Optimal. Leaf size=278 \[ -\frac {7 d (7 b c-15 a d) (b c-a d) \sqrt {a+b x}}{24 c^4 (c+d x)^{3/2}}-\frac {3 a (b c-a d) \sqrt {a+b x}}{4 c^2 x^2 (c+d x)^{3/2}}-\frac {(11 b c-21 a d) (b c-a d) \sqrt {a+b x}}{8 c^3 x (c+d x)^{3/2}}-\frac {a (a+b x)^{3/2}}{3 c x^3 (c+d x)^{3/2}}-\frac {d \left (113 b^2 c^2-420 a b c d+315 a^2 d^2\right ) \sqrt {a+b x}}{24 c^5 \sqrt {c+d x}}-\frac {5 (b c-a d) \left (b^2 c^2-14 a b c d+21 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 \sqrt {a} c^{11/2}} \]
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Rubi [A]
time = 0.24, antiderivative size = 278, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {100, 154, 156,
157, 12, 95, 214} \begin {gather*} -\frac {5 (b c-a d) \left (21 a^2 d^2-14 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 \sqrt {a} c^{11/2}}-\frac {d \sqrt {a+b x} \left (315 a^2 d^2-420 a b c d+113 b^2 c^2\right )}{24 c^5 \sqrt {c+d x}}-\frac {7 d \sqrt {a+b x} (7 b c-15 a d) (b c-a d)}{24 c^4 (c+d x)^{3/2}}-\frac {\sqrt {a+b x} (11 b c-21 a d) (b c-a d)}{8 c^3 x (c+d x)^{3/2}}-\frac {3 a \sqrt {a+b x} (b c-a d)}{4 c^2 x^2 (c+d x)^{3/2}}-\frac {a (a+b x)^{3/2}}{3 c x^3 (c+d x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 95
Rule 100
Rule 154
Rule 156
Rule 157
Rule 214
Rubi steps
\begin {align*} \int \frac {(a+b x)^{5/2}}{x^4 (c+d x)^{5/2}} \, dx &=-\frac {a (a+b x)^{3/2}}{3 c x^3 (c+d x)^{3/2}}-\frac {\int \frac {\sqrt {a+b x} \left (-\frac {9}{2} a (b c-a d)-3 b (b c-a d) x\right )}{x^3 (c+d x)^{5/2}} \, dx}{3 c}\\ &=-\frac {3 a (b c-a d) \sqrt {a+b x}}{4 c^2 x^2 (c+d x)^{3/2}}-\frac {a (a+b x)^{3/2}}{3 c x^3 (c+d x)^{3/2}}-\frac {\int \frac {-\frac {3}{4} a (11 b c-21 a d) (b c-a d)-\frac {3}{2} b (4 b c-9 a d) (b c-a d) x}{x^2 \sqrt {a+b x} (c+d x)^{5/2}} \, dx}{6 c^2}\\ &=-\frac {3 a (b c-a d) \sqrt {a+b x}}{4 c^2 x^2 (c+d x)^{3/2}}-\frac {(11 b c-21 a d) (b c-a d) \sqrt {a+b x}}{8 c^3 x (c+d x)^{3/2}}-\frac {a (a+b x)^{3/2}}{3 c x^3 (c+d x)^{3/2}}+\frac {\int \frac {\frac {15}{8} a (b c-a d) \left (b^2 c^2-14 a b c d+21 a^2 d^2\right )-\frac {3}{2} a b d (11 b c-21 a d) (b c-a d) x}{x \sqrt {a+b x} (c+d x)^{5/2}} \, dx}{6 a c^3}\\ &=-\frac {7 d (7 b c-15 a d) (b c-a d) \sqrt {a+b x}}{24 c^4 (c+d x)^{3/2}}-\frac {3 a (b c-a d) \sqrt {a+b x}}{4 c^2 x^2 (c+d x)^{3/2}}-\frac {(11 b c-21 a d) (b c-a d) \sqrt {a+b x}}{8 c^3 x (c+d x)^{3/2}}-\frac {a (a+b x)^{3/2}}{3 c x^3 (c+d x)^{3/2}}-\frac {\int \frac {-\frac {45}{16} a (b c-a d)^2 \left (b^2 c^2-14 a b c d+21 a^2 d^2\right )+\frac {21}{8} a b d (7 b c-15 a d) (b c-a d)^2 x}{x \sqrt {a+b x} (c+d x)^{3/2}} \, dx}{9 a c^4 (b c-a d)}\\ &=-\frac {7 d (7 b c-15 a d) (b c-a d) \sqrt {a+b x}}{24 c^4 (c+d x)^{3/2}}-\frac {3 a (b c-a d) \sqrt {a+b x}}{4 c^2 x^2 (c+d x)^{3/2}}-\frac {(11 b c-21 a d) (b c-a d) \sqrt {a+b x}}{8 c^3 x (c+d x)^{3/2}}-\frac {a (a+b x)^{3/2}}{3 c x^3 (c+d x)^{3/2}}-\frac {d \left (113 b^2 c^2-420 a b c d+315 a^2 d^2\right ) \sqrt {a+b x}}{24 c^5 \sqrt {c+d x}}+\frac {2 \int \frac {45 a (b c-a d)^3 \left (b^2 c^2-14 a b c d+21 a^2 d^2\right )}{32 x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{9 a c^5 (b c-a d)^2}\\ &=-\frac {7 d (7 b c-15 a d) (b c-a d) \sqrt {a+b x}}{24 c^4 (c+d x)^{3/2}}-\frac {3 a (b c-a d) \sqrt {a+b x}}{4 c^2 x^2 (c+d x)^{3/2}}-\frac {(11 b c-21 a d) (b c-a d) \sqrt {a+b x}}{8 c^3 x (c+d x)^{3/2}}-\frac {a (a+b x)^{3/2}}{3 c x^3 (c+d x)^{3/2}}-\frac {d \left (113 b^2 c^2-420 a b c d+315 a^2 d^2\right ) \sqrt {a+b x}}{24 c^5 \sqrt {c+d x}}+\frac {\left (5 (b c-a d) \left (b^2 c^2-14 a b c d+21 a^2 d^2\right )\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{16 c^5}\\ &=-\frac {7 d (7 b c-15 a d) (b c-a d) \sqrt {a+b x}}{24 c^4 (c+d x)^{3/2}}-\frac {3 a (b c-a d) \sqrt {a+b x}}{4 c^2 x^2 (c+d x)^{3/2}}-\frac {(11 b c-21 a d) (b c-a d) \sqrt {a+b x}}{8 c^3 x (c+d x)^{3/2}}-\frac {a (a+b x)^{3/2}}{3 c x^3 (c+d x)^{3/2}}-\frac {d \left (113 b^2 c^2-420 a b c d+315 a^2 d^2\right ) \sqrt {a+b x}}{24 c^5 \sqrt {c+d x}}+\frac {\left (5 (b c-a d) \left (b^2 c^2-14 a b c d+21 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{8 c^5}\\ &=-\frac {7 d (7 b c-15 a d) (b c-a d) \sqrt {a+b x}}{24 c^4 (c+d x)^{3/2}}-\frac {3 a (b c-a d) \sqrt {a+b x}}{4 c^2 x^2 (c+d x)^{3/2}}-\frac {(11 b c-21 a d) (b c-a d) \sqrt {a+b x}}{8 c^3 x (c+d x)^{3/2}}-\frac {a (a+b x)^{3/2}}{3 c x^3 (c+d x)^{3/2}}-\frac {d \left (113 b^2 c^2-420 a b c d+315 a^2 d^2\right ) \sqrt {a+b x}}{24 c^5 \sqrt {c+d x}}-\frac {5 (b c-a d) \left (b^2 c^2-14 a b c d+21 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 \sqrt {a} c^{11/2}}\\ \end {align*}
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Mathematica [A]
time = 10.31, size = 198, normalized size = 0.71 \begin {gather*} \frac {-8 a c^{9/2} (a+b x)^{7/2}-2 c^{7/2} (b c-9 a d) x (a+b x)^{7/2}-\left (b^2 c^2-14 a b c d+21 a^2 d^2\right ) x^2 \left (3 c^{5/2} (a+b x)^{5/2}-5 (b c-a d) x \left (\sqrt {c} \sqrt {a+b x} (4 a c+b c x+3 a d x)-3 a^{3/2} (c+d x)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )\right )\right )}{24 a^2 c^{11/2} x^3 (c+d x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1008\) vs.
