Optimal. Leaf size=204 \[ -\frac {2 (a+b x)^{9/2}}{3 d (c+d x)^{3/2}}-\frac {6 b (a+b x)^{7/2}}{d^2 \sqrt {c+d x}}+\frac {105 b^2 (b c-a d)^2 \sqrt {a+b x} \sqrt {c+d x}}{8 d^5}-\frac {35 b^2 (b c-a d) (a+b x)^{3/2} \sqrt {c+d x}}{4 d^4}+\frac {7 b^2 (a+b x)^{5/2} \sqrt {c+d x}}{d^3}-\frac {105 b^{3/2} (b c-a d)^3 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 d^{11/2}} \]
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Rubi [A]
time = 0.07, antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {49, 52, 65, 223,
212} \begin {gather*} -\frac {105 b^{3/2} (b c-a d)^3 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 d^{11/2}}+\frac {105 b^2 \sqrt {a+b x} \sqrt {c+d x} (b c-a d)^2}{8 d^5}-\frac {35 b^2 (a+b x)^{3/2} \sqrt {c+d x} (b c-a d)}{4 d^4}+\frac {7 b^2 (a+b x)^{5/2} \sqrt {c+d x}}{d^3}-\frac {6 b (a+b x)^{7/2}}{d^2 \sqrt {c+d x}}-\frac {2 (a+b x)^{9/2}}{3 d (c+d x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 49
Rule 52
Rule 65
Rule 212
Rule 223
Rubi steps
\begin {align*} \int \frac {(a+b x)^{9/2}}{(c+d x)^{5/2}} \, dx &=-\frac {2 (a+b x)^{9/2}}{3 d (c+d x)^{3/2}}+\frac {(3 b) \int \frac {(a+b x)^{7/2}}{(c+d x)^{3/2}} \, dx}{d}\\ &=-\frac {2 (a+b x)^{9/2}}{3 d (c+d x)^{3/2}}-\frac {6 b (a+b x)^{7/2}}{d^2 \sqrt {c+d x}}+\frac {\left (21 b^2\right ) \int \frac {(a+b x)^{5/2}}{\sqrt {c+d x}} \, dx}{d^2}\\ &=-\frac {2 (a+b x)^{9/2}}{3 d (c+d x)^{3/2}}-\frac {6 b (a+b x)^{7/2}}{d^2 \sqrt {c+d x}}+\frac {7 b^2 (a+b x)^{5/2} \sqrt {c+d x}}{d^3}-\frac {\left (35 b^2 (b c-a d)\right ) \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}} \, dx}{2 d^3}\\ &=-\frac {2 (a+b x)^{9/2}}{3 d (c+d x)^{3/2}}-\frac {6 b (a+b x)^{7/2}}{d^2 \sqrt {c+d x}}-\frac {35 b^2 (b c-a d) (a+b x)^{3/2} \sqrt {c+d x}}{4 d^4}+\frac {7 b^2 (a+b x)^{5/2} \sqrt {c+d x}}{d^3}+\frac {\left (105 b^2 (b c-a d)^2\right ) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{8 d^4}\\ &=-\frac {2 (a+b x)^{9/2}}{3 d (c+d x)^{3/2}}-\frac {6 b (a+b x)^{7/2}}{d^2 \sqrt {c+d x}}+\frac {105 b^2 (b c-a d)^2 \sqrt {a+b x} \sqrt {c+d x}}{8 d^5}-\frac {35 b^2 (b c-a d) (a+b x)^{3/2} \sqrt {c+d x}}{4 d^4}+\frac {7 b^2 (a+b x)^{5/2} \sqrt {c+d x}}{d^3}-\frac {\left (105 b^2 (b c-a d)^3\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{16 d^5}\\ &=-\frac {2 (a+b x)^{9/2}}{3 d (c+d x)^{3/2}}-\frac {6 b (a+b x)^{7/2}}{d^2 \sqrt {c+d x}}+\frac {105 b^2 (b c-a d)^2 \sqrt {a+b x} \sqrt {c+d x}}{8 d^5}-\frac {35 b^2 (b c-a d) (a+b x)^{3/2} \sqrt {c+d x}}{4 d^4}+\frac {7 b^2 (a+b x)^{5/2} \sqrt {c+d x}}{d^3}-\frac {\left (105 b (b c-a d)^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{8 d^5}\\ &=-\frac {2 (a+b x)^{9/2}}{3 d (c+d x)^{3/2}}-\frac {6 b (a+b x)^{7/2}}{d^2 \sqrt {c+d x}}+\frac {105 b^2 (b c-a d)^2 \sqrt {a+b x} \sqrt {c+d x}}{8 d^5}-\frac {35 b^2 (b c-a d) (a+b x)^{3/2} \sqrt {c+d x}}{4 d^4}+\frac {7 b^2 (a+b x)^{5/2} \sqrt {c+d x}}{d^3}-\frac {\left (105 b (b c-a d)^3\right ) \text {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{8 d^5}\\ &=-\frac {2 (a+b x)^{9/2}}{3 d (c+d x)^{3/2}}-\frac {6 b (a+b x)^{7/2}}{d^2 \sqrt {c+d x}}+\frac {105 b^2 (b c-a d)^2 \sqrt {a+b x} \sqrt {c+d x}}{8 d^5}-\frac {35 b^2 (b c-a d) (a+b x)^{3/2} \sqrt {c+d x}}{4 d^4}+\frac {7 b^2 (a+b x)^{5/2} \sqrt {c+d x}}{d^3}-\frac {105 b^{3/2} (b c-a d)^3 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 d^{11/2}}\\ \end {align*}
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Mathematica [A]
time = 0.37, size = 165, normalized size = 0.81 \begin {gather*} -\frac {(a+b x)^{9/2} \left (16 d^4+\frac {144 b d^3 (c+d x)}{a+b x}-\frac {693 b^2 d^2 (c+d x)^2}{(a+b x)^2}+\frac {840 b^3 d (c+d x)^3}{(a+b x)^3}-\frac {315 b^4 (c+d x)^4}{(a+b x)^4}\right )}{24 d^5 (c+d x)^{3/2}}-\frac {105 b^{3/2} (b c-a d)^3 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{8 d^{11/2}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [F(-1)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {\left (b x +a \right )^{\frac {9}{2}}}{\left (d x +c \right )^{\frac {5}{2}}}\, dx\]
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. 427 vs.
\(2 (164) = 328\).
time = 0.82, size = 879, normalized size = 4.31 \begin {gather*} \left [-\frac {315 \, {\left (b^{4} c^{5} - 3 \, a b^{3} c^{4} d + 3 \, a^{2} b^{2} c^{3} d^{2} - a^{3} b c^{2} d^{3} + {\left (b^{4} c^{3} d^{2} - 3 \, a b^{3} c^{2} d^{3} + 3 \, a^{2} b^{2} c d^{4} - a^{3} b d^{5}\right )} x^{2} + 2 \, {\left (b^{4} c^{4} d - 3 \, a b^{3} c^{3} d^{2} + 3 \, a^{2} b^{2} c^{2} d^{3} - a^{3} b c d^{4}\right )} x\right )} \sqrt {\frac {b}{d}} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d^{2} x + b c d + a d^{2}\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {b}{d}} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \, {\left (8 \, b^{4} d^{4} x^{4} + 315 \, b^{4} c^{4} - 840 \, a b^{3} c^{3} d + 693 \, a^{2} b^{2} c^{2} d^{2} - 144 \, a^{3} b c d^{3} - 16 \, a^{4} d^{4} - 2 \, {\left (9 \, b^{4} c d^{3} - 25 \, a b^{3} d^{4}\right )} x^{3} + 3 \, {\left (21 \, b^{4} c^{2} d^{2} - 60 \, a b^{3} c d^{3} + 55 \, a^{2} b^{2} d^{4}\right )} x^{2} + 2 \, {\left (210 \, b^{4} c^{3} d - 567 \, a b^{3} c^{2} d^{2} + 477 \, a^{2} b^{2} c d^{3} - 104 \, a^{3} b d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{96 \, {\left (d^{7} x^{2} + 2 \, c d^{6} x + c^{2} d^{5}\right )}}, \frac {315 \, {\left (b^{4} c^{5} - 3 \, a b^{3} c^{4} d + 3 \, a^{2} b^{2} c^{3} d^{2} - a^{3} b c^{2} d^{3} + {\left (b^{4} c^{3} d^{2} - 3 \, a b^{3} c^{2} d^{3} + 