Optimal. Leaf size=170 \[ -\frac {2 (a+b x)^{7/2}}{3 d (c+d x)^{3/2}}-\frac {14 b (a+b x)^{5/2}}{3 d^2 \sqrt {c+d x}}-\frac {35 b^2 (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{4 d^4}+\frac {35 b^2 (a+b x)^{3/2} \sqrt {c+d x}}{6 d^3}+\frac {35 b^{3/2} (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 d^{9/2}} \]
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Rubi [A]
time = 0.06, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps
used = 7, 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 {35 b^{3/2} (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 d^{9/2}}-\frac {35 b^2 \sqrt {a+b x} \sqrt {c+d x} (b c-a d)}{4 d^4}+\frac {35 b^2 (a+b x)^{3/2} \sqrt {c+d x}}{6 d^3}-\frac {14 b (a+b x)^{5/2}}{3 d^2 \sqrt {c+d x}}-\frac {2 (a+b x)^{7/2}}{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)^{7/2}}{(c+d x)^{5/2}} \, dx &=-\frac {2 (a+b x)^{7/2}}{3 d (c+d x)^{3/2}}+\frac {(7 b) \int \frac {(a+b x)^{5/2}}{(c+d x)^{3/2}} \, dx}{3 d}\\ &=-\frac {2 (a+b x)^{7/2}}{3 d (c+d x)^{3/2}}-\frac {14 b (a+b x)^{5/2}}{3 d^2 \sqrt {c+d x}}+\frac {\left (35 b^2\right ) \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}} \, dx}{3 d^2}\\ &=-\frac {2 (a+b x)^{7/2}}{3 d (c+d x)^{3/2}}-\frac {14 b (a+b x)^{5/2}}{3 d^2 \sqrt {c+d x}}+\frac {35 b^2 (a+b x)^{3/2} \sqrt {c+d x}}{6 d^3}-\frac {\left (35 b^2 (b c-a d)\right ) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{4 d^3}\\ &=-\frac {2 (a+b x)^{7/2}}{3 d (c+d x)^{3/2}}-\frac {14 b (a+b x)^{5/2}}{3 d^2 \sqrt {c+d x}}-\frac {35 b^2 (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{4 d^4}+\frac {35 b^2 (a+b x)^{3/2} \sqrt {c+d x}}{6 d^3}+\frac {\left (35 b^2 (b c-a d)^2\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{8 d^4}\\ &=-\frac {2 (a+b x)^{7/2}}{3 d (c+d x)^{3/2}}-\frac {14 b (a+b x)^{5/2}}{3 d^2 \sqrt {c+d x}}-\frac {35 b^2 (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{4 d^4}+\frac {35 b^2 (a+b x)^{3/2} \sqrt {c+d x}}{6 d^3}+\frac {\left (35 b (b c-a d)^2\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 )}{4 d^4}\\ &=-\frac {2 (a+b x)^{7/2}}{3 d (c+d x)^{3/2}}-\frac {14 b (a+b x)^{5/2}}{3 d^2 \sqrt {c+d x}}-\frac {35 b^2 (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{4 d^4}+\frac {35 b^2 (a+b x)^{3/2} \sqrt {c+d x}}{6 d^3}+\frac {\left (35 b (b c-a d)^2\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 )}{4 d^4}\\ &=-\frac {2 (a+b x)^{7/2}}{3 d (c+d x)^{3/2}}-\frac {14 b (a+b x)^{5/2}}{3 d^2 \sqrt {c+d x}}-\frac {35 b^2 (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{4 d^4}+\frac {35 b^2 (a+b x)^{3/2} \sqrt {c+d x}}{6 d^3}+\frac {35 b^{3/2} (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 d^{9/2}}\\ \end {align*}
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Mathematica [A]
time = 0.25, size = 166, normalized size = 0.98 \begin {gather*} -\frac {\sqrt {a+b x} \left (8 a^3 d^3+8 a^2 b d^2 (7 c+10 d x)-a b^2 d \left (175 c^2+238 c d x+39 d^2 x^2\right )+b^3 \left (105 c^3+140 c^2 d x+21 c d^2 x^2-6 d^3 x^3\right )\right )}{12 d^4 (c+d x)^{3/2}}+\frac {35 b^{3/2} (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{4 d^{9/2}} \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 \[\int \frac {\left (b x +a \right )^{\frac {7}{2}}}{\left (d x +c \right )^{\frac {5}{2}}}\, dx\]
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. 316 vs.
