Optimal. Leaf size=200 \[ -\frac {35 d^3}{8 (b c-a d)^4 (c+d x)^{3/2}}-\frac {1}{3 (b c-a d) (a+b x)^3 (c+d x)^{3/2}}+\frac {3 d}{4 (b c-a d)^2 (a+b x)^2 (c+d x)^{3/2}}-\frac {21 d^2}{8 (b c-a d)^3 (a+b x) (c+d x)^{3/2}}-\frac {105 b d^3}{8 (b c-a d)^5 \sqrt {c+d x}}+\frac {105 b^{3/2} d^3 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{8 (b c-a d)^{11/2}} \]
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Rubi [A]
time = 0.09, antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {44, 53, 65, 214}
\begin {gather*} \frac {105 b^{3/2} d^3 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{8 (b c-a d)^{11/2}}-\frac {105 b d^3}{8 \sqrt {c+d x} (b c-a d)^5}-\frac {35 d^3}{8 (c+d x)^{3/2} (b c-a d)^4}-\frac {21 d^2}{8 (a+b x) (c+d x)^{3/2} (b c-a d)^3}+\frac {3 d}{4 (a+b x)^2 (c+d x)^{3/2} (b c-a d)^2}-\frac {1}{3 (a+b x)^3 (c+d x)^{3/2} (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 44
Rule 53
Rule 65
Rule 214
Rubi steps
\begin {align*} \int \frac {1}{(a+b x)^4 (c+d x)^{5/2}} \, dx &=-\frac {1}{3 (b c-a d) (a+b x)^3 (c+d x)^{3/2}}-\frac {(3 d) \int \frac {1}{(a+b x)^3 (c+d x)^{5/2}} \, dx}{2 (b c-a d)}\\ &=-\frac {1}{3 (b c-a d) (a+b x)^3 (c+d x)^{3/2}}+\frac {3 d}{4 (b c-a d)^2 (a+b x)^2 (c+d x)^{3/2}}+\frac {\left (21 d^2\right ) \int \frac {1}{(a+b x)^2 (c+d x)^{5/2}} \, dx}{8 (b c-a d)^2}\\ &=-\frac {1}{3 (b c-a d) (a+b x)^3 (c+d x)^{3/2}}+\frac {3 d}{4 (b c-a d)^2 (a+b x)^2 (c+d x)^{3/2}}-\frac {21 d^2}{8 (b c-a d)^3 (a+b x) (c+d x)^{3/2}}-\frac {\left (105 d^3\right ) \int \frac {1}{(a+b x) (c+d x)^{5/2}} \, dx}{16 (b c-a d)^3}\\ &=-\frac {35 d^3}{8 (b c-a d)^4 (c+d x)^{3/2}}-\frac {1}{3 (b c-a d) (a+b x)^3 (c+d x)^{3/2}}+\frac {3 d}{4 (b c-a d)^2 (a+b x)^2 (c+d x)^{3/2}}-\frac {21 d^2}{8 (b c-a d)^3 (a+b x) (c+d x)^{3/2}}-\frac {\left (105 b d^3\right ) \int \frac {1}{(a+b x) (c+d x)^{3/2}} \, dx}{16 (b c-a d)^4}\\ &=-\frac {35 d^3}{8 (b c-a d)^4 (c+d x)^{3/2}}-\frac {1}{3 (b c-a d) (a+b x)^3 (c+d x)^{3/2}}+\frac {3 d}{4 (b c-a d)^2 (a+b x)^2 (c+d x)^{3/2}}-\frac {21 d^2}{8 (b c-a d)^3 (a+b x) (c+d x)^{3/2}}-\frac {105 b d^3}{8 (b c-a d)^5 \sqrt {c+d x}}-\frac {\left (105 b^2 d^3\right ) \int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx}{16 (b c-a d)^5}\\ &=-\frac {35 d^3}{8 (b c-a d)^4 (c+d x)^{3/2}}-\frac {1}{3 (b c-a d) (a+b x)^3 (c+d x)^{3/2}}+\frac {3 d}{4 (b c-a d)^2 (a+b x)^2 (c+d x)^{3/2}}-\frac {21 d^2}{8 (b c-a d)^3 (a+b x) (c+d x)^{3/2}}-\frac {105 b d^3}{8 (b c-a d)^5 \sqrt {c+d x}}-\frac {\left (105 b^2 d^2\right ) \text {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)^5}\\ &=-\frac {35 d^3}{8 (b c-a d)^4 (c+d x)^{3/2}}-\frac {1}{3 (b c-a d) (a+b x)^3 (c+d x)^{3/2}}+\frac {3 d}{4 (b c-a d)^2 (a+b x)^2 (c+d x)^{3/2}}-\frac {21 d^2}{8 (b c-a d)^3 (a+b x) (c+d x)^{3/2}}-\frac {105 b d^3}{8 (b c-a d)^5 \sqrt {c+d x}}+\frac {105 b^{3/2} d^3 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{8 (b c-a d)^{11/2}}\\ \end {align*}
