Optimal. Leaf size=208 \[ -\frac {3 d \sqrt {c+d x}}{40 b^2 (a+b x)^4}-\frac {d^2 \sqrt {c+d x}}{80 b^2 (b c-a d) (a+b x)^3}+\frac {d^3 \sqrt {c+d x}}{64 b^2 (b c-a d)^2 (a+b x)^2}-\frac {3 d^4 \sqrt {c+d x}}{128 b^2 (b c-a d)^3 (a+b x)}-\frac {(c+d x)^{3/2}}{5 b (a+b x)^5}+\frac {3 d^5 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{128 b^{5/2} (b c-a d)^{7/2}} \]
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Rubi [A]
time = 0.07, antiderivative size = 208, 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 = {43, 44, 65, 214}
\begin {gather*} \frac {3 d^5 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{128 b^{5/2} (b c-a d)^{7/2}}-\frac {3 d^4 \sqrt {c+d x}}{128 b^2 (a+b x) (b c-a d)^3}+\frac {d^3 \sqrt {c+d x}}{64 b^2 (a+b x)^2 (b c-a d)^2}-\frac {d^2 \sqrt {c+d x}}{80 b^2 (a+b x)^3 (b c-a d)}-\frac {3 d \sqrt {c+d x}}{40 b^2 (a+b x)^4}-\frac {(c+d x)^{3/2}}{5 b (a+b x)^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 44
Rule 65
Rule 214
Rubi steps
\begin {align*} \int \frac {(c+d x)^{3/2}}{(a+b x)^6} \, dx &=-\frac {(c+d x)^{3/2}}{5 b (a+b x)^5}+\frac {(3 d) \int \frac {\sqrt {c+d x}}{(a+b x)^5} \, dx}{10 b}\\ &=-\frac {3 d \sqrt {c+d x}}{40 b^2 (a+b x)^4}-\frac {(c+d x)^{3/2}}{5 b (a+b x)^5}+\frac {\left (3 d^2\right ) \int \frac {1}{(a+b x)^4 \sqrt {c+d x}} \, dx}{80 b^2}\\ &=-\frac {3 d \sqrt {c+d x}}{40 b^2 (a+b x)^4}-\frac {d^2 \sqrt {c+d x}}{80 b^2 (b c-a d) (a+b x)^3}-\frac {(c+d x)^{3/2}}{5 b (a+b x)^5}-\frac {d^3 \int \frac {1}{(a+b x)^3 \sqrt {c+d x}} \, dx}{32 b^2 (b c-a d)}\\ &=-\frac {3 d \sqrt {c+d x}}{40 b^2 (a+b x)^4}-\frac {d^2 \sqrt {c+d x}}{80 b^2 (b c-a d) (a+b x)^3}+\frac {d^3 \sqrt {c+d x}}{64 b^2 (b c-a d)^2 (a+b x)^2}-\frac {(c+d x)^{3/2}}{5 b (a+b x)^5}+\frac {\left (3 d^4\right ) \int \frac {1}{(a+b x)^2 \sqrt {c+d x}} \, dx}{128 b^2 (b c-a d)^2}\\ &=-\frac {3 d \sqrt {c+d x}}{40 b^2 (a+b x)^4}-\frac {d^2 \sqrt {c+d x}}{80 b^2 (b c-a d) (a+b x)^3}+\frac {d^3 \sqrt {c+d x}}{64 b^2 (b c-a d)^2 (a+b x)^2}-\frac {3 d^4 \sqrt {c+d x}}{128 b^2 (b c-a d)^3 (a+b x)}-\frac {(c+d x)^{3/2}}{5 b (a+b x)^5}-\frac {\left (3 d^5\right ) \int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx}{256 b^2 (b c-a d)^3}\\ &=-\frac {3 d \sqrt {c+d x}}{40 b^2 (a+b x)^4}-\frac {d^2 \sqrt {c+d x}}{80 b^2 (b c-a d) (a+b x)^3}+\frac {d^3 \sqrt {c+d x}}{64 b^2 (b c-a d)^2 (a+b x)^2}-\frac {3 d^4 \sqrt {c+d x}}{128 b^2 (b c-a d)^3 (a+b x)}-\frac {(c+d x)^{3/2}}{5 b (a+b x)^5}-\frac {\left (3 d^4\right ) \text {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{128 b^2 (b c-a d)^3}\\ &=-\frac {3 d \sqrt {c+d x}}{40 b^2 (a+b x)^4}-\frac {d^2 \sqrt {c+d x}}{80 b^2 (b c-a d) (a+b x)^3}+\frac {d^3 \sqrt {c+d x}}{64 b^2 (b c-a d)^2 (a+b x)^2}-\frac {3 d^4 \sqrt {c+d x}}{128 b^2 (b c-a d)^3 (a+b x)}-\frac {(c+d x)^{3/2}}{5 b (a+b x)^5}+\frac {3 d^5 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{128 b^{5/2} (b c-a d)^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 1.57, size = 223, normalized size = 1.07 \begin {gather*} \frac {\frac {\sqrt {b} \sqrt {c+d x} \left (-15 a^4 d^4-10 a^3 b d^3 (c+7 d x)+2 a^2 b^2 d^2 \left (124 c^2+233 c d x+64 d^2 x^2\right )-2 a b^3 d \left (168 c^3+256 c^2 d x+23 c d^2 x^2-35 d^3 x^3\right )+b^4 \left (128 c^4+176 c^3 d x+8 c^2 d^2 x^2-10 c d^3 x^3+15 d^4 x^4\right )\right )}{(-b c+a d)^3 (a+b x)^5}+\frac {15 d^5 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{(-b c+a d)^{7/2}}}{640 b^{5/2}} \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 [A]
time = 0.16, size = 237, normalized size = 1.14
method | result | size |
derivativedivides | \(2 d^{5} \left (\frac {\frac {3 b^{2} \left (d x +c \right )^{\frac {9}{2}}}{256 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {7 b \left (d x +c \right )^{\frac {7}{2}}}{128 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {\left (d x +c \right )^{\frac {5}{2}}}{10 a d -10 b c}-\frac {7 \left (d x +c \right )^{\frac {3}{2}}}{128 b}-\frac {3 \left (a d -b c \right ) \sqrt {d x +c}}{256 b^{2}}}{\left (\left (d x +c \right ) b +a d -b c \right )^{5}}+\frac {3 \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{256 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b^{2} \sqrt {\left (a d -b c \right ) b}}\right )\) | \(237\) |
default | \(2 d^{5} \left (\frac {\frac {3 b^{2} \left (d x +c \right )^{\frac {9}{2}}}{256 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {7 b \left (d x +c \right )^{\frac {7}{2}}}{128 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {\left (d x +c \right )^{\frac {5}{2}}}{10 a d -10 b c}-\frac {7 \left (d x +c \right )^{\frac {3}{2}}}{128 b}-\frac {3 \left (a d -b c \right ) \sqrt {d x +c}}{256 b^{2}}}{\left (\left (d x +c \right ) b +a d -b c \right )^{5}}+\frac {3 \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{256 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b^{2} \sqrt {\left (a d -b c \right ) b}}\right )\) | \(237\) |
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. 739 vs.
