Optimal. Leaf size=172 \[ -\frac {d \sqrt {c+d x}}{8 b^2 (a+b x)^3}-\frac {d^2 \sqrt {c+d x}}{32 b^2 (b c-a d) (a+b x)^2}+\frac {3 d^3 \sqrt {c+d x}}{64 b^2 (b c-a d)^2 (a+b x)}-\frac {(c+d x)^{3/2}}{4 b (a+b x)^4}-\frac {3 d^4 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{64 b^{5/2} (b c-a d)^{5/2}} \]
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Rubi [A]
time = 0.05, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {43, 44, 65, 214}
\begin {gather*} -\frac {3 d^4 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{64 b^{5/2} (b c-a d)^{5/2}}+\frac {3 d^3 \sqrt {c+d x}}{64 b^2 (a+b x) (b c-a d)^2}-\frac {d^2 \sqrt {c+d x}}{32 b^2 (a+b x)^2 (b c-a d)}-\frac {d \sqrt {c+d x}}{8 b^2 (a+b x)^3}-\frac {(c+d x)^{3/2}}{4 b (a+b x)^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 44
Rule 65
Rule 214
Rubi steps
\begin {align*} \int \frac {(c+d x)^{3/2}}{(a+b x)^5} \, dx &=-\frac {(c+d x)^{3/2}}{4 b (a+b x)^4}+\frac {(3 d) \int \frac {\sqrt {c+d x}}{(a+b x)^4} \, dx}{8 b}\\ &=-\frac {d \sqrt {c+d x}}{8 b^2 (a+b x)^3}-\frac {(c+d x)^{3/2}}{4 b (a+b x)^4}+\frac {d^2 \int \frac {1}{(a+b x)^3 \sqrt {c+d x}} \, dx}{16 b^2}\\ &=-\frac {d \sqrt {c+d x}}{8 b^2 (a+b x)^3}-\frac {d^2 \sqrt {c+d x}}{32 b^2 (b c-a d) (a+b x)^2}-\frac {(c+d x)^{3/2}}{4 b (a+b x)^4}-\frac {\left (3 d^3\right ) \int \frac {1}{(a+b x)^2 \sqrt {c+d x}} \, dx}{64 b^2 (b c-a d)}\\ &=-\frac {d \sqrt {c+d x}}{8 b^2 (a+b x)^3}-\frac {d^2 \sqrt {c+d x}}{32 b^2 (b c-a d) (a+b x)^2}+\frac {3 d^3 \sqrt {c+d x}}{64 b^2 (b c-a d)^2 (a+b x)}-\frac {(c+d x)^{3/2}}{4 b (a+b x)^4}+\frac {\left (3 d^4\right ) \int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx}{128 b^2 (b c-a d)^2}\\ &=-\frac {d \sqrt {c+d x}}{8 b^2 (a+b x)^3}-\frac {d^2 \sqrt {c+d x}}{32 b^2 (b c-a d) (a+b x)^2}+\frac {3 d^3 \sqrt {c+d x}}{64 b^2 (b c-a d)^2 (a+b x)}-\frac {(c+d x)^{3/2}}{4 b (a+b x)^4}+\frac {\left (3 d^3\right ) \text {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{64 b^2 (b c-a d)^2}\\ &=-\frac {d \sqrt {c+d x}}{8 b^2 (a+b x)^3}-\frac {d^2 \sqrt {c+d x}}{32 b^2 (b c-a d) (a+b x)^2}+\frac {3 d^3 \sqrt {c+d x}}{64 b^2 (b c-a d)^2 (a+b x)}-\frac {(c+d x)^{3/2}}{4 b (a+b x)^4}-\frac {3 d^4 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{64 b^{5/2} (b c-a d)^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.86, size = 171, normalized size = 0.99 \begin {gather*} -\frac {\sqrt {c+d x} \left (3 a^3 d^3+a^2 b d^2 (2 c+11 d x)-a b^2 d \left (24 c^2+44 c d x+11 d^2 x^2\right )+b^3 \left (16 c^3+24 c^2 d x+2 c d^2 x^2-3 d^3 x^3\right )\right )}{64 b^2 (b c-a d)^2 (a+b x)^4}+\frac {3 d^4 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{64 b^{5/2} (-b c+a d)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [F(-1)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.16, size = 172, normalized size = 1.00
method | result | size |
derivativedivides | \(2 d^{4} \left (\frac {\frac {3 b \left (d x +c \right )^{\frac {7}{2}}}{128 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {11 \left (d x +c \right )^{\frac {5}{2}}}{128 \left (a d -b c \right )}-\frac {11 \left (d x +c \right )^{\frac {3}{2}}}{128 b}-\frac {3 \left (a d -b c \right ) \sqrt {d x +c}}{128 b^{2}}}{\left (\left (d x +c \right ) b +a d -b c \right )^{4}}+\frac {3 \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{128 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b^{2} \sqrt {\left (a d -b c \right ) b}}\right )\) | \(172\) |
default | \(2 d^{4} \left (\frac {\frac {3 b \left (d x +c \right )^{\frac {7}{2}}}{128 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {11 \left (d x +c \right )^{\frac {5}{2}}}{128 \left (a d -b c \right )}-\frac {11 \left (d x +c \right )^{\frac {3}{2}}}{128 b}-\frac {3 \left (a d -b c \right ) \sqrt {d x +c}}{128 b^{2}}}{\left (\left (d x +c \right ) b +a d -b c \right )^{4}}+\frac {3 \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{128 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b^{2} \sqrt {\left (a d -b c \right ) b}}\right )\) | \(172\) |
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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 515 vs.