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Mathematica [A]
time = 0.99, size = 220, normalized size = 1.10 \begin {gather*} \frac {1}{24} \left (\frac {-16 a^4 d^4+16 a^3 b d^3 (13 c+9 d x)+3 a^2 b^2 d^2 \left (55 c^2+318 c d x+231 d^2 x^2\right )+2 a b^3 d \left (-25 c^3+90 c^2 d x+567 c d^2 x^2+420 d^3 x^3\right )+b^4 \left (8 c^4-18 c^3 d x+63 c^2 d^2 x^2+420 c d^3 x^3+315 d^4 x^4\right )}{(-b c+a d)^5 (a+b x)^3 (c+d x)^{3/2}}+\frac {315 b^{3/2} d^3 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{(-b c+a d)^{11/2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 177, normalized size = 0.88
method | result | size |
derivativedivides | \(2 d^{3} \left (\frac {b^{2} \left (\frac {\frac {41 \left (d x +c \right )^{\frac {5}{2}} b^{2}}{16}+\frac {35 \left (a d -b c \right ) b \left (d x +c \right )^{\frac {3}{2}}}{6}+\left (\frac {55}{16} a^{2} d^{2}-\frac {55}{8} a b c d +\frac {55}{16} b^{2} c^{2}\right ) \sqrt {d x +c}}{\left (\left (d x +c \right ) b +a d -b c \right )^{3}}+\frac {105 \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{16 \sqrt {\left (a d -b c \right ) b}}\right )}{\left (a d -b c \right )^{5}}-\frac {1}{3 \left (a d -b c \right )^{4} \left (d x +c \right )^{\frac {3}{2}}}+\frac {4 b}{\left (a d -b c \right )^{5} \sqrt {d x +c}}\right )\) | \(177\) |
default | \(2 d^{3} \left (\frac {b^{2} \left (\frac {\frac {41 \left (d x +c \right )^{\frac {5}{2}} b^{2}}{16}+\frac {35 \left (a d -b c \right ) b \left (d x +c \right )^{\frac {3}{2}}}{6}+\left (\frac {55}{16} a^{2} d^{2}-\frac {55}{8} a b c d +\frac {55}{16} b^{2} c^{2}\right ) \sqrt {d x +c}}{\left (\left (d x +c \right ) b +a d -b c \right )^{3}}+\frac {105 \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{16 \sqrt {\left (a d -b c \right ) b}}\right )}{\left (a d -b c \right )^{5}}-\frac {1}{3 \left (a d -b c \right )^{4} \left (d x +c \right )^{\frac {3}{2}}}+\frac {4 b}{\left (a d -b c \right )^{5} \sqrt {d x +c}}\right )\) | \(177\) |
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. 915 vs.
\(2 (168) = 336\).
time = 0.47, size = 1840, normalized size = 9.20 \begin {gather*} \left [-\frac {315 \, {\left (b^{4} d^{5} x^{5} + a^{3} b c^{2} d^{3} + {\left (2 \, b^{4} c d^{4} + 3 \, a b^{3} d^{5}\right )} x^{4} + {\left (b^{4} c^{2} d^{3} + 6 \, a b^{3} c d^{4} + 3 \, a^{2} b^{2} d^{5}\right )} x^{3} + {\left (3 \, a b^{3} c^{2} d^{3} + 6 \, a^{2} b^{2} c d^{4} + a^{3} b d^{5}\right )} x^{2} + {\left (3 \, a^{2} b^{2} c^{2} d^{3} + 2 \, a^{3} b c 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 (315 \, b^{4} d^{4} x^{4} + 8 \, b^{4} c^{4} - 50 \, a b^{3} c^{3} d + 165 \, a^{2} b^{2} c^{2} d^{2} + 208 \, a^{3} b c d^{3} - 16 \, a^{4} d^{4} + 420 \, {\left (b^{4} c d^{3} + 2 \, a b^{3} d^{4}\right )} x^{3} + 63 \, {\left (b^{4} c^{2} d^{2} + 18 \, a b^{3} c d^{3} + 11 \, a^{2} b^{2} d^{4}\right )} x^{2} - 