Optimal. Leaf size=172 \[ -\frac {256 b d^3 \sqrt {a+b x}}{15 \sqrt {c+d x} (b c-a d)^5}-\frac {128 d^3 \sqrt {a+b x}}{15 (c+d x)^{3/2} (b c-a d)^4}-\frac {32 d^2}{5 \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)^3}+\frac {16 d}{15 (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)^2}-\frac {2}{5 (a+b x)^{5/2} (c+d x)^{3/2} (b c-a d)} \]
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Rubi [A] time = 0.04, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {45, 37} \[ -\frac {256 b d^3 \sqrt {a+b x}}{15 \sqrt {c+d x} (b c-a d)^5}-\frac {128 d^3 \sqrt {a+b x}}{15 (c+d x)^{3/2} (b c-a d)^4}-\frac {32 d^2}{5 \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)^3}+\frac {16 d}{15 (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)^2}-\frac {2}{5 (a+b x)^{5/2} (c+d x)^{3/2} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rubi steps
\begin {align*} \int \frac {1}{(a+b x)^{7/2} (c+d x)^{5/2}} \, dx &=-\frac {2}{5 (b c-a d) (a+b x)^{5/2} (c+d x)^{3/2}}-\frac {(8 d) \int \frac {1}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx}{5 (b c-a d)}\\ &=-\frac {2}{5 (b c-a d) (a+b x)^{5/2} (c+d x)^{3/2}}+\frac {16 d}{15 (b c-a d)^2 (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {\left (16 d^2\right ) \int \frac {1}{(a+b x)^{3/2} (c+d x)^{5/2}} \, dx}{5 (b c-a d)^2}\\ &=-\frac {2}{5 (b c-a d) (a+b x)^{5/2} (c+d x)^{3/2}}+\frac {16 d}{15 (b c-a d)^2 (a+b x)^{3/2} (c+d x)^{3/2}}-\frac {32 d^2}{5 (b c-a d)^3 \sqrt {a+b x} (c+d x)^{3/2}}-\frac {\left (64 d^3\right ) \int \frac {1}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx}{5 (b c-a d)^3}\\ &=-\frac {2}{5 (b c-a d) (a+b x)^{5/2} (c+d x)^{3/2}}+\frac {16 d}{15 (b c-a d)^2 (a+b x)^{3/2} (c+d x)^{3/2}}-\frac {32 d^2}{5 (b c-a d)^3 \sqrt {a+b x} (c+d x)^{3/2}}-\frac {128 d^3 \sqrt {a+b x}}{15 (b c-a d)^4 (c+d x)^{3/2}}-\frac {\left (128 b d^3\right ) \int \frac {1}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx}{15 (b c-a d)^4}\\ &=-\frac {2}{5 (b c-a d) (a+b x)^{5/2} (c+d x)^{3/2}}+\frac {16 d}{15 (b c-a d)^2 (a+b x)^{3/2} (c+d x)^{3/2}}-\frac {32 d^2}{5 (b c-a d)^3 \sqrt {a+b x} (c+d x)^{3/2}}-\frac {128 d^3 \sqrt {a+b x}}{15 (b c-a d)^4 (c+d x)^{3/2}}-\frac {256 b d^3 \sqrt {a+b x}}{15 (b c-a d)^5 \sqrt {c+d x}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 170, normalized size = 0.99 \[ -\frac {2 \left (-5 a^4 d^4+20 a^3 b d^3 (3 c+2 d x)+30 a^2 b^2 d^2 \left (3 c^2+12 c d x+8 d^2 x^2\right )+20 a b^3 d \left (-c^3+6 c^2 d x+24 c d^2 x^2+16 d^3 x^3\right )+b^4 \left (3 c^4-8 c^3 d x+48 c^2 d^2 x^2+192 c d^3 x^3+128 d^4 x^4\right )\right )}{15 (a+b x)^{5/2} (c+d x)^{3/2} (b c-a d)^5} \]
Antiderivative was successfully verified.