\(2 (144) = 288\).
time = 0.32, size = 1043, normalized size = 6.06 \begin {gather*} \left [\frac {3 \, {\left (b^{4} d^{4} x^{4} + 4 \, a b^{3} d^{4} x^{3} + 6 \, a^{2} b^{2} d^{4} x^{2} + 4 \, a^{3} b d^{4} x + a^{4} d^{4}\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 (16 \, b^{5} c^{4} - 40 \, a b^{4} c^{3} d + 26 \, a^{2} b^{3} c^{2} d^{2} + a^{3} b^{2} c d^{3} - 3 \, a^{4} b d^{4} - 3 \, {\left (b^{5} c d^{3} - a b^{4} d^{4}\right )} x^{3} + {\left (2 \, b^{5} c^{2} d^{2} - 13 \, a b^{4} c d^{3} + 11 \, a^{2} b^{3} d^{4}\right )} x^{2} + {\left (24 \, b^{5} c^{3} d - 68 \, a b^{4} c^{2} d^{2} + 55 \, a^{2} b^{3} c d^{3} - 11 \, a^{3} b^{2} d^{4}\right )} x\right )} \sqrt {d x + c}}{128 \, {\left (a^{4} b^{6} c^{3} - 3 \, a^{5} b^{5} c^{2} d + 3 \, a^{6} b^{4} c d^{2} - a^{7} b^{3} d^{3} + {\left (b^{10} c^{3} - 3 \, a b^{9} c^{2} d + 3 \, a^{2} b^{8} c d^{2} - a^{3} b^{7} d^{3}\right )} x^{4} + 4 \, {\left (a b^{9} c^{3} - 3 \, a^{2} b^{8} c^{2} d + 3 \, a^{3} b^{7} c d^{2} - a^{4} b^{6} d^{3}\right )} x^{3} + 6 \, {\left (a^{2} b^{8} c^{3} - 3 \, a^{3} b^{7} c^{2} d + 3 \, a^{4} b^{6} c d^{2} - a^{5} b^{5} d^{3}\right )} x^{2} + 4 \, {\left (a^{3} b^{7} c^{3} - 3 \, a^{4} b^{6} c^{2} d + 3 \, a^{5} b^{5} c d^{2} - a^{6} b^{4} d^{3}\right )} x\right )}}, \frac {3 \, {\left (b^{4} d^{4} x^{4} + 4 \, a b^{3} d^{4} x^{3} + 6 \, a^{2} b^{2} d^{4} x^{2} + 4 \, a^{3} b d^{4} x + a^{4} d^{4}\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 (16 \, b^{5} c^{4} - 40 \, a b^{4} c^{3} d + 26 \, a^{2} b^{3} c^{2} d^{2} + a^{3} b^{2} c d^{3} - 3 \, a^{4} b d^{4} - 3 \, {\left (b^{5} c d^{3} - a b^{4} d^{4}\right )} x^{3} + {\left (2 \, b^{5} c^{2} d^{2} - 13 \, a b^{4} c d^{3} + 11 \, a^{2} b^{3} d^{4}\right )} x^{2} + {\left (24 \, b^{5} c^{3} d - 68 \, a b^{4} c^{2} d^{2} + 55 \, a^{2} b^{3} c d^{3} - 11 \, a^{3} b^{2} d^{4}\right )} x\right )} \sqrt {d x + c}}{64 \, {\left (a^{4} b^{6} c^{3} - 3 \, a^{5} b^{5} c^{2} d + 3 \, a^{6} b^{4} c d^{2} - a^{7} b^{3} d^{3} + {\left (b^{10} c^{3} - 3 \, a b^{9} c^{2} d + 3 \, a^{2} b^{8} c d^{2} - a^{3} b^{7} d^{3}\right )} x^{4} + 4 \, {\left (a b^{9} c^{3} - 3 \, a^{2} b^{8} c^{2} d + 3 \, a^{3} b^{7} c d^{2} - a^{4} b^{6} d^{3}\right )} x^{3} + 