18 \, {\left (b^{4} c^{3} d - 10 \, a b^{3} c^{2} d^{2} - 53 \, a^{2} b^{2} c d^{3} - 8 \, a^{3} b d^{4}\right )} x\right )} \sqrt {d x + c}}{48 \, {\left (a^{3} b^{5} c^{7} - 5 \, a^{4} b^{4} c^{6} d + 10 \, a^{5} b^{3} c^{5} d^{2} - 10 \, a^{6} b^{2} c^{4} d^{3} + 5 \, a^{7} b c^{3} d^{4} - a^{8} c^{2} d^{5} + {\left (b^{8} c^{5} d^{2} - 5 \, a b^{7} c^{4} d^{3} + 10 \, a^{2} b^{6} c^{3} d^{4} - 10 \, a^{3} b^{5} c^{2} d^{5} + 5 \, a^{4} b^{4} c d^{6} - a^{5} b^{3} d^{7}\right )} x^{5} + {\left (2 \, b^{8} c^{6} d - 7 \, a b^{7} c^{5} d^{2} + 5 \, a^{2} b^{6} c^{4} d^{3} + 10 \, a^{3} b^{5} c^{3} d^{4} - 20 \, a^{4} b^{4} c^{2} d^{5} + 13 \, a^{5} b^{3} c d^{6} - 3 \, a^{6} b^{2} d^{7}\right )} x^{4} + {\left (b^{8} c^{7} + a b^{7} c^{6} d - 17 \, a^{2} b^{6} c^{5} d^{2} + 35 \, a^{3} b^{5} c^{4} d^{3} - 25 \, a^{4} b^{4} c^{3} d^{4} - a^{5} b^{3} c^{2} d^{5} + 9 \, a^{6} b^{2} c d^{6} - 3 \, a^{7} b d^{7}\right )} x^{3} + {\left (3 \, a b^{7} c^{7} - 9 \, a^{2} b^{6} c^{6} d + a^{3} b^{5} c^{5} d^{2} + 25 \, a^{4} b^{4} c^{4} d^{3} - 35 \, a^{5} b^{3} c^{3} d^{4} + 17 \, a^{6} b^{2} c^{2} d^{5} - a^{7} b c d^{6} - a^{8} d^{7}\right )} x^{2} + {\left (3 \, a^{2} b^{6} c^{7} - 13 \, a^{3} b^{5} c^{6} d + 20 \, a^{4} b^{4} c^{5} d^{2} - 10 \, a^{5} b^{3} c^{4} d^{3} - 5 \, a^{6} b^{2} c^{3} d^{4} + 7 \, a^{7} b c^{2} d^{5} - 2 \, a^{8} c d^{6}\right )} x\right )}}, \frac {315 \, {\left (b^{4} d^{5} x^{5} + a^{3} b c^{2} d^{3} + {\left (2 \, b^{4} c d^{4} + 3 \, a b^{3} d^{5}\right )} x^{4} + {\left (b^{4} c^{2} d^{3} + 6 \, a b^{3} c d^{4} + 3 \, a^{2} b^{2} d^{5}\right )} x^{3} + {\left (3 \, a b^{3} c^{2} d^{3} + 6 \, a^{2} b^{2} c d^{4} + a^{3} b d^{5}\right )} x^{2} + {\left (3 \, a^{2} b^{2} c^{2} d^{3} + 2 \, a^{3} b c 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 (315 \, b^{4} d^{4} x^{4} + 8 \, b^{4} c^{4} - 50 \, a b^{3} c^{3} d + 165 \, a^{2} b^{2} c^{2} d^{2} + 208 \, a^{3} b c d^{3} - 16 \, a^{4} d^{4} + 420 \, {\left (b^{4} c d^{3} + 2 \, a b^{3} d^{4}\right )} x^{3} + 63 \, {\left (b^{4} c^{2} d^{2} + 18 \, a b^{3} c d^{3} + 11 \, a^{2} b^{2} d^{4}\right )} x^{2} - 18 \, {\left (b^{4} c^{3} d - 10 \, a b^{3} c^{2} d^{2} - 53 \, a^{2} b^{2} c d^{3} - 8 \, a^{3} b d^{4}\right )} x\right )} \sqrt {d x + c}}{24 \, {\left (a^{3} b^{5} c^{7} - 5 \, a^{4} b^{4} c^{6} d + 10 \, a^{5} b^{3} c^{5} d^{2} - 10 \, a^{6} b^{2} c^{4} d^{3} + 5 \, a^{7} b c^{3} d^{4} - a^{8} c^{2} d^{5} + {\left (b^{8} c^{5} d^{2} - 5 \, a b^{7} c^{4} d^{3} + 10 \, a^{2} b^{6} c^{3} d^{4} - 10 \, a^{3} b^{5} c^{2} d^{5} + 5 \, a^{4} b^{4} c d^{6} - a^{5} b^{3} d^{7}\right )} x^{5} + {\left (2 \, b^{8} c^{6} d - 7 \, a b^{7} c^{5} d^{2} + 5 \, a^{2} b^{6} c^{4} d^{3} + 10 \, a^{3} b^{5} c^{3} d^{4} - 20 \, a^{4} b^{4} c^{2} d^{5} + 13 \, a^{5} b^{3} c d^{6} - 3 \, a^{6} b^{2} d^{7}\right )} x^{4} + {\left (b^{8} c^{7} + a b^{7} c^{6} d - 17 \, a^{2} b^{6} c^{5} d^{2} + 35 \, a^{3} b^{5} c^{4} d^{3} - 25 \, a^{4} b^{4} c^{3} d^{4} - a^{5} b^{3} c^{2} d^{5} + 9 \, a^{6} b^{2} c d^{6} - 3 \, a^{7} b d^{7}\right )} x^{3} + {\left (3 \, a b^{7} c^{7} - 9 \, a^{2} b^{6} c^{6} d + a^{3} b^{5} c^{5} d^{2} + 25 \, a^{4} b^{4} c^{4} d^{3} - 35 \, a^{5} b^{3} c^{3} d^{4} + 17 \, a^{6} b^{2} c^{2} d^{5} - a^{7} b c d^{6} - a^{8} d^{7}\right )} x^{2} + {\left (3 \, a^{2} b^{6} c^{7} - 13 \, a^{3} b^{5} c^{6} d + 20 \, a^{4} b^{4} c^{5} d^{2} - 10 \, a^{5} b^{3} c^{4} d^{3} - 5 \, a^{6} b^{2} c^{3} d^{4} + 7 \, a^{7} b c^{2} d^{5} - 2 \, a^{8} c d^{6}\right )} x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 432 vs.
\(2 (168) = 336\).
time = 1.40, size = 432, normalized size = 2.16 \begin {gather*} -\frac {105 \, b^{2} d^{3} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{8 \, {\left (b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}\right )} \sqrt {-b^{2} c + a b d}} - \frac {315 \, {\left (d x + c\right )}^{4} b^{4} d^{3} - 840 \, {\left (d x + c\right )}^{3} b^{4} c d^{3} + 693 \, {\left (d x + c\right )}^{2} b^{4} c^{2} d^{3} - 144 \, {\left (d x + c\right )} b^{4} c^{3} d^{3} - 16 \, b^{4} c^{4} d^{3} + 840 \, {\left (d x + c\right )}^{3} a b^{3} d^{4} - 1386 \, {\left (d x + c\right )}^{2} a b^{3} c d^{4} + 432 \, {\left (d x + c\right )} a b^{3} c^{2} d^{4} + 64 \, a b^{3} c^{3} d^{4} + 693 \, {\left (d x + c\right )}^{2} a^{2} b^{2} d^{5} - 432 \, {\left (d x + c\right )} a^{2} b^{2} c d^{5} - 96 \, a^{2} b^{2} c^{2} d^{5} + 144 \, {\left (d x + c\right )} a^{3} b d^{6} + 64 \, a^{3} b c d^{6} - 16 \, a^{4} d^{7}}{24 \, {\left (b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}\right )} {\left ({\left (d x + c\right )}^{\frac {3}{2}} b - \sqrt {d x + c} b c + \sqrt {d x + c} a d\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.64, size = 334, normalized size = 1.67 \begin {gather*} \frac {\frac {231\,b^2\,d^3\,{\left (c+d\,x\right )}^2}{8\,{\left (a\,d-b\,c\right )}^3}-\frac {2\,d^3}{3\,\left (a\,d-b\,c\right )}+\frac {35\,b^3\,d^3\,{\left (c+d\,x\right )}^3}{{\left (a\,d-b\,c\right )}^4}+\frac {105\,b^4\,d^3\,{\left (c+d\,x\right )}^4}{8\,{\left (a\,d-b\,c\right )}^5}+\frac {6\,b\,d^3\,\left (c+d\,x\right )}{{\left (a\,d-b\,c\right )}^2}}{{\left (c+d\,x\right )}^{3/2}\,\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 )}^{9/2}-\left (3\,b^3\,c-3\,a\,b^2\,d\right )\,{\left (c+d\,x\right )}^{7/2}+{\left (c+d\,x\right )}^{5/2}\,\left (3\,a^2\,b\,d^2-6\,a\,b^2\,c\,d+3\,b^3\,c^2\right )}+\frac {105\,b^{3/2}\,d^3\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {c+d\,x}\,\left (a^5\,d^5-5\,a^4\,b\,c\,d^4+10\,a^3\,b^2\,c^2\,d^3-10\,a^2\,b^3\,c^3\,d^2+5\,a\,b^4\,c^4\,d-b^5\,c^5\right )}{{\left (a\,d-b\,c\right )}^{11/2}}\right )}{8\,{\left (a\,d-b\,c\right )}^{11/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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