\(2 (176) = 352\).
time = 0.32, size = 1492, normalized size = 7.17
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. 410 vs.
\(2 (176) = 352\).
time = 0.02, size = 523, normalized size = 2.51 \begin {gather*} \frac {15 \sqrt {c+d x} \left (c+d x\right )^{4} d^{5} b^{4}+70 \sqrt {c+d x} \left (c+d x\right )^{3} d^{6} b^{3} a-70 \sqrt {c+d x} \left (c+d x\right )^{3} d^{5} c b^{4}+128 \sqrt {c+d x} \left (c+d x\right )^{2} d^{7} b^{2} a^{2}-256 \sqrt {c+d x} \left (c+d x\right )^{2} d^{6} c b^{3} a+128 \sqrt {c+d x} \left (c+d x\right )^{2} d^{5} c^{2} b^{4}-70 \sqrt {c+d x} \left (c+d x\right ) d^{8} b a^{3}+210 \sqrt {c+d x} \left (c+d x\right ) d^{7} c b^{2} a^{2}-210 \sqrt {c+d x} \left (c+d x\right ) d^{6} c^{2} b^{3} a+70 \sqrt {c+d x} \left (c+d x\right ) d^{5} c^{3} b^{4}-15 \sqrt {c+d x} d^{9} a^{4}+60 \sqrt {c+d x} d^{8} c b a^{3}-90 \sqrt {c+d x} d^{7} c^{2} b^{2} a^{2}+60 \sqrt {c+d x} d^{6} c^{3} b^{3} a-15 \sqrt {c+d x} d^{5} c^{4} b^{4}}{\left (640 d^{3} b^{2} a^{3}-1920 d^{2} c b^{3} a^{2}+1920 d c^{2} b^{4} a-640 c^{3} b^{5}\right ) \left (\left (c+d x\right ) b+d a-c b\right )^{5}}+\frac {3 d^{5} \arctan \left (\frac {b \sqrt {c+d x}}{\sqrt {-b^{2} c+a b d}}\right )}{2 \left (64 d^{3} b^{2} a^{3}-192 d^{2} c b^{3} a^{2}+192 d c^{2} b^{4} a-64 c^{3} b^{5}\right ) \sqrt {-b^{2} c+a b d}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.47, size = 398, normalized size = 1.91 \begin {gather*} \frac {\frac {d^5\,{\left (c+d\,x\right )}^{5/2}}{5\,\left (a\,d-b\,c\right )}-\frac {7\,d^5\,{\left (c+d\,x\right )}^{3/2}}{64\,b}+\frac {3\,b^2\,d^5\,{\left (c+d\,x\right )}^{9/2}}{128\,{\left (a\,d-b\,c\right )}^3}-\frac {3\,d^5\,\left (a\,d-b\,c\right )\,\sqrt {c+d\,x}}{128\,b^2}+\frac {7\,b\,d^5\,{\left (c+d\,x\right )}^{7/2}}{64\,{\left (a\,d-b\,c\right )}^2}}{b^5\,{\left (c+d\,x\right )}^5-{\left (c+d\,x\right )}^2\,\left (-10\,a^3\,b^2\,d^3+30\,a^2\,b^3\,c\,d^2-30\,a\,b^4\,c^2\,d+10\,b^5\,c^3\right )-\left (5\,b^5\,c-5\,a\,b^4\,d\right )\,{\left (c+d\,x\right )}^4+a^5\,d^5-b^5\,c^5+{\left (c+d\,x\right )}^3\,\left (10\,a^2\,b^3\,d^2-20\,a\,b^4\,c\,d+10\,b^5\,c^2\right )+\left (c+d\,x\right )\,\left (5\,a^4\,b\,d^4-20\,a^3\,b^2\,c\,d^3+30\,a^2\,b^3\,c^2\,d^2-20\,a\,b^4\,c^3\,d+5\,b^5\,c^4\right )-10\,a^2\,b^3\,c^3\,d^2+10\,a^3\,b^2\,c^2\,d^3+5\,a\,b^4\,c^4\,d-5\,a^4\,b\,c\,d^4}+\frac {3\,d^5\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {c+d\,x}}{\sqrt {a\,d-b\,c}}\right )}{128\,b^{5/2}\,{\left (a\,d-b\,c\right )}^{7/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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