\(2 (132) = 264\).
time = 0.57, size = 657, normalized size = 3.86 \begin {gather*} \left [\frac {105 \, {\left (b^{3} c^{4} - 2 \, a b^{2} c^{3} d + a^{2} b c^{2} d^{2} + {\left (b^{3} c^{2} d^{2} - 2 \, a b^{2} c d^{3} + a^{2} b d^{4}\right )} x^{2} + 2 \, {\left (b^{3} c^{3} d - 2 \, a b^{2} c^{2} d^{2} + a^{2} b c d^{3}\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 (6 \, b^{3} d^{3} x^{3} - 105 \, b^{3} c^{3} + 175 \, a b^{2} c^{2} d - 56 \, a^{2} b c d^{2} - 8 \, a^{3} d^{3} - 3 \, {\left (7 \, b^{3} c d^{2} - 13 \, a b^{2} d^{3}\right )} x^{2} - 2 \, {\left (70 \, b^{3} c^{2} d - 119 \, a b^{2} c d^{2} + 40 \, a^{2} b d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, {\left (d^{6} x^{2} + 2 \, c d^{5} x + c^{2} d^{4}\right )}}, -\frac {105 \, {\left (b^{3} c^{4} - 2 \, a b^{2} c^{3} d + a^{2} b c^{2} d^{2} + {\left (b^{3} c^{2} d^{2} - 2 \, a b^{2} c d^{3} + a^{2} b d^{4}\right )} x^{2} + 2 \, {\left (b^{3} c^{3} d - 2 \, a b^{2} c^{2} d^{2} + a^{2} b c d^{3}\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 (6 \, b^{3} d^{3} x^{3} - 105 \, b^{3} c^{3} + 175 \, a b^{2} c^{2} d - 56 \, a^{2} b c d^{2} - 8 \, a^{3} d^{3} - 3 \, {\left (7 \, b^{3} c d^{2} - 13 \, a b^{2} d^{3}\right )} x^{2} - 2 \, {\left (70 \, b^{3} c^{2} d - 119 \, a b^{2} c d^{2} + 40 \, a^{2} b d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{24 \, {\left (d^{6} x^{2} + 2 \, c d^{5} x + c^{2} d^{4}\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. 380 vs.
\(2 (132) = 264\).
time = 0.08, size = 475, normalized size = 2.79 \begin {gather*} \frac {2 \left (\left (\left (-\frac {\left (-18 b^{6} d^{6} c+18 b^{5} d^{7} a\right ) \sqrt {a+b x} \sqrt {a+b x}}{72 b^{2} d^{7} \left |b\right | c-72 b d^{8} \left |b\right | a}-\frac {63 b^{7} d^{5} c^{2}-126 b^{6} d^{6} a c+63 b^{5} d^{7} a^{2}}{72 b^{2} d^{7} \left |b\right | c-72 b d^{8} \left |b\right | a}\right ) \sqrt {a+b x} \sqrt {a+b x}-\frac {420 b^{8} d^{4} c^{3}-1260 b^{7} d^{5} a c^{2}+1260 b^{6} d^{6} a^{2} c-420 b^{5} d^{7} a^{3}}{72 b^{2} d^{7} \left |b\right | c-72 b d^{8} \left |b\right | a}\right ) \sqrt {a+b x} \sqrt {a+b x}-\frac {315 b^{9} d^{3} c^{4}-1260 b^{8} d^{4} a c^{3}+1890 b^{7} d^{5} a^{2} c^{2}-1260 b^{6} d^{6} a^{3} c+315 b^{5} d^{7} a^{4}}{72 b^{2} d^{7} \left |b\right | c-72 b d^{8} \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 (-35 a^{2} b^{3} d^{2}+70 a b^{4} c d-35 b^{5} c^{2}\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 |}{8 d^{4} \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 = -0.01 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^{7/2}}{{\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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