Optimal. Leaf size=200 \[ -\frac {35 d^3}{8 (b c-a d)^4 (c+d x)^{3/2}}-\frac {1}{3 (b c-a d) (a+b x)^3 (c+d x)^{3/2}}+\frac {3 d}{4 (b c-a d)^2 (a+b x)^2 (c+d x)^{3/2}}-\frac {21 d^2}{8 (b c-a d)^3 (a+b x) (c+d x)^{3/2}}-\frac {105 b d^3}{8 (b c-a d)^5 \sqrt {c+d x}}+\frac {105 b^{3/2} d^3 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{8 (b c-a d)^{11/2}} \]
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Rubi [A]
time = 0.10, antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {44, 53, 65, 214}
\begin {gather*} \frac {105 b^{3/2} d^3 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{8 (b c-a d)^{11/2}}-\frac {105 b d^3}{8 \sqrt {c+d x} (b c-a d)^5}-\frac {35 d^3}{8 (c+d x)^{3/2} (b c-a d)^4}-\frac {21 d^2}{8 (a+b x) (c+d x)^{3/2} (b c-a d)^3}+\frac {3 d}{4 (a+b x)^2 (c+d x)^{3/2} (b c-a d)^2}-\frac {1}{3 (a+b x)^3 (c+d x)^{3/2} (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 44
Rule 53
Rule 65
Rule 214
Rubi steps
\begin {align*} \int \frac {1}{(a+b x)^4 (c+d x)^{5/2}} \, dx &=-\frac {1}{3 (b c-a d) (a+b x)^3 (c+d x)^{3/2}}-\frac {(3 d) \int \frac {1}{(a+b x)^3 (c+d x)^{5/2}} \, dx}{2 (b c-a d)}\\ &=-\frac {1}{3 (b c-a d) (a+b x)^3 (c+d x)^{3/2}}+\frac {3 d}{4 (b c-a d)^2 (a+b x)^2 (c+d x)^{3/2}}+\frac {\left (21 d^2\right ) \int \frac {1}{(a+b x)^2 (c+d x)^{5/2}} \, dx}{8 (b c-a d)^2}\\ &=-\frac {1}{3 (b c-a d) (a+b x)^3 (c+d x)^{3/2}}+\frac {3 d}{4 (b c-a d)^2 (a+b x)^2 (c+d x)^{3/2}}-\frac {21 d^2}{8 (b c-a d)^3 (a+b x) (c+d x)^{3/2}}-\frac {\left (105 d^3\right ) \int \frac {1}{(a+b x) (c+d x)^{5/2}} \, dx}{16 (b c-a d)^3}\\ &=-\frac {35 d^3}{8 (b c-a d)^4 (c+d x)^{3/2}}-\frac {1}{3 (b c-a d) (a+b x)^3 (c+d x)^{3/2}}+\frac {3 d}{4 (b c-a d)^2 (a+b x)^2 (c+d x)^{3/2}}-\frac {21 d^2}{8 (b c-a d)^3 (a+b x) (c+d x)^{3/2}}-\frac {\left (105 b d^3\right ) \int \frac {1}{(a+b x) (c+d x)^{3/2}} \, dx}{16 (b c-a d)^4}\\ &=-\frac {35 d^3}{8 (b c-a d)^4 (c+d x)^{3/2}}-\frac {1}{3 (b c-a d) (a+b x)^3 (c+d x)^{3/2}}+\frac {3 d}{4 (b c-a d)^2 (a+b x)^2 (c+d x)^{3/2}}-\frac {21 d^2}{8 (b c-a d)^3 (a+b x) (c+d x)^{3/2}}-\frac {105 b d^3}{8 (b c-a d)^5 \sqrt {c+d x}}-\frac {\left (105 b^2 d^3\right ) \int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx}{16 (b c-a d)^5}\\ &=-\frac {35 d^3}{8 (b c-a d)^4 (c+d x)^{3/2}}-\frac {1}{3 (b c-a d) (a+b x)^3 (c+d x)^{3/2}}+\frac {3 d}{4 (b c-a d)^2 (a+b x)^2 (c+d x)^{3/2}}-\frac {21 d^2}{8 (b c-a d)^3 (a+b x) (c+d x)^{3/2}}-\frac {105 b d^3}{8 (b c-a d)^5 \sqrt {c+d x}}-\frac {\left (105 b^2 d^2\right ) \text {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{8 (b c-a d)^5}\\ &=-\frac {35 d^3}{8 (b c-a d)^4 (c+d x)^{3/2}}-\frac {1}{3 (b c-a d) (a+b x)^3 (c+d x)^{3/2}}+\frac {3 d}{4 (b c-a d)^2 (a+b x)^2 (c+d x)^{3/2}}-\frac {21 d^2}{8 (b c-a d)^3 (a+b x) (c+d x)^{3/2}}-\frac {105 b d^3}{8 (b c-a d)^5 \sqrt {c+d x}}+\frac {105 b^{3/2} d^3 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{8 (b c-a d)^{11/2}}\\ \end {align*}
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Mathematica [A]
time = 0.91, size = 220, normalized size = 1.10 \begin {gather*} \frac {1}{24} \left (\frac {-16 a^4 d^4+16 a^3 b d^3 (13 c+9 d x)+3 a^2 b^2 d^2 \left (55 c^2+318 c d x+231 d^2 x^2\right )+2 a b^3 d \left (-25 c^3+90 c^2 d x+567 c d^2 x^2+420 d^3 x^3\right )+b^4 \left (8 c^4-18 c^3 d x+63 c^2 d^2 x^2+420 c d^3 x^3+315 d^4 x^4\right )}{(-b c+a d)^5 (a+b x)^3 (c+d x)^{3/2}}+\frac {315 b^{3/2} d^3 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{(-b c+a d)^{11/2}}\right ) \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 [A]
time = 0.17, size = 177, normalized size = 0.88
method | result | size |
