Optimal. Leaf size=173 \[ -\frac {35 d^3}{8 \sqrt {c+d x} (b c-a d)^4}+\frac {35 \sqrt {b} d^3 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{8 (b c-a d)^{9/2}}-\frac {35 d^2}{24 (a+b x) \sqrt {c+d x} (b c-a d)^3}+\frac {7 d}{12 (a+b x)^2 \sqrt {c+d x} (b c-a d)^2}-\frac {1}{3 (a+b x)^3 \sqrt {c+d x} (b c-a d)} \]
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Rubi [A] time = 0.07, antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {51, 63, 208} \begin {gather*} -\frac {35 d^3}{8 \sqrt {c+d x} (b c-a d)^4}-\frac {35 d^2}{24 (a+b x) \sqrt {c+d x} (b c-a d)^3}+\frac {35 \sqrt {b} d^3 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{8 (b c-a d)^{9/2}}+\frac {7 d}{12 (a+b x)^2 \sqrt {c+d x} (b c-a d)^2}-\frac {1}{3 (a+b x)^3 \sqrt {c+d x} (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rubi steps
\begin {align*} \int \frac {1}{(a+b x)^4 (c+d x)^{3/2}} \, dx &=-\frac {1}{3 (b c-a d) (a+b x)^3 \sqrt {c+d x}}-\frac {(7 d) \int \frac {1}{(a+b x)^3 (c+d x)^{3/2}} \, dx}{6 (b c-a d)}\\ &=-\frac {1}{3 (b c-a d) (a+b x)^3 \sqrt {c+d x}}+\frac {7 d}{12 (b c-a d)^2 (a+b x)^2 \sqrt {c+d x}}+\frac {\left (35 d^2\right ) \int \frac {1}{(a+b x)^2 (c+d x)^{3/2}} \, dx}{24 (b c-a d)^2}\\ &=-\frac {1}{3 (b c-a d) (a+b x)^3 \sqrt {c+d x}}+\frac {7 d}{12 (b c-a d)^2 (a+b x)^2 \sqrt {c+d x}}-\frac {35 d^2}{24 (b c-a d)^3 (a+b x) \sqrt {c+d x}}-\frac {\left (35 d^3\right ) \int \frac {1}{(a+b x) (c+d x)^{3/2}} \, dx}{16 (b c-a d)^3}\\ &=-\frac {35 d^3}{8 (b c-a d)^4 \sqrt {c+d x}}-\frac {1}{3 (b c-a d) (a+b x)^3 \sqrt {c+d x}}+\frac {7 d}{12 (b c-a d)^2 (a+b x)^2 \sqrt {c+d x}}-\frac {35 d^2}{24 (b c-a d)^3 (a+b x) \sqrt {c+d x}}-\frac {\left (35 b d^3\right ) \int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx}{16 (b c-a d)^4}\\ &=-\frac {35 d^3}{8 (b c-a d)^4 \sqrt {c+d x}}-\frac {1}{3 (b c-a d) (a+b x)^3 \sqrt {c+d x}}+\frac {7 d}{12 (b c-a d)^2 (a+b x)^2 \sqrt {c+d x}}-\frac {35 d^2}{24 (b c-a d)^3 (a+b x) \sqrt {c+d x}}-\frac {\left (35 b 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 )}{8 (b c-a d)^4}\\ &=-\frac {35 d^3}{8 (b c-a d)^4 \sqrt {c+d x}}-\frac {1}{3 (b c-a d) (a+b x)^3 \sqrt {c+d x}}+\frac {7 d}{12 (b c-a d)^2 (a+b x)^2 \sqrt {c+d x}}-\frac {35 d^2}{24 (b c-a d)^3 (a+b x) \sqrt {c+d x}}+\frac {35 \sqrt {b} d^3 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{8 (b c-a d)^{9/2}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 50, normalized size = 0.29 \begin {gather*} -\frac {2 d^3 \, _2F_1\left (-\frac {1}{2},4;\frac {1}{2};-\frac {b (c+d x)}{a d-b c}\right )}{\sqrt {c+d x} (a d-b c)^4} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.68, size = 223, normalized size = 1.29 \begin {gather*} \frac {d^3 \left (48 a^3 d^3+231 a^2 b d^2 (c+d x)-144 a^2 b c d^2+144 a b^2 c^2 d+280 a b^2 d (c+d x)^2-462 a b^2 c d (c+d x)-48 b^3 c^3+231 b^3 c^2 (c+d x)+105 b^3 (c+d x)^3-280 b^3 c (c+d x)^2\right )}{24 \sqrt {c+d x} (b c-a d)^4 (-a d-b (c+d x)+b c)^3}+\frac {35 \sqrt {b} d^3 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x} \sqrt {a d-b c}}{b c-a d}\right )}{8 (a d-b c)^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.53, size = 1204, normalized size = 6.96 \begin {gather*} \left [\frac {105 \, {\left (b^{3} d^{4} x^{4} + a^{3} c d^{3} + {\left (b^{3} c d^{3} + 3 \, a b^{2} d^{4}\right )} x^{3} + 3 \, {\left (a b^{2} c d^{3} + a^{2} b d^{4}\right )} x^{2} + {\left (3 \, a^{2} b c d^{3} + a^{3} d^{4}\right )} x\right )} \sqrt {\frac {b}{b c - a d}} \log \left (\frac {b d x + 2 \, b c - a d + 2 \, {\left (b c - a d\right )} \sqrt {d x + c} \sqrt {\frac {b}{b c - a d}}}{b x + a}\right ) - 2 \, {\left (105 \, b^{3} d^{3} x^{3} + 8 \, b^{3} c^{3} - 38 \, a b^{2} c^{2} d + 87 \, a^{2} b c d^{2} + 48 \, a^{3} d^{3} + 35 \, {\left (b^{3} c d^{2} + 8 \, a b^{2} d^{3}\right )} x^{2} - 7 \, {\left (2 \, b^{3} c^{2} d - 14 \, a b^{2} c d^{2} - 33 \, a^{2} b d^{3}\right )} x\right )} \sqrt {d x + c}}{48 \, {\left (a^{3} b^{4} c^{5} - 4 \, a^{4} b^{3} c^{4} d + 6 \, a^{5} b^{2} c^{3} d^{2} - 4 \, a^{6} b c^{2} d^{3} + a^{7} c d^{4} + {\left (b^{7} c^{4} d - 4 \, a b^{6} c^{3} d^{2} + 6 \, a^{2} b^{5} c^{2} d^{3} - 4 \, a^{3} b^{4} c d^{4} + a^{4} b^{3} d^{5}\right )} x^{4} + {\left (b^{7} c^{5} - a b^{6} c^{4} d - 6 \, a^{2} b^{5} c^{3} d^{2} + 14 \, a^{3} b^{4} c^{2} d^{3} - 11 \, a^{4} b^{3} c d^{4} + 3 \, a^{5} b^{2} d^{5}\right )} x^{3} + 3 \, {\left (a b^{6} c^{5} - 3 \, a^{2} b^{5} c^{4} d + 2 \, a^{3} b^{4} c^{3} d^{2} + 2 \, a^{4} b^{3} c^{2} d^{3} - 3 \, a^{5} b^{2} c d^{4} + a^{6} b d^{5}\right )} x^{2} + {\left (3 \, a^{2} b^{5} c^{5} - 11 \, a^{3} b^{4} c^{4} d + 14 \, a^{4} b^{3} c^{3} d^{2} - 6 \, a^{5} b^{2} c^{2} d^{3} - a^{6} b c d^{4} + a^{7} d^{5}\right )} x\right )}}, \frac {105 \, {\left (b^{3} d^{4} x^{4} + a^{3} c d^{3} + {\left (b^{3} c d^{3} + 3 \, a b^{2} d^{4}\right )} x^{3} + 3 \, {\left (a b^{2} c d^{3} + a^{2} b d^{4}\right )} x^{2} + {\left (3 \, a^{2} b c d^{3} + a^{3} d^{4}\right )} x\right )} \sqrt {-\frac {b}{b c - a d}} \arctan \left (-\frac {{\left (b c - a d\right )} \sqrt {d x + c} \sqrt {-\frac {b}{b c - a d}}}{b d x + b c}\right ) - {\left (105 \, b^{3} d^{3} x^{3} + 8 \, b^{3} c^{3} - 38 \, a b^{2} c^{2} d + 87 \, a^{2} b c d^{2} + 48 \, a^{3} d^{3} + 35 \, {\left (b^{3} c d^{2} + 8 \, a b^{2} d^{3}\right )} x^{2} - 7 \, {\left (2 \, b^{3} c^{2} d - 14 \, a b^{2} c d^{2} - 33 \, a^{2} b d^{3}\right )} x\right )} \sqrt {d x + c}}{24 \, {\left (a^{3} b^{4} c^{5} - 4 \, a^{4} b^{3} c^{4} d + 6 \, a^{5} b^{2} c^{3} d^{2} - 4 \, a^{6} b c^{2} d^{3} + a^{7} c d^{4} + {\left (b^{7} c^{4} d - 4 \, a b^{6} c^{3} d^{2} + 6 \, a^{2} b^{5} c^{2} d^{3} - 4 \, a^{3} b^{4} c d^{4} + a^{4} b^{3} d^{5}\right )} x^{4} + {\left (b^{7} c^{5} - a b^{6} c^{4} d - 6 \, a^{2} b^{5} c^{3} d^{2} + 14 \, a^{3} b^{4} c^{2} d^{3} - 11 \, a^{4} b^{3} c d^{4} + 3 \, a^{5} b^{2} d^{5}\right )} x^{3} + 3 \, {\left (a b^{6} c^{5} - 3 \, a^{2} b^{5} c^{4} d + 2 \, a^{3} b^{4} c^{3} d^{2} + 2 \, a^{4} b^{3} c^{2} d^{3} - 3 \, a^{5} b^{2} c d^{4} + a^{6} b d^{5}\right )} x^{2} + {\left (3 \, a^{2} b^{5} c^{5} - 11 \, a^{3} b^{4} c^{4} d + 14 \, a^{4} b^{3} c^{3} d^{2} - 6 \, a^{5} b^{2} c^{2} d^{3} - a^{6} b c d^{4} + a^{7} d^{5}\right )} x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.31, size = 326, normalized size = 1.88 \begin {gather*} -\frac {35 \, b d^{3} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{8 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \sqrt {-b^{2} c + a b d}} - \frac {2 \, d^{3}}{{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \sqrt {d x + c}} - \frac {57 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{3} d^{3} - 136 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} c d^{3} + 87 \, \sqrt {d x + c} b^{3} c^{2} d^{3} + 136 \, {\left (d x + c\right )}^{\frac {3}{2}} a b^{2} d^{4} - 174 \, \sqrt {d x + c} a b^{2} c d^{4} + 87 \, \sqrt {d x + c} a^{2} b d^{5}}{24 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} {\left ({\left (d x + c\right )} b - b c + a d\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 292, normalized size = 1.69 \begin {gather*} -\frac {29 \sqrt {d x +c}\, a^{2} b \,d^{5}}{8 \left (a d -b c \right )^{4} \left (b d x +a d \right )^{3}}+\frac {29 \sqrt {d x +c}\, a \,b^{2} c \,d^{4}}{4 \left (a d -b c \right )^{4} \left (b d x +a d \right )^{3}}-\frac {29 \sqrt {d x +c}\, b^{3} c^{2} d^{3}}{8 \left (a d -b c \right )^{4} \left (b d x +a d \right )^{3}}-\frac {17 \left (d x +c \right )^{\frac {3}{2}} a \,b^{2} d^{4}}{3 \left (a d -b c \right )^{4} \left (b d x +a d \right )^{3}}+\frac {17 \left (d x +c \right )^{\frac {3}{2}} b^{3} c \,d^{3}}{3 \left (a d -b c \right )^{4} \left (b d x +a d \right )^{3}}-\frac {19 \left (d x +c \right )^{\frac {5}{2}} b^{3} d^{3}}{8 \left (a d -b c \right )^{4} \left (b d x +a d \right )^{3}}-\frac {35 b \,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 )^{4} \sqrt {\left (a d -b c \right ) b}}-\frac {2 d^{3}}{\left (a d -b c \right )^{4} \sqrt {d x +c}} \end {gather*}
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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.54, size = 294, normalized size = 1.70 \begin {gather*} -\frac {\frac {2\,d^3}{a\,d-b\,c}+\frac {35\,b^2\,d^3\,{\left (c+d\,x\right )}^2}{3\,{\left (a\,d-b\,c\right )}^3}+\frac {35\,b^3\,d^3\,{\left (c+d\,x\right )}^3}{8\,{\left (a\,d-b\,c\right )}^4}+\frac {77\,b\,d^3\,\left (c+d\,x\right )}{8\,{\left (a\,d-b\,c\right )}^2}}{\sqrt {c+d\,x}\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )+b^3\,{\left (c+d\,x\right )}^{7/2}-\left (3\,b^3\,c-3\,a\,b^2\,d\right )\,{\left (c+d\,x\right )}^{5/2}+{\left (c+d\,x\right )}^{3/2}\,\left (3\,a^2\,b\,d^2-6\,a\,b^2\,c\,d+3\,b^3\,c^2\right )}-\frac {35\,\sqrt {b}\,d^3\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {c+d\,x}\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}{{\left (a\,d-b\,c\right )}^{9/2}}\right )}{8\,{\left (a\,d-b\,c\right )}^{9/2}} \end {gather*}
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 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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