3 \, a^{2} b^{2} c d^{4} - a^{3} b d^{5}\right )} x^{2} + 2 \, {\left (b^{4} c^{4} d - 3 \, a b^{3} c^{3} d^{2} + 3 \, a^{2} b^{2} c^{2} d^{3} - a^{3} b c d^{4}\right )} x\right )} \sqrt {-\frac {b}{d}} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {b}{d}}}{2 \, {\left (b^{2} d x^{2} + a b c + {\left (b^{2} c + a b d\right )} x\right )}}\right ) + 2 \, {\left (8 \, b^{4} d^{4} x^{4} + 315 \, b^{4} c^{4} - 840 \, a b^{3} c^{3} d + 693 \, a^{2} b^{2} c^{2} d^{2} - 144 \, a^{3} b c d^{3} - 16 \, a^{4} d^{4} - 2 \, {\left (9 \, b^{4} c d^{3} - 25 \, a b^{3} d^{4}\right )} x^{3} + 3 \, {\left (21 \, b^{4} c^{2} d^{2} - 60 \, a b^{3} c d^{3} + 55 \, a^{2} b^{2} d^{4}\right )} x^{2} + 2 \, {\left (210 \, b^{4} c^{3} d - 567 \, a b^{3} c^{2} d^{2} + 477 \, a^{2} b^{2} c d^{3} - 104 \, a^{3} b d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, {\left (d^{7} x^{2} + 2 \, c d^{6} x + c^{2} d^{5}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)]
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. 500 vs.
\(2 (164) = 328\).
time = 0.10, size = 613, normalized size = 3.00 \begin {gather*} \frac {2 \left (\left (\left (\left (-\frac {\left (-216 b^{6} d^{8} c+216 b^{5} d^{9} a\right ) \sqrt {a+b x} \sqrt {a+b x}}{1296 b^{2} d^{9} \left |b\right | c-1296 b d^{10} \left |b\right | a}-\frac {486 b^{7} d^{7} c^{2}-972 b^{6} d^{8} a c+486 b^{5} d^{9} a^{2}}{1296 b^{2} d^{9} \left |b\right | c-1296 b d^{10} \left |b\right | a}\right ) \sqrt {a+b x} \sqrt {a+b x}-\frac {-1701 b^{8} d^{6} c^{3}+5103 b^{7} d^{7} a c^{2}-5103 b^{6} d^{8} a^{2} c+1701 b^{5} d^{9} a^{3}}{1296 b^{2} d^{9} \left |b\right | c-1296 b d^{10} \left |b\right | a}\right ) \sqrt {a+b x} \sqrt {a+b x}-\frac {-11340 b^{9} d^{5} c^{4}+45360 b^{8} d^{6} a c^{3}-68040 b^{7} d^{7} a^{2} c^{2}+45360 b^{6} d^{8} a^{3} c-11340 b^{5} d^{9} a^{4}}{1296 b^{2} d^{9} \left |b\right | c-1296 b d^{10} \left |b\right | a}\right ) \sqrt {a+b x} \sqrt {a+b x}-\frac {-8505 b^{10} d^{4} c^{5}+42525 b^{9} d^{5} a c^{4}-85050 b^{8} d^{6} a^{2} c^{3}+85050 b^{7} d^{7} a^{3} c^{2}-42525 b^{6} d^{8} a^{4} c+8505 b^{5} d^{9} a^{5}}{1296 b^{2} d^{9} \left |b\right | c-1296 b d^{10} \left |b\right | a}\right ) \sqrt {a+b x} \sqrt {-a b d+b^{2} c+b d \left (a+b x\right )}}{\left (-a b d+b^{2} c+b d \left (a+b x\right )\right )^{2}}+\frac {2 \left (-105 a^{3} b^{3} d^{3}+315 a^{2} b^{4} c d^{2}-315 a b^{5} c^{2} d+105 b^{6} c^{3}\right ) \ln \left |\sqrt {-a b d+b^{2} c+b d \left (a+b x\right )}-\sqrt {b d} \sqrt {a+b x}\right |}{16 d^{5} \sqrt {b d} \left |b\right |} \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 )}^{9/2}}{{\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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