\(2(234)=468\).
time = 0.08, size = 1009, normalized size = 3.63
method | result | size |
default | \(\frac {\sqrt {b x +a}\, \left (315 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{3} d^{5} x^{5}-525 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{2} b c \,d^{4} x^{5}+225 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a \,b^{2} c^{2} d^{3} x^{5}-15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) b^{3} c^{3} d^{2} x^{5}+630 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{3} c \,d^{4} x^{4}-1050 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{2} b \,c^{2} d^{3} x^{4}+450 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a \,b^{2} c^{3} d^{2} x^{4}-30 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) b^{3} c^{4} d \,x^{4}+315 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{3} c^{2} d^{3} x^{3}-525 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{2} b \,c^{3} d^{2} x^{3}+225 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a \,b^{2} c^{4} d \,x^{3}-15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) b^{3} c^{5} x^{3}-630 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {a c}\, a^{2} d^{4} x^{4}+840 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {a c}\, a b c \,d^{3} x^{4}-226 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {a c}\, b^{2} c^{2} d^{2} x^{4}-840 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {a c}\, a^{2} c \,d^{3} x^{3}+1148 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {a c}\, a b \,c^{2} d^{2} x^{3}-324 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {a c}\, b^{2} c^{3} d \,x^{3}-126 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {a c}\, a^{2} c^{2} d^{2} x^{2}+192 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {a c}\, a b \,c^{3} d \,x^{2}-66 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {a c}\, b^{2} c^{4} x^{2}+36 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {a c}\, a^{2} c^{3} d x -52 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {a c}\, a b \,c^{4} x -16 a^{2} c^{4} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\right )}{48 c^{5} \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, x^{3} \sqrt {a c}\, \left (d x +c \right )^{\frac {3}{2}}}\) | \(1009\) |
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 [A]
time = 8.22, size = 842, normalized size = 3.03 \begin {gather*} \left [-\frac {15 \, {\left ({\left (b^{3} c^{3} d^{2} - 15 \, a b^{2} c^{2} d^{3} + 35 \, a^{2} b c d^{4} - 21 \, a^{3} d^{5}\right )} x^{5} + 2 \, {\left (b^{3} c^{4} d - 15 \, a b^{2} c^{3} d^{2} + 35 \, a^{2} b c^{2} d^{3} - 21 \, a^{3} c d^{4}\right )} x^{4} + {\left (b^{3} c^{5} - 15 \, a b^{2} c^{4} d + 35 \, a^{2} b c^{3} d^{2} - 21 \, a^{3} c^{2} d^{3}\right )} x^{3}\right )} \sqrt {a c} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \, {\left (8 \, a^{3} c^{5} + {\left (113 \, a b^{2} c^{3} d^{2} - 420 \, a^{2} b c^{2} d^{3} + 315 \, a^{3} c d^{4}\right )} x^{4} + 2 \, {\left (81 \, a b^{2} c^{4} d - 287 \, a^{2} b c^{3} d^{2} + 210 \, a^{3} c^{2} d^{3}\right )} x^{3} + 3 \, {\left (11 \, a b^{2} c^{5} - 32 \, a^{2} b c^{4} d + 21 \, a^{3} c^{3} d^{2}\right )} x^{2} + 2 \, {\left (13 \, a^{2} b c^{5} - 9 \, a^{3} c^{4} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{96 \, {\left (a c^{6} d^{2} x^{5} + 2 \, a c^{7} d x^{4} + a c^{8} x^{3}\right )}}, \frac {15 \, {\left ({\left (b^{3} c^{3} d^{2} - 15 \, a b^{2} c^{2} d^{3} + 35 \, a^{2} b c d^{4} - 21 \, a^{3} d^{5}\right )} x^{5} + 2 \, {\left (b^{3} c^{4} d - 15 \, a b^{2} c^{3} d^{2} + 35 \, a^{2} b c^{2} d^{3} - 21 \, a^{3} c d^{4}\right )} x^{4} + {\left (b^{3} c^{5} - 15 \, a b^{2} c^{4} d + 35 \, a^{2} b c^{3} d^{2} - 21 \, a^{3} c^{2} d^{3}\right )} x^{3}\right )} \sqrt {-a c} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) - 2 \, {\left (8 \, a^{3} c^{5} + {\left (113 \, a b^{2} c^{3} d^{2} - 420 \, a^{2} b c^{2} d^{3} + 315 \, a^{3} c d^{4}\right )} x^{4} + 2 \, {\left (81 \, a b^{2} c^{4} d - 287 \, a^{2} b c^{3} d^{2} + 210 \, a^{3} c^{2} d^{3}\right )} x^{3} + 3 \, {\left (11 \, a b^{2} c^{5} - 32 \, a^{2} b c^{4} d + 21 \, a^{3} c^{3} d^{2}\right )} x^{2} + 2 \, {\left (13 \, a^{2} b c^{5} - 9 \, a^{3} c^{4} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, {\left (a c^{6} d^{2} x^{5} + 2 \, a c^{7} d x^{4} + a c^{8} x^{3}\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. 2354 vs.
\(2 (234) = 468\).
time = 10.78, size = 2354, normalized size = 8.47 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^{5/2}}{x^4\,{\left (c+d\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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