\frac {3}{2}}-\frac {a^{3} b^{3} \left (d x +c \right )^{2}}{4}-a^{2} b^{4} c \left (d x +c \right )^{\frac {3}{2}}-\frac {3 a \,b^{5} c^{2} \left (d x +c \right )}{2}-b^{6} c^{3} \sqrt {d x +c}+\frac {a^{4} b^{2} \left (d x +c \right )^{\frac {3}{2}}}{3}+\frac {3 a^{3} b^{3} c \left (d x +c \right )}{2}+3 a^{2} b^{4} c^{2} \sqrt {d x +c}-\frac {a^{5} b \left (d x +c \right )}{2}-3 a^{4} b^{2} c \sqrt {d x +c}+a^{6} \sqrt {d x +c}\right )}{b^{7}}-\frac {2 a \left (-b^{6} c^{3}+3 a^{2} b^{4} c^{2}-3 a^{4} b^{2} c +a^{6}\right ) \ln \left (a +b \sqrt {d x +c}\right )}{b^{8}}}{d^{4}}\) | \(288\) |
default | \(\frac {\frac {2 \left (\frac {\left (d x +c \right )^{\frac {7}{2}} b^{6}}{7}-\frac {a \left (d x +c \right )^{3} b^{5}}{6}-\frac {3 b^{6} c \left (d x +c \right )^{\frac {5}{2}}}{5}+\frac {a^{2} b^{4} \left (d x +c \right )^{\frac {5}{2}}}{5}+\frac {3 a \,b^{5} c \left (d x +c \right )^{2}}{4}+b^{6} c^{2} \left (d x +c \right )^{\frac {3}{2}}-\frac {a^{3} b^{3} \left (d x +c \right )^{2}}{4}-a^{2} b^{4} c \left (d x +c \right )^{\frac {3}{2}}-\frac {3 a \,b^{5} c^{2} \left (d x +c \right )}{2}-b^{6} c^{3} \sqrt {d x +c}+\frac {a^{4} b^{2} \left (d x +c \right )^{\frac {3}{2}}}{3}+\frac {3 a^{3} b^{3} c \left (d x +c \right )}{2}+3 a^{2} b^{4} c^{2} \sqrt {d x +c}-\frac {a^{5} b \left (d x +c \right )}{2}-3 a^{4} b^{2} c \sqrt {d x +c}+a^{6} \sqrt {d x +c}\right )}{b^{7}}-\frac {2 a \left (-b^{6} c^{3}+3 a^{2} b^{4} c^{2}-3 a^{4} b^{2} c +a^{6}\right ) \ln \left (a +b \sqrt {d x +c}\right )}{b^{8}}}{d^{4}}\) | \(288\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 243, normalized size = 1.06 \begin {gather*} \frac {\frac {60 \, {\left (d x + c\right )}^{\frac {7}{2}} b^{6} - 70 \, {\left (d x + c\right )}^{3} a b^{5} - 84 \, {\left (3 \, b^{6} c - a^{2} b^{4}\right )} {\left (d x + c\right )}^{\frac {5}{2}} + 105 \, {\left (3 \, a b^{5} c - a^{3} b^{3}\right )} {\left (d x + c\right )}^{2} + 140 \, {\left (3 \, b^{6} c^{2} - 3 \, a^{2} b^{4} c + a^{4} b^{2}\right )} {\left (d x + c\right )}^{\frac {3}{2}} - 210 \, {\left (3 \, a b^{5} c^{2} - 3 \, a^{3} b^{3} c + a^{5} b\right )} {\left (d x + c\right )} - 420 \, {\left (b^{6} c^{3} - 3 \, a^{2} b^{4} c^{2} + 3 \, a^{4} b^{2} c - a^{6}\right )} \sqrt {d x + c}}{b^{7}} + \frac {420 \, {\left (a b^{6} c^{3} - 3 \, a^{3} b^{4} c^{2} + 3 \, a^{5} b^{2} c - a^{7}\right )} \log \left (\sqrt {d x + c} b + a\right )}{b^{8}}}{210 \, d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 228, normalized size = 0.99 \begin {gather*} -\frac {70 \, a b^{6} d^{3} x^{3} - 105 \, {\left (a b^{6} c - a^{3} b^{4}\right )} d^{2} x^{2} + 210 \, {\left (a b^{6} c^{2} - 2 \, a^{3} b^{4} c + a^{5} b^{2}\right )} d x - 420 \, {\left (a b^{6} c^{3} - 3 \, a^{3} b^{4} c^{2} + 3 \, a^{5} b^{2} c - a^{7}\right )} \log \left (\sqrt {d x + c} b + a\right ) - 4 \, {\left (15 \, b^{7} d^{3} x^{3} - 48 \, b^{7} c^{3} + 231 \, a^{2} b^{5} c^{2} - 280 \, a^{4} b^{3} c + 105 \, a^{6} b - 3 \, {\left (6 \, b^{7} c - 7 \, a^{2} b^{5}\right )} d^{2} x^{2} + {\left (24 \, b^{7} c^{2} - 63 \, a^{2} b^{5} c + 35 \, a^{4} b^{3}\right )} d x\right )} \sqrt {d x + c}}{210 \, b^{8} d^{4}} \end {gather*}
