Optimal. Leaf size=136 \[ \frac {d^3 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{8 b^{5/2} (b c-a d)^{3/2}}-\frac {d^2 \sqrt {c+d x}}{8 b^2 (a+b x) (b c-a d)}-\frac {d \sqrt {c+d x}}{4 b^2 (a+b x)^2}-\frac {(c+d x)^{3/2}}{3 b (a+b x)^3} \]
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Rubi [A] time = 0.06, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {47, 51, 63, 208} \[ -\frac {d^2 \sqrt {c+d x}}{8 b^2 (a+b x) (b c-a d)}+\frac {d^3 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{8 b^{5/2} (b c-a d)^{3/2}}-\frac {d \sqrt {c+d x}}{4 b^2 (a+b x)^2}-\frac {(c+d x)^{3/2}}{3 b (a+b x)^3} \]
Antiderivative was successfully verified.
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Rule 47
Rule 51
Rule 63
Rule 208
Rubi steps
\begin {align*} \int \frac {(c+d x)^{3/2}}{(a+b x)^4} \, dx &=-\frac {(c+d x)^{3/2}}{3 b (a+b x)^3}+\frac {d \int \frac {\sqrt {c+d x}}{(a+b x)^3} \, dx}{2 b}\\ &=-\frac {d \sqrt {c+d x}}{4 b^2 (a+b x)^2}-\frac {(c+d x)^{3/2}}{3 b (a+b x)^3}+\frac {d^2 \int \frac {1}{(a+b x)^2 \sqrt {c+d x}} \, dx}{8 b^2}\\ &=-\frac {d \sqrt {c+d x}}{4 b^2 (a+b x)^2}-\frac {d^2 \sqrt {c+d x}}{8 b^2 (b c-a d) (a+b x)}-\frac {(c+d x)^{3/2}}{3 b (a+b x)^3}-\frac {d^3 \int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx}{16 b^2 (b c-a d)}\\ &=-\frac {d \sqrt {c+d x}}{4 b^2 (a+b x)^2}-\frac {d^2 \sqrt {c+d x}}{8 b^2 (b c-a d) (a+b x)}-\frac {(c+d x)^{3/2}}{3 b (a+b x)^3}-\frac {d^2 \operatorname {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{8 b^2 (b c-a d)}\\ &=-\frac {d \sqrt {c+d x}}{4 b^2 (a+b x)^2}-\frac {d^2 \sqrt {c+d x}}{8 b^2 (b c-a d) (a+b x)}-\frac {(c+d x)^{3/2}}{3 b (a+b x)^3}+\frac {d^3 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{8 b^{5/2} (b c-a d)^{3/2}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 52, normalized size = 0.38 \[ \frac {2 d^3 (c+d x)^{5/2} \, _2F_1\left (\frac {5}{2},4;\frac {7}{2};-\frac {b (c+d x)}{a d-b c}\right )}{5 (a d-b c)^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.50, size = 666, normalized size = 4.90 \[ \left [-\frac {3 \, {\left (b^{3} d^{3} x^{3} + 3 \, a b^{2} d^{3} x^{2} + 3 \, a^{2} b d^{3} x + a^{3} d^{3}\right )} \sqrt {b^{2} c - a b d} \log \left (\frac {b d x + 2 \, b c - a d - 2 \, \sqrt {b^{2} c - a b d} \sqrt {d x + c}}{b x + a}\right ) + 2 \, {\left (8 \, b^{4} c^{3} - 10 \, a b^{3} c^{2} d - a^{2} b^{2} c d^{2} + 3 \, a^{3} b d^{3} + 3 \, {\left (b^{4} c d^{2} - a b^{3} d^{3}\right )} x^{2} + 2 \, {\left (7 \, b^{4} c^{2} d - 11 \, a b^{3} c d^{2} + 4 \, a^{2} b^{2} d^{3}\right )} x\right )} \sqrt {d x + c}}{48 \, {\left (a^{3} b^{5} c^{2} - 2 \, a^{4} b^{4} c d + a^{5} b^{3} d^{2} + {\left (b^{8} c^{2} - 2 \, a b^{7} c d + a^{2} b^{6} d^{2}\right )} x^{3} + 3 \, {\left (a b^{7} c^{2} - 2 \, a^{2} b^{6} c d + a^{3} b^{5} d^{2}\right )} x^{2} + 3 \, {\left (a^{2} b^{6} c^{2} - 2 \, a^{3} b^{5} c d + a^{4} b^{4} d^{2}\right )} x\right )}}, -\frac {3 \, {\left (b^{3} d^{3} x^{3} + 3 \, a b^{2} d^{3} x^{2} + 3 \, a^{2} b d^{3} x + a^{3} d^{3}\right )} \sqrt {-b^{2} c + a b d} \arctan \left (\frac {\sqrt {-b^{2} c + a b d} \sqrt {d x + c}}{b d x + b c}\right ) + {\left (8 \, b^{4} c^{3} - 10 \, a b^{3} c^{2} d - a^{2} b^{2} c d^{2} + 3 \, a^{3} b d^{3} + 3 \, {\left (b^{4} c d^{2} - a b^{3} d^{3}\right )} x^{2} + 2 \, {\left (7 \, b^{4} c^{2} d - 11 \, a b^{3} c d^{2} + 4 \, a^{2} b^{2} d^{3}\right )} x\right )} \sqrt {d x + c}}{24 \, {\left (a^{3} b^{5} c^{2} - 2 \, a^{4} b^{4} c d + a^{5} b^{3} d^{2} + {\left (b^{8} c^{2} - 2 \, a b^{7} c d + a^{2} b^{6} d^{2}\right )} x^{3} + 3 \, {\left (a b^{7} c^{2} - 2 \, a^{2} b^{6} c d + a^{3} b^{5} d^{2}\right )} x^{2} + 3 \, {\left (a^{2} b^{6} c^{2} - 2 \, a^{3} b^{5} c d + a^{4} b^{4} d^{2}\right )} x\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.40, size = 185, normalized size = 1.36 \[ -\frac {d^{3} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{8 \, {\left (b^{3} c - a b^{2} d\right )} \sqrt {-b^{2} c + a b d}} - \frac {3 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{2} d^{3} + 8 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{2} c d^{3} - 3 \, \sqrt {d x + c} b^{2} c^{2} d^{3} - 8 \, {\left (d x + c\right )}^{\frac {3}{2}} a b d^{4} + 6 \, \sqrt {d x + c} a b c d^{4} - 3 \, \sqrt {d x + c} a^{2} d^{5}}{24 \, {\left (b^{3} c - a b^{2} d\right )} {\left ({\left (d x + c\right )} b - b c + a d\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 163, normalized size = 1.20 \[ -\frac {\sqrt {d x +c}\, a \,d^{4}}{8 \left (b d x +a d \right )^{3} b^{2}}+\frac {\sqrt {d x +c}\, c \,d^{3}}{8 \left (b d x +a d \right )^{3} b}+\frac {\left (d x +c \right )^{\frac {5}{2}} d^{3}}{8 \left (b d x +a d \right )^{3} \left (a d -b c \right )}-\frac {\left (d x +c \right )^{\frac {3}{2}} d^{3}}{3 \left (b d x +a d \right )^{3} b}+\frac {d^{3} \arctan \left (\frac {\sqrt {d x +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )}{8 \left (a d -b c \right ) \sqrt {\left (a d -b c \right ) b}\, b^{2}} \]
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 = 0.34, size = 209, normalized size = 1.54 \[ \frac {d^3\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {c+d\,x}}{\sqrt {a\,d-b\,c}}\right )}{8\,b^{5/2}\,{\left (a\,d-b\,c\right )}^{3/2}}-\frac {\frac {d^3\,{\left (c+d\,x\right )}^{3/2}}{3\,b}-\frac {d^3\,{\left (c+d\,x\right )}^{5/2}}{8\,\left (a\,d-b\,c\right )}+\frac {d^3\,\left (a\,d-b\,c\right )\,\sqrt {c+d\,x}}{8\,b^2}}{\left (c+d\,x\right )\,\left (3\,a^2\,b\,d^2-6\,a\,b^2\,c\,d+3\,b^3\,c^2\right )+b^3\,{\left (c+d\,x\right )}^3-\left (3\,b^3\,c-3\,a\,b^2\,d\right )\,{\left (c+d\,x\right )}^2+a^3\,d^3-b^3\,c^3+3\,a\,b^2\,c^2\,d-3\,a^2\,b\,c\,d^2} \]
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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