6 \, {\left (a^{2} b^{8} c^{3} - 3 \, a^{3} b^{7} c^{2} d + 3 \, a^{4} b^{6} c d^{2} - a^{5} b^{5} d^{3}\right )} x^{2} + 4 \, {\left (a^{3} b^{7} c^{3} - 3 \, a^{4} b^{6} c^{2} d + 3 \, a^{5} b^{5} c d^{2} - a^{6} b^{4} d^{3}\right )} x\right )}}\right ] \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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Giac [A]
time = 0.01, size = 360, normalized size = 2.09 \begin {gather*} \frac {3 \sqrt {c+d x} \left (c+d x\right )^{3} d^{4} b^{3}+11 \sqrt {c+d x} \left (c+d x\right )^{2} d^{5} b^{2} a-11 \sqrt {c+d x} \left (c+d x\right )^{2} d^{4} c b^{3}-11 \sqrt {c+d x} \left (c+d x\right ) d^{6} b a^{2}+22 \sqrt {c+d x} \left (c+d x\right ) d^{5} c b^{2} a-11 \sqrt {c+d x} \left (c+d x\right ) d^{4} c^{2} b^{3}-3 \sqrt {c+d x} d^{7} a^{3}+9 \sqrt {c+d x} d^{6} c b a^{2}-9 \sqrt {c+d x} d^{5} c^{2} b^{2} a+3 \sqrt {c+d x} d^{4} c^{3} b^{3}}{\left (64 d^{2} b^{2} a^{2}-128 d c b^{3} a+64 c^{2} b^{4}\right ) \left (\left (c+d x\right ) b+d a-c b\right )^{4}}+\frac {3 d^{4} \arctan \left (\frac {b \sqrt {c+d x}}{\sqrt {-b^{2} c+a b d}}\right )}{2 \left (32 d^{2} b^{2} a^{2}-64 d c b^{3} a+32 c^{2} b^{4}\right ) \sqrt {-b^{2} c+a b d}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.37, size = 296, normalized size = 1.72 \begin {gather*} \frac {3\,d^4\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {c+d\,x}}{\sqrt {a\,d-b\,c}}\right )}{64\,b^{5/2}\,{\left (a\,d-b\,c\right )}^{5/2}}-\frac {\frac {11\,d^4\,{\left (c+d\,x\right )}^{3/2}}{64\,b}-\frac {11\,d^4\,{\left (c+d\,x\right )}^{5/2}}{64\,\left (a\,d-b\,c\right )}+\frac {3\,d^4\,\left (a\,d-b\,c\right )\,\sqrt {c+d\,x}}{64\,b^2}-\frac {3\,b\,d^4\,{\left (c+d\,x\right )}^{7/2}}{64\,{\left (a\,d-b\,c\right )}^2}}{b^4\,{\left (c+d\,x\right )}^4-\left (4\,b^4\,c-4\,a\,b^3\,d\right )\,{\left (c+d\,x\right )}^3-\left (c+d\,x\right )\,\left (-4\,a^3\,b\,d^3+12\,a^2\,b^2\,c\,d^2-12\,a\,b^3\,c^2\,d+4\,b^4\,c^3\right )+a^4\,d^4+b^4\,c^4+{\left (c+d\,x\right )}^2\,\left (6\,a^2\,b^2\,d^2-12\,a\,b^3\,c\,d+6\,b^4\,c^2\right )+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d-4\,a^3\,b\,c\,d^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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