Optimal. Leaf size=135 \[ -\frac {2}{3 (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {4 d}{(b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/2}}+\frac {16 d^2 \sqrt {a+b x}}{3 (b c-a d)^3 (c+d x)^{3/2}}+\frac {32 b d^2 \sqrt {a+b x}}{3 (b c-a d)^4 \sqrt {c+d x}} \]
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Rubi [A]
time = 0.02, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {47, 37}
\begin {gather*} \frac {32 b d^2 \sqrt {a+b x}}{3 \sqrt {c+d x} (b c-a d)^4}+\frac {16 d^2 \sqrt {a+b x}}{3 (c+d x)^{3/2} (b c-a d)^3}+\frac {4 d}{\sqrt {a+b x} (c+d x)^{3/2} (b c-a d)^2}-\frac {2}{3 (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rubi steps
\begin {align*} \int \frac {1}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx &=-\frac {2}{3 (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}-\frac {(2 d) \int \frac {1}{(a+b x)^{3/2} (c+d x)^{5/2}} \, dx}{b c-a d}\\ &=-\frac {2}{3 (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {4 d}{(b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/2}}+\frac {\left (8 d^2\right ) \int \frac {1}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx}{(b c-a d)^2}\\ &=-\frac {2}{3 (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {4 d}{(b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/2}}+\frac {16 d^2 \sqrt {a+b x}}{3 (b c-a d)^3 (c+d x)^{3/2}}+\frac {\left (16 b d^2\right ) \int \frac {1}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx}{3 (b c-a d)^3}\\ &=-\frac {2}{3 (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {4 d}{(b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/2}}+\frac {16 d^2 \sqrt {a+b x}}{3 (b c-a d)^3 (c+d x)^{3/2}}+\frac {32 b d^2 \sqrt {a+b x}}{3 (b c-a d)^4 \sqrt {c+d x}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 92, normalized size = 0.68 \begin {gather*} -\frac {2 (a+b x)^{3/2} \left (d^3-\frac {9 b d^2 (c+d x)}{a+b x}-\frac {9 b^2 d (c+d x)^2}{(a+b x)^2}+\frac {b^3 (c+d x)^3}{(a+b x)^3}\right )}{3 (b c-a d)^4 (c+d x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 135, normalized size = 1.00
method | result | size |
default | \(-\frac {2}{3 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {3}{2}} \left (d x +c \right )^{\frac {3}{2}}}-\frac {2 d \left (-\frac {2}{\left (-a d +b c \right ) \left (d x +c \right )^{\frac {3}{2}} \sqrt {b x +a}}-\frac {4 d \left (-\frac {2 \sqrt {b x +a}}{3 \left (a d -b c \right ) \left (d x +c \right )^{\frac {3}{2}}}+\frac {4 b \sqrt {b x +a}}{3 \left (a d -b c \right )^{2} \sqrt {d x +c}}\right )}{-a d +b c}\right )}{-a d +b c}\) | \(135\) |
gosper | \(-\frac {2 \left (-16 b^{3} d^{3} x^{3}-24 a \,b^{2} d^{3} x^{2}-24 b^{3} c \,d^{2} x^{2}-6 a^{2} b \,d^{3} x -36 a \,b^{2} c \,d^{2} x -6 b^{3} c^{2} d x +a^{3} d^{3}-9 a^{2} b c \,d^{2}-9 a \,b^{2} c^{2} d +b^{3} c^{3}\right )}{3 \left (d x +c \right )^{\frac {3}{2}} \left (b x +a \right )^{\frac {3}{2}} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}\) | \(169\) |
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. 447 vs.
\(2 (113) = 226\).
time = 1.75, size = 447, normalized size = 3.31 \begin {gather*} \frac {2 \, {\left (16 \, b^{3} d^{3} x^{3} - b^{3} c^{3} + 9 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} - a^{3} d^{3} + 24 \, {\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + 6 \, {\left (b^{3} c^{2} d + 6 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3 \, {\left (a^{2} b^{4} c^{6} - 4 \, a^{3} b^{3} c^{5} d + 6 \, a^{4} b^{2} c^{4} d^{2} - 4 \, a^{5} b c^{3} d^{3} + a^{6} c^{2} d^{4} + {\left (b^{6} c^{4} d^{2} - 4 \, a b^{5} c^{3} d^{3} + 6 \, a^{2} b^{4} c^{2} d^{4} - 4 \, a^{3} b^{3} c d^{5} + a^{4} b^{2} d^{6}\right )} x^{4} + 2 \, {\left (b^{6} c^{5} d - 3 \, a b^{5} c^{4} d^{2} + 2 \, a^{2} b^{4} c^{3} d^{3} + 2 \, a^{3} b^{3} c^{2} d^{4} - 3 \, a^{4} b^{2} c d^{5} + a^{5} b d^{6}\right )} x^{3} + {\left (b^{6} c^{6} - 9 \, a^{2} b^{4} c^{4} d^{2} + 16 \, a^{3} b^{3} c^{3} d^{3} - 9 \, a^{4} b^{2} c^{2} d^{4} + a^{6} d^{6}\right )} x^{2} + 2 \, {\left (a b^{5} c^{6} - 3 \, a^{2} b^{4} c^{5} d + 2 \, a^{3} b^{3} c^{4} d^{2} + 2 \, a^{4} b^{2} c^{3} d^{3} - 3 \, a^{5} b c^{2} d^{4} + a^{6} c d^{5}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a + b x\right )^{\frac {5}{2}} \left (c + d x\right )^{\frac {5}{2}}}\, dx \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. 670 vs.