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 \begin {gather*} \int \frac {x^{3}}{a + b \sqrt {c + d x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.62, size = 341, normalized size = 1.48 \begin {gather*} \frac {2 \, {\left (a b^{6} c^{3} - 3 \, a^{3} b^{4} c^{2} + 3 \, a^{5} b^{2} c - a^{7}\right )} \log \left ({\left | \sqrt {d x + c} b + a \right |}\right )}{b^{8} d^{4}} + \frac {60 \, {\left (d x + c\right )}^{\frac {7}{2}} b^{6} d^{24} - 252 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{6} c d^{24} + 420 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{6} c^{2} d^{24} - 420 \, \sqrt {d x + c} b^{6} c^{3} d^{24} - 70 \, {\left (d x + c\right )}^{3} a b^{5} d^{24} + 315 \, {\left (d x + c\right )}^{2} a b^{5} c d^{24} - 630 \, {\left (d x + c\right )} a b^{5} c^{2} d^{24} + 84 \, {\left (d x + c\right )}^{\frac {5}{2}} a^{2} b^{4} d^{24} - 420 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{2} b^{4} c d^{24} + 1260 \, \sqrt {d x + c} a^{2} b^{4} c^{2} d^{24} - 105 \, {\left (d x + c\right )}^{2} a^{3} b^{3} d^{24} + 630 \, {\left (d x + c\right )} a^{3} b^{3} c d^{24} + 140 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{4} b^{2} d^{24} - 1260 \, \sqrt {d x + c} a^{4} b^{2} c d^{24} - 210 \, {\left (d x + c\right )} a^{5} b d^{24} + 420 \, \sqrt {d x + c} a^{6} d^{24}}{210 \, b^{7} d^{28}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 317, normalized size = 1.38 \begin {gather*} \frac {2\,{\left (c+d\,x\right )}^{7/2}}{7\,b\,d^4}-\left (\frac {a^2\,\left (\frac {a^2\,\left (\frac {6\,c}{b\,d^4}-\frac {2\,a^2}{b^3\,d^4}\right )}{b^2}-\frac {6\,c^2}{b\,d^4}\right )}{b^2}+\frac {2\,c^3}{b\,d^4}\right )\,\sqrt {c+d\,x}-\left (\frac {a^2\,\left (\frac {6\,c}{b\,d^4}-\frac {2\,a^2}{b^3\,d^4}\right )}{3\,b^2}-\frac {2\,c^2}{b\,d^4}\right )\,{\left (c+d\,x\right )}^{3/2}-\left (\frac {6\,c}{5\,b\,d^4}-\frac {2\,a^2}{5\,b^3\,d^4}\right )\,{\left (c+d\,x\right )}^{5/2}+\frac {a\,\left (\frac {6\,c}{b\,d^4}-\frac {2\,a^2}{b^3\,d^4}\right )\,{\left (c+d\,x\right )}^2}{4\,b}-\frac {a\,{\left (c+d\,x\right )}^3}{3\,b^2\,d^4}-\frac {\ln \left (a+b\,\sqrt {c+d\,x}\right )\,\left (2\,a^7-6\,a^5\,b^2\,c+6\,a^3\,b^4\,c^2-2\,a\,b^6\,c^3\right )}{b^8\,d^4}+\frac {a\,d\,x\,\left (\frac {a^2\,\left (\frac {6\,c}{b\,d^4}-\frac {2\,a^2}{b^3\,d^4}\right )}{b^2}-\frac {6\,c^2}{b\,d^4}\right )}{2\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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