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fricas [B] time = 3.43, size = 715, normalized size = 4.16 \[ -\frac {2 \, {\left (128 \, b^{4} d^{4} x^{4} + 3 \, b^{4} c^{4} - 20 \, a b^{3} c^{3} d + 90 \, a^{2} b^{2} c^{2} d^{2} + 60 \, a^{3} b c d^{3} - 5 \, a^{4} d^{4} + 64 \, {\left (3 \, b^{4} c d^{3} + 5 \, a b^{3} d^{4}\right )} x^{3} + 48 \, {\left (b^{4} c^{2} d^{2} + 10 \, a b^{3} c d^{3} + 5 \, a^{2} b^{2} d^{4}\right )} x^{2} - 8 \, {\left (b^{4} c^{3} d - 15 \, a b^{3} c^{2} d^{2} - 45 \, a^{2} b^{2} c d^{3} - 5 \, a^{3} b d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{15 \, {\left (a^{3} b^{5} c^{7} - 5 \, a^{4} b^{4} c^{6} d + 10 \, a^{5} b^{3} c^{5} d^{2} - 10 \, a^{6} b^{2} c^{4} d^{3} + 5 \, a^{7} b c^{3} d^{4} - a^{8} c^{2} d^{5} + {\left (b^{8} c^{5} d^{2} - 5 \, a b^{7} c^{4} d^{3} + 10 \, a^{2} b^{6} c^{3} d^{4} - 10 \, a^{3} b^{5} c^{2} d^{5} + 5 \, a^{4} b^{4} c d^{6} - a^{5} b^{3} d^{7}\right )} x^{5} + {\left (2 \, b^{8} c^{6} d - 7 \, a b^{7} c^{5} d^{2} + 5 \, a^{2} b^{6} c^{4} d^{3} + 10 \, a^{3} b^{5} c^{3} d^{4} - 20 \, a^{4} b^{4} c^{2} d^{5} + 13 \, a^{5} b^{3} c d^{6} - 3 \, a^{6} b^{2} d^{7}\right )} x^{4} + {\left (b^{8} c^{7} + a b^{7} c^{6} d - 17 \, a^{2} b^{6} c^{5} d^{2} + 35 \, a^{3} b^{5} c^{4} d^{3} - 25 \, a^{4} b^{4} c^{3} d^{4} - a^{5} b^{3} c^{2} d^{5} + 9 \, a^{6} b^{2} c d^{6} - 3 \, a^{7} b d^{7}\right )} x^{3} + {\left (3 \, a b^{7} c^{7} - 9 \, a^{2} b^{6} c^{6} d + a^{3} b^{5} c^{5} d^{2} + 25 \, a^{4} b^{4} c^{4} d^{3} - 35 \, a^{5} b^{3} c^{3} d^{4} + 17 \, a^{6} b^{2} c^{2} d^{5} - a^{7} b c d^{6} - a^{8} d^{7}\right )} x^{2} + {\left (3 \, a^{2} b^{6} c^{7} - 13 \, a^{3} b^{5} c^{6} d + 20 \, a^{4} b^{4} c^{5} d^{2} - 10 \, a^{5} b^{3} c^{4} d^{3} - 5 \, a^{6} b^{2} c^{3} d^{4} + 7 \, a^{7} b c^{2} d^{5} - 2 \, a^{8} c d^{6}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 3.92, size = 1203, normalized size = 6.99 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 256, normalized size = 1.49 \[ -\frac {2 \left (-128 b^{4} x^{4} d^{4}-320 a \,b^{3} d^{4} x^{3}-192 b^{4} c \,d^{3} x^{3}-240 a^{2} b^{2} d^{4} x^{2}-480 a \,b^{3} c \,d^{3} x^{2}-48 b^{4} c^{2} d^{2} x^{2}-40 a^{3} b \,d^{4} x -360 a^{2} b^{2} c \,d^{3} x -120 a \,b^{3} c^{2} d^{2} x +8 b^{4} c^{3} d x +5 a^{4} d^{4}-60 a^{3} b c \,d^{3}-90 a^{2} b^{2} c^{2} d^{2}+20 a \,b^{3} c^{3} d -3 b^{4} c^{4}\right )}{15 \left (b x +a \right )^{\frac {5}{2}} \left (d x +c \right )^{\frac {3}{2}} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right )} \]
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 = 1.53, size = 346, normalized size = 2.01 \[ \frac {\sqrt {c+d\,x}\,\left (\frac {32\,x^2\,\left (5\,a^2\,d^2+10\,a\,b\,c\,d+b^2\,c^2\right )}{5\,{\left (a\,d-b\,c\right )}^5}+\frac {256\,b^2\,d^2\,x^4}{15\,{\left (a\,d-b\,c\right )}^5}+\frac {-10\,a^4\,d^4+120\,a^3\,b\,c\,d^3+180\,a^2\,b^2\,c^2\,d^2-40\,a\,b^3\,c^3\,d+6\,b^4\,c^4}{15\,b^2\,d^2\,{\left (a\,d-b\,c\right )}^5}+\frac {x\,\left (80\,a^3\,b\,d^4+720\,a^2\,b^2\,c\,d^3+240\,a\,b^3\,c^2\,d^2-16\,b^4\,c^3\,d\right )}{15\,b^2\,d^2\,{\left (a\,d-b\,c\right )}^5}+\frac {128\,b\,d\,x^3\,\left (5\,a\,d+3\,b\,c\right )}{15\,{\left (a\,d-b\,c\right )}^5}\right )}{x^4\,\sqrt {a+b\,x}+\frac {x^2\,\sqrt {a+b\,x}\,\left (a^2\,d^2+4\,a\,b\,c\,d+b^2\,c^2\right )}{b^2\,d^2}+\frac {2\,x^3\,\left (a\,d+b\,c\right )\,\sqrt {a+b\,x}}{b\,d}+\frac {a^2\,c^2\,\sqrt {a+b\,x}}{b^2\,d^2}+\frac {2\,a\,c\,x\,\left (a\,d+b\,c\right )\,\sqrt {a+b\,x}}{b^2\,d^2}} \]
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 \[ \int \frac {1}{\left (a + b x\right )^{\frac {7}{2}} \left (c + d x\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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