derivativedivides | \(2 d^{3} \left (\frac {b^{2} \left (\frac {\frac {41 \left (d x +c \right )^{\frac {5}{2}} b^{2}}{16}+\frac {35 \left (a d -b c \right ) b \left (d x +c \right )^{\frac {3}{2}}}{6}+\left (\frac {55}{16} a^{2} d^{2}-\frac {55}{8} a b c d +\frac {55}{16} b^{2} c^{2}\right ) \sqrt {d x +c}}{\left (\left (d x +c \right ) b +a d -b c \right )^{3}}+\frac {105 \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{16 \sqrt {\left (a d -b c \right ) b}}\right )}{\left (a d -b c \right )^{5}}-\frac {1}{3 \left (a d -b c \right )^{4} \left (d x +c \right )^{\frac {3}{2}}}+\frac {4 b}{\left (a d -b c \right )^{5} \sqrt {d x +c}}\right )\) | \(177\) |
default | \(2 d^{3} \left (\frac {b^{2} \left (\frac {\frac {41 \left (d x +c \right )^{\frac {5}{2}} b^{2}}{16}+\frac {35 \left (a d -b c \right ) b \left (d x +c \right )^{\frac {3}{2}}}{6}+\left (\frac {55}{16} a^{2} d^{2}-\frac {55}{8} a b c d +\frac {55}{16} b^{2} c^{2}\right ) \sqrt {d x +c}}{\left (\left (d x +c \right ) b +a d -b c \right )^{3}}+\frac {105 \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{16 \sqrt {\left (a d -b c \right ) b}}\right )}{\left (a d -b c \right )^{5}}-\frac {1}{3 \left (a d -b c \right )^{4} \left (d x +c \right )^{\frac {3}{2}}}+\frac {4 b}{\left (a d -b c \right )^{5} \sqrt {d x +c}}\right )\) | \(177\) |
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. 915 vs.
\(2 (168) = 336\).
time = 0.32, size = 1840, normalized size = 9.20
result too large to display
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. 432 vs.
\(2 (168) = 336\).
time = 0.01, size = 469, normalized size = 2.34 \begin {gather*} 2 \left (\frac {315 \left (c+d x\right )^{4} b^{4} d^{3}-840 \left (c+d x\right )^{3} b^{4} d^{3} c+840 \left (c+d x\right )^{3} b^{3} d^{4} a+693 \left (c+d x\right )^{2} b^{4} d^{3} c^{2}-1386 \left (c+d x\right )^{2} b^{3} d^{4} c a+693 \left (c+d x\right )^{2} b^{2} d^{5} a^{2}-144 \left (c+d x\right ) b^{4} d^{3} c^{3}+432 \left (c+d x\right ) b^{3} d^{4} c^{2} a-432 \left (c+d x\right ) b^{2} d^{5} c a^{2}+144 \left (c+d x\right ) b d^{6} a^{3}-16 b^{4} d^{3} c^{4}+64 b^{3} d^{4} c^{3} a-96 b^{2} d^{5} c^{2} a^{2}+64 b d^{6} c a^{3}-16 d^{7} a^{4}}{\left (48 b^{5} c^{5}-240 b^{4} d c^{4} a+480 b^{3} d^{2} c^{3} a^{2}-480 b^{2} d^{3} c^{2} a^{3}+240 b d^{4} c a^{4}-48 d^{5} a^{5}\right ) \left (-\sqrt {c+d x} \left (c+d x\right ) b+\sqrt {c+d x} b c-\sqrt {c+d x} d a\right )^{3}}+\frac {105 b^{2} d^{3} \arctan \left (\frac {b \sqrt {c+d x}}{\sqrt {-b^{2} c+a b d}}\right )}{2 \left (-8 b^{5} c^{5}+40 b^{4} d c^{4} a-80 b^{3} d^{2} c^{3} a^{2}+80 b^{2} d^{3} c^{2} a^{3}-40 b d^{4} c a^{4}+8 d^{5} a^{5}\right ) \sqrt {-b^{2} c+a b d}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.64, size = 334, normalized size = 1.67 \begin {gather*} \frac {\frac {231\,b^2\,d^3\,{\left (c+d\,x\right )}^2}{8\,{\left (a\,d-b\,c\right )}^3}-\frac {2\,d^3}{3\,\left (a\,d-b\,c\right )}+\frac {35\,b^3\,d^3\,{\left (c+d\,x\right )}^3}{{\left (a\,d-b\,c\right )}^4}+\frac {105\,b^4\,d^3\,{\left (c+d\,x\right )}^4}{8\,{\left (a\,d-b\,c\right )}^5}+\frac {6\,b\,d^3\,\left (c+d\,x\right )}{{\left (a\,d-b\,c\right )}^2}}{{\left (c+d\,x\right )}^{3/2}\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )+b^3\,{\left (c+d\,x\right )}^{9/2}-\left (3\,b^3\,c-3\,a\,b^2\,d\right )\,{\left (c+d\,x\right )}^{7/2}+{\left (c+d\,x\right )}^{5/2}\,\left (3\,a^2\,b\,d^2-6\,a\,b^2\,c\,d+3\,b^3\,c^2\right )}+\frac {105\,b^{3/2}\,d^3\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {c+d\,x}\,\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 )}{{\left (a\,d-b\,c\right )}^{11/2}}\right )}{8\,{\left (a\,d-b\,c\right )}^{11/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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