Optimal. Leaf size=135 \[ \frac {32 b d^2 \sqrt {a+b x}}{3 \sqrt {c+d x} (b c-a d)^4}+\frac {16 d^2 \sqrt {a+b x}}{3 (c+d x)^{3/2} (b c-a d)^3}+\frac {4 d}{\sqrt {a+b x} (c+d x)^{3/2} (b c-a d)^2}-\frac {2}{3 (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)} \]
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Rubi [A] time = 0.03, 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 = {45, 37} \[ \frac {32 b d^2 \sqrt {a+b x}}{3 \sqrt {c+d x} (b c-a d)^4}+\frac {16 d^2 \sqrt {a+b x}}{3 (c+d x)^{3/2} (b c-a d)^3}+\frac {4 d}{\sqrt {a+b x} (c+d x)^{3/2} (b c-a d)^2}-\frac {2}{3 (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
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 = 0.04, size = 118, normalized size = 0.87 \[ \frac {-2 a^3 d^3+6 a^2 b d^2 (3 c+2 d x)+6 a b^2 d \left (3 c^2+12 c d x+8 d^2 x^2\right )+b^3 \left (-2 c^3+12 c^2 d x+48 c d^2 x^2+32 d^3 x^3\right )}{3 (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 2.68, size = 447, normalized size = 3.31 \[ \frac {2 \, {\left (16 \, b^{3} d^{3} x^{3} - b^{3} c^{3} + 9 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} - a^{3} d^{3} + 24 \, {\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + 6 \, {\left (b^{3} c^{2} d + 6 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3 \, {\left (a^{2} b^{4} c^{6} - 4 \, a^{3} b^{3} c^{5} d + 6 \, a^{4} b^{2} c^{4} d^{2} - 4 \, a^{5} b c^{3} d^{3} + a^{6} c^{2} d^{4} + {\left (b^{6} c^{4} d^{2} - 4 \, a b^{5} c^{3} d^{3} + 6 \, a^{2} b^{4} c^{2} d^{4} - 4 \, a^{3} b^{3} c d^{5} + a^{4} b^{2} d^{6}\right )} x^{4} + 2 \, {\left (b^{6} c^{5} d - 3 \, a b^{5} c^{4} d^{2} + 2 \, a^{2} b^{4} c^{3} d^{3} + 2 \, a^{3} b^{3} c^{2} d^{4} - 3 \, a^{4} b^{2} c d^{5} + a^{5} b d^{6}\right )} x^{3} + {\left (b^{6} c^{6} - 9 \, a^{2} b^{4} c^{4} d^{2} + 16 \, a^{3} b^{3} c^{3} d^{3} - 9 \, a^{4} b^{2} c^{2} d^{4} + a^{6} d^{6}\right )} x^{2} + 2 \, {\left (a b^{5} c^{6} - 3 \, a^{2} b^{4} c^{5} d + 2 \, a^{3} b^{3} c^{4} d^{2} + 2 \, a^{4} b^{2} c^{3} d^{3} - 3 \, a^{5} b c^{2} d^{4} + a^{6} c d^{5}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.66, size = 670, normalized size = 4.96 \[ \frac {2 \, \sqrt {b x + a} {\left (\frac {8 \, {\left (b^{7} c^{3} d^{4} {\left | b \right |} - 3 \, a b^{6} c^{2} d^{5} {\left | b \right |} + 3 \, a^{2} b^{5} c d^{6} {\left | b \right |} - a^{3} b^{4} d^{7} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{9} c^{7} d - 7 \, a b^{8} c^{6} d^{2} + 21 \, a^{2} b^{7} c^{5} d^{3} - 35 \, a^{3} b^{6} c^{4} d^{4} + 35 \, a^{4} b^{5} c^{3} d^{5} - 21 \, a^{5} b^{4} c^{2} d^{6} + 7 \, a^{6} b^{3} c d^{7} - a^{7} b^{2} d^{8}} + \frac {9 \, {\left (b^{8} c^{4} d^{3} {\left | b \right |} - 4 \, a b^{7} c^{3} d^{4} {\left | b \right |} + 6 \, a^{2} b^{6} c^{2} d^{5} {\left | b \right |} - 4 \, a^{3} b^{5} c d^{6} {\left | b \right |} + a^{4} b^{4} d^{7} {\left | b \right |}\right )}}{b^{9} c^{7} d - 7 \, a b^{8} c^{6} d^{2} + 21 \, a^{2} b^{7} c^{5} d^{3} - 35 \, a^{3} b^{6} c^{4} d^{4} + 35 \, a^{4} b^{5} c^{3} d^{5} - 21 \, a^{5} b^{4} c^{2} d^{6} + 7 \, a^{6} b^{3} c d^{7} - a^{7} b^{2} d^{8}}\right )}}{3 \, {\left (b^{2} c + {\left (b x + a\right )} b d - a b d\right )}^{\frac {3}{2}}} + \frac {8 \, {\left (4 \, \sqrt {b d} b^{7} c^{2} d - 8 \, \sqrt {b d} a b^{6} c d^{2} + 4 \, \sqrt {b d} a^{2} b^{5} d^{3} - 9 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{5} c d + 9 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b^{4} d^{2} + 3 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} b^{3} d\right )}}{3 \, {\left (b^{3} c^{3} {\left | b \right |} - 3 \, a b^{2} c^{2} d {\left | b \right |} + 3 \, a^{2} b c d^{2} {\left | b \right |} - a^{3} d^{3} {\left | b \right |}\right )} {\left (b^{2} c - a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 169, normalized size = 1.25 \[ -\frac {2 \left (-16 b^{3} d^{3} x^{3}-24 a \,b^{2} d^{3} x^{2}-24 b^{3} c \,d^{2} x^{2}-6 a^{2} b \,d^{3} x -36 a \,b^{2} c \,d^{2} x -6 b^{3} c^{2} d x +a^{3} d^{3}-9 a^{2} b c \,d^{2}-9 a \,b^{2} c^{2} d +b^{3} c^{3}\right )}{3 \left (b x +a \right )^{\frac {3}{2}} \left (d x +c \right )^{\frac {3}{2}} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )} \]
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 \[ \text {Exception raised: ValueError} \]
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 \[ \frac {\sqrt {c+d\,x}\,\left (\frac {16\,b\,x^2\,\left (a\,d+b\,c\right )}{{\left (a\,d-b\,c\right )}^4}-\frac {2\,a^3\,d^3-18\,a^2\,b\,c\,d^2-18\,a\,b^2\,c^2\,d+2\,b^3\,c^3}{3\,b\,d^2\,{\left (a\,d-b\,c\right )}^4}+\frac {32\,b^2\,d\,x^3}{3\,{\left (a\,d-b\,c\right )}^4}+\frac {4\,x\,\left (a^2\,d^2+6\,a\,b\,c\,d+b^2\,c^2\right )}{d\,{\left (a\,d-b\,c\right )}^4}\right )}{x^3\,\sqrt {a+b\,x}+\frac {a\,c^2\,\sqrt {a+b\,x}}{b\,d^2}+\frac {x^2\,\left (a\,d+2\,b\,c\right )\,\sqrt {a+b\,x}}{b\,d}+\frac {c\,x\,\left (2\,a\,d+b\,c\right )\,\sqrt {a+b\,x}}{b\,d^2}} \]
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 \[ \int \frac {1}{\left (a + b x\right )^{\frac {5}{2}} \left (c + d x\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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