\(2 (113) = 226\).
time = 2.94, size = 670, normalized size = 4.96 \begin {gather*} \frac {2 \, \sqrt {b x + a} {\left (\frac {8 \, {\left (b^{7} c^{3} d^{4} {\left | b \right |} - 3 \, a b^{6} c^{2} d^{5} {\left | b \right |} + 3 \, a^{2} b^{5} c d^{6} {\left | b \right |} - a^{3} b^{4} d^{7} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{9} c^{7} d - 7 \, a b^{8} c^{6} d^{2} + 21 \, a^{2} b^{7} c^{5} d^{3} - 35 \, a^{3} b^{6} c^{4} d^{4} + 35 \, a^{4} b^{5} c^{3} d^{5} - 21 \, a^{5} b^{4} c^{2} d^{6} + 7 \, a^{6} b^{3} c d^{7} - a^{7} b^{2} d^{8}} + \frac {9 \, {\left (b^{8} c^{4} d^{3} {\left | b \right |} - 4 \, a b^{7} c^{3} d^{4} {\left | b \right |} + 6 \, a^{2} b^{6} c^{2} d^{5} {\left | b \right |} - 4 \, a^{3} b^{5} c d^{6} {\left | b \right |} + a^{4} b^{4} d^{7} {\left | b \right |}\right )}}{b^{9} c^{7} d - 7 \, a b^{8} c^{6} d^{2} + 21 \, a^{2} b^{7} c^{5} d^{3} - 35 \, a^{3} b^{6} c^{4} d^{4} + 35 \, a^{4} b^{5} c^{3} d^{5} - 21 \, a^{5} b^{4} c^{2} d^{6} + 7 \, a^{6} b^{3} c d^{7} - a^{7} b^{2} d^{8}}\right )}}{3 \, {\left (b^{2} c + {\left (b x + a\right )} b d - a b d\right )}^{\frac {3}{2}}} + \frac {8 \, {\left (4 \, \sqrt {b d} b^{7} c^{2} d - 8 \, \sqrt {b d} a b^{6} c d^{2} + 4 \, \sqrt {b d} a^{2} b^{5} d^{3} - 9 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{5} c d + 9 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b^{4} d^{2} + 3 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} b^{3} d\right )}}{3 \, {\left (b^{3} c^{3} {\left | b \right |} - 3 \, a b^{2} c^{2} d {\left | b \right |} + 3 \, a^{2} b c d^{2} {\left | b \right |} - a^{3} d^{3} {\left | b \right |}\right )} {\left (b^{2} c - a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.85, size = 224, normalized size = 1.66 \begin {gather*} \frac {\sqrt {c+d\,x}\,\left (\frac {16\,b\,x^2\,\left (a\,d+b\,c\right )}{{\left (a\,d-b\,c\right )}^4}-\frac {2\,a^3\,d^3-18\,a^2\,b\,c\,d^2-18\,a\,b^2\,c^2\,d+2\,b^3\,c^3}{3\,b\,d^2\,{\left (a\,d-b\,c\right )}^4}+\frac {32\,b^2\,d\,x^3}{3\,{\left (a\,d-b\,c\right )}^4}+\frac {4\,x\,\left (a^2\,d^2+6\,a\,b\,c\,d+b^2\,c^2\right )}{d\,{\left (a\,d-b\,c\right )}^4}\right )}{x^3\,\sqrt {a+b\,x}+\frac {a\,c^2\,\sqrt {a+b\,x}}{b\,d^2}+\frac {x^2\,\left (a\,d+2\,b\,c\right )\,\sqrt {a+b\,x}}{b\,d}+\frac {c\,x\,\left (2\,a\,d+b\,c\right )\,\sqrt {a+b\,x}}{b\,d^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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