Optimal. Leaf size=96 \[ -\frac {6 b^2 (c+d x)^{5/2} (b c-a d)}{5 d^4}+\frac {2 b (c+d x)^{3/2} (b c-a d)^2}{d^4}-\frac {2 \sqrt {c+d x} (b c-a d)^3}{d^4}+\frac {2 b^3 (c+d x)^{7/2}}{7 d^4} \]
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Rubi [A] time = 0.03, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {43} \[ -\frac {6 b^2 (c+d x)^{5/2} (b c-a d)}{5 d^4}+\frac {2 b (c+d x)^{3/2} (b c-a d)^2}{d^4}-\frac {2 \sqrt {c+d x} (b c-a d)^3}{d^4}+\frac {2 b^3 (c+d x)^{7/2}}{7 d^4} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int \frac {(a+b x)^3}{\sqrt {c+d x}} \, dx &=\int \left (\frac {(-b c+a d)^3}{d^3 \sqrt {c+d x}}+\frac {3 b (b c-a d)^2 \sqrt {c+d x}}{d^3}-\frac {3 b^2 (b c-a d) (c+d x)^{3/2}}{d^3}+\frac {b^3 (c+d x)^{5/2}}{d^3}\right ) \, dx\\ &=-\frac {2 (b c-a d)^3 \sqrt {c+d x}}{d^4}+\frac {2 b (b c-a d)^2 (c+d x)^{3/2}}{d^4}-\frac {6 b^2 (b c-a d) (c+d x)^{5/2}}{5 d^4}+\frac {2 b^3 (c+d x)^{7/2}}{7 d^4}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 79, normalized size = 0.82 \[ \frac {2 \sqrt {c+d x} \left (-21 b^2 (c+d x)^2 (b c-a d)+35 b (c+d x) (b c-a d)^2-35 (b c-a d)^3+5 b^3 (c+d x)^3\right )}{35 d^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 115, normalized size = 1.20 \[ \frac {2 \, {\left (5 \, b^{3} d^{3} x^{3} - 16 \, b^{3} c^{3} + 56 \, a b^{2} c^{2} d - 70 \, a^{2} b c d^{2} + 35 \, a^{3} d^{3} - 3 \, {\left (2 \, b^{3} c d^{2} - 7 \, a b^{2} d^{3}\right )} x^{2} + {\left (8 \, b^{3} c^{2} d - 28 \, a b^{2} c d^{2} + 35 \, a^{2} b d^{3}\right )} x\right )} \sqrt {d x + c}}{35 \, d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.01, size = 137, normalized size = 1.43 \[ \frac {2 \, {\left (35 \, \sqrt {d x + c} a^{3} + \frac {35 \, {\left ({\left (d x + c\right )}^{\frac {3}{2}} - 3 \, \sqrt {d x + c} c\right )} a^{2} b}{d} + \frac {7 \, {\left (3 \, {\left (d x + c\right )}^{\frac {5}{2}} - 10 \, {\left (d x + c\right )}^{\frac {3}{2}} c + 15 \, \sqrt {d x + c} c^{2}\right )} a b^{2}}{d^{2}} + \frac {{\left (5 \, {\left (d x + c\right )}^{\frac {7}{2}} - 21 \, {\left (d x + c\right )}^{\frac {5}{2}} c + 35 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{2} - 35 \, \sqrt {d x + c} c^{3}\right )} b^{3}}{d^{3}}\right )}}{35 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 116, normalized size = 1.21 \[ \frac {2 \sqrt {d x +c}\, \left (5 b^{3} x^{3} d^{3}+21 a \,b^{2} d^{3} x^{2}-6 b^{3} c \,d^{2} x^{2}+35 a^{2} b \,d^{3} x -28 a \,b^{2} c \,d^{2} x +8 b^{3} c^{2} d x +35 a^{3} d^{3}-70 a^{2} b c \,d^{2}+56 a \,b^{2} c^{2} d -16 b^{3} c^{3}\right )}{35 d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.38, size = 137, normalized size = 1.43 \[ \frac {2 \, {\left (35 \, \sqrt {d x + c} a^{3} + \frac {35 \, {\left ({\left (d x + c\right )}^{\frac {3}{2}} - 3 \, \sqrt {d x + c} c\right )} a^{2} b}{d} + \frac {7 \, {\left (3 \, {\left (d x + c\right )}^{\frac {5}{2}} - 10 \, {\left (d x + c\right )}^{\frac {3}{2}} c + 15 \, \sqrt {d x + c} c^{2}\right )} a b^{2}}{d^{2}} + \frac {{\left (5 \, {\left (d x + c\right )}^{\frac {7}{2}} - 21 \, {\left (d x + c\right )}^{\frac {5}{2}} c + 35 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{2} - 35 \, \sqrt {d x + c} c^{3}\right )} b^{3}}{d^{3}}\right )}}{35 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.26, size = 87, normalized size = 0.91 \[ \frac {2\,b^3\,{\left (c+d\,x\right )}^{7/2}}{7\,d^4}-\frac {\left (6\,b^3\,c-6\,a\,b^2\,d\right )\,{\left (c+d\,x\right )}^{5/2}}{5\,d^4}+\frac {2\,{\left (a\,d-b\,c\right )}^3\,\sqrt {c+d\,x}}{d^4}+\frac {2\,b\,{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^{3/2}}{d^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 37.06, size = 366, normalized size = 3.81 \[ \begin {cases} \frac {- \frac {2 a^{3} c}{\sqrt {c + d x}} - 2 a^{3} \left (- \frac {c}{\sqrt {c + d x}} - \sqrt {c + d x}\right ) - \frac {6 a^{2} b c \left (- \frac {c}{\sqrt {c + d x}} - \sqrt {c + d x}\right )}{d} - \frac {6 a^{2} b \left (\frac {c^{2}}{\sqrt {c + d x}} + 2 c \sqrt {c + d x} - \frac {\left (c + d x\right )^{\frac {3}{2}}}{3}\right )}{d} - \frac {6 a b^{2} c \left (\frac {c^{2}}{\sqrt {c + d x}} + 2 c \sqrt {c + d x} - \frac {\left (c + d x\right )^{\frac {3}{2}}}{3}\right )}{d^{2}} - \frac {6 a b^{2} \left (- \frac {c^{3}}{\sqrt {c + d x}} - 3 c^{2} \sqrt {c + d x} + c \left (c + d x\right )^{\frac {3}{2}} - \frac {\left (c + d x\right )^{\frac {5}{2}}}{5}\right )}{d^{2}} - \frac {2 b^{3} c \left (- \frac {c^{3}}{\sqrt {c + d x}} - 3 c^{2} \sqrt {c + d x} + c \left (c + d x\right )^{\frac {3}{2}} - \frac {\left (c + d x\right )^{\frac {5}{2}}}{5}\right )}{d^{3}} - \frac {2 b^{3} \left (\frac {c^{4}}{\sqrt {c + d x}} + 4 c^{3} \sqrt {c + d x} - 2 c^{2} \left (c + d x\right )^{\frac {3}{2}} + \frac {4 c \left (c + d x\right )^{\frac {5}{2}}}{5} - \frac {\left (c + d x\right )^{\frac {7}{2}}}{7}\right )}{d^{3}}}{d} & \text {for}\: d \neq 0 \\\frac {\begin {cases} a^{3} x & \text {for}\: b = 0 \\\frac {\left (a + b x\right )^{4}}{4 b} & \text {otherwise} \end {cases}}{\sqrt {c}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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