Optimal. Leaf size=241 \[ -\frac {b \left (105 b^2 c^2-100 a b c d+3 a^2 d^2\right ) \sqrt {c+d x}}{24 a^4 c \sqrt {a+b x}}-\frac {c \sqrt {c+d x}}{3 a x^3 \sqrt {a+b x}}+\frac {7 (b c-a d) \sqrt {c+d x}}{12 a^2 x^2 \sqrt {a+b x}}-\frac {(35 b c-3 a d) (b c-a d) \sqrt {c+d x}}{24 a^3 c x \sqrt {a+b x}}+\frac {(b c-a d) \left (35 b^2 c^2-10 a b c d-a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 a^{9/2} c^{3/2}} \]
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Rubi [A]
time = 0.15, antiderivative size = 241, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {100, 156, 157,
12, 95, 214} \begin {gather*} -\frac {\sqrt {c+d x} (35 b c-3 a d) (b c-a d)}{24 a^3 c x \sqrt {a+b x}}+\frac {7 \sqrt {c+d x} (b c-a d)}{12 a^2 x^2 \sqrt {a+b x}}+\frac {(b c-a d) \left (-a^2 d^2-10 a b c d+35 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 a^{9/2} c^{3/2}}-\frac {b \sqrt {c+d x} \left (3 a^2 d^2-100 a b c d+105 b^2 c^2\right )}{24 a^4 c \sqrt {a+b x}}-\frac {c \sqrt {c+d x}}{3 a x^3 \sqrt {a+b x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 95
Rule 100
Rule 156
Rule 157
Rule 214
Rubi steps
\begin {align*} \int \frac {(c+d x)^{3/2}}{x^4 (a+b x)^{3/2}} \, dx &=-\frac {c \sqrt {c+d x}}{3 a x^3 \sqrt {a+b x}}-\frac {\int \frac {\frac {7}{2} c (b c-a d)+3 d (b c-a d) x}{x^3 (a+b x)^{3/2} \sqrt {c+d x}} \, dx}{3 a}\\ &=-\frac {c \sqrt {c+d x}}{3 a x^3 \sqrt {a+b x}}+\frac {7 (b c-a d) \sqrt {c+d x}}{12 a^2 x^2 \sqrt {a+b x}}+\frac {\int \frac {\frac {1}{4} c (35 b c-3 a d) (b c-a d)+7 b c d (b c-a d) x}{x^2 (a+b x)^{3/2} \sqrt {c+d x}} \, dx}{6 a^2 c}\\ &=-\frac {c \sqrt {c+d x}}{3 a x^3 \sqrt {a+b x}}+\frac {7 (b c-a d) \sqrt {c+d x}}{12 a^2 x^2 \sqrt {a+b x}}-\frac {(35 b c-3 a d) (b c-a d) \sqrt {c+d x}}{24 a^3 c x \sqrt {a+b x}}-\frac {\int \frac {\frac {3}{8} c (b c-a d) \left (35 b^2 c^2-10 a b c d-a^2 d^2\right )+\frac {1}{4} b c d (35 b c-3 a d) (b c-a d) x}{x (a+b x)^{3/2} \sqrt {c+d x}} \, dx}{6 a^3 c^2}\\ &=-\frac {b \left (105 b^2 c^2-100 a b c d+3 a^2 d^2\right ) \sqrt {c+d x}}{24 a^4 c \sqrt {a+b x}}-\frac {c \sqrt {c+d x}}{3 a x^3 \sqrt {a+b x}}+\frac {7 (b c-a d) \sqrt {c+d x}}{12 a^2 x^2 \sqrt {a+b x}}-\frac {(35 b c-3 a d) (b c-a d) \sqrt {c+d x}}{24 a^3 c x \sqrt {a+b x}}-\frac {\int \frac {3 c (b c-a d)^2 \left (35 b^2 c^2-10 a b c d-a^2 d^2\right )}{16 x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{3 a^4 c^2 (b c-a d)}\\ &=-\frac {b \left (105 b^2 c^2-100 a b c d+3 a^2 d^2\right ) \sqrt {c+d x}}{24 a^4 c \sqrt {a+b x}}-\frac {c \sqrt {c+d x}}{3 a x^3 \sqrt {a+b x}}+\frac {7 (b c-a d) \sqrt {c+d x}}{12 a^2 x^2 \sqrt {a+b x}}-\frac {(35 b c-3 a d) (b c-a d) \sqrt {c+d x}}{24 a^3 c x \sqrt {a+b x}}-\frac {\left ((b c-a d) \left (35 b^2 c^2-10 a b c d-a^2 d^2\right )\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{16 a^4 c}\\ &=-\frac {b \left (105 b^2 c^2-100 a b c d+3 a^2 d^2\right ) \sqrt {c+d x}}{24 a^4 c \sqrt {a+b x}}-\frac {c \sqrt {c+d x}}{3 a x^3 \sqrt {a+b x}}+\frac {7 (b c-a d) \sqrt {c+d x}}{12 a^2 x^2 \sqrt {a+b x}}-\frac {(35 b c-3 a d) (b c-a d) \sqrt {c+d x}}{24 a^3 c x \sqrt {a+b x}}-\frac {\left ((b c-a d) \left (35 b^2 c^2-10 a b c d-a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{8 a^4 c}\\ &=-\frac {b \left (105 b^2 c^2-100 a b c d+3 a^2 d^2\right ) \sqrt {c+d x}}{24 a^4 c \sqrt {a+b x}}-\frac {c \sqrt {c+d x}}{3 a x^3 \sqrt {a+b x}}+\frac {7 (b c-a d) \sqrt {c+d x}}{12 a^2 x^2 \sqrt {a+b x}}-\frac {(35 b c-3 a d) (b c-a d) \sqrt {c+d x}}{24 a^3 c x \sqrt {a+b x}}+\frac {(b c-a d) \left (35 b^2 c^2-10 a b c d-a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 a^{9/2} c^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 10.13, size = 190, normalized size = 0.79 \begin {gather*} -\frac {\sqrt {c+d x} \left (105 b^3 c^2 x^3+5 a b^2 c x^2 (7 c-20 d x)+a^2 b x \left (-14 c^2-38 c d x+3 d^2 x^2\right )+a^3 \left (8 c^2+14 c d x+3 d^2 x^2\right )\right )}{24 a^4 c x^3 \sqrt {a+b x}}+\frac {\left (35 b^3 c^3-45 a b^2 c^2 d+9 a^2 b c d^2+a^3 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 a^{9/2} c^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(706\) vs.
\(2(203)=406\).
time = 0.08, size = 707, normalized size = 2.93
method | result | size |
default | \(\frac {\sqrt {d x +c}\, \left (3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{3} b \,d^{3} x^{4}+27 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{2} b^{2} c \,d^{2} x^{4}-135 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a \,b^{3} c^{2} d \,x^{4}+105 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) b^{4} c^{3} x^{4}+3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{4} d^{3} x^{3}+27 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{3} b c \,d^{2} x^{3}-135 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{2} b^{2} c^{2} d \,x^{3}+105 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a \,b^{3} c^{3} x^{3}-6 a^{2} b \,d^{2} x^{3} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+200 a \,b^{2} c d \,x^{3} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}-210 b^{3} c^{2} x^{3} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}-6 a^{3} d^{2} x^{2} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+76 a^{2} b c d \,x^{2} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}-70 a \,b^{2} c^{2} x^{2} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}-28 a^{3} c d x \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+28 a^{2} b \,c^{2} x \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}-16 a^{3} c^{2} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\right )}{48 a^{4} c \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, x^{3} \sqrt {a c}\, \sqrt {b x +a}}\) | \(707\) |
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 [A]
time = 3.01, size = 638, normalized size = 2.65 \begin {gather*} \left [\frac {3 \, {\left ({\left (35 \, b^{4} c^{3} - 45 \, a b^{3} c^{2} d + 9 \, a^{2} b^{2} c d^{2} + a^{3} b d^{3}\right )} x^{4} + {\left (35 \, a b^{3} c^{3} - 45 \, a^{2} b^{2} c^{2} d + 9 \, a^{3} b c d^{2} + a^{4} d^{3}\right )} x^{3}\right )} \sqrt {a c} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \, {\left (8 \, a^{4} c^{3} + {\left (105 \, a b^{3} c^{3} - 100 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2}\right )} x^{3} + {\left (35 \, a^{2} b^{2} c^{3} - 38 \, a^{3} b c^{2} d + 3 \, a^{4} c d^{2}\right )} x^{2} - 14 \, {\left (a^{3} b c^{3} - a^{4} c^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{96 \, {\left (a^{5} b c^{2} x^{4} + a^{6} c^{2} x^{3}\right )}}, -\frac {3 \, {\left ({\left (35 \, b^{4} c^{3} - 45 \, a b^{3} c^{2} d + 9 \, a^{2} b^{2} c d^{2} + a^{3} b d^{3}\right )} x^{4} + {\left (35 \, a b^{3} c^{3} - 45 \, a^{2} b^{2} c^{2} d + 9 \, a^{3} b c d^{2} + a^{4} d^{3}\right )} x^{3}\right )} \sqrt {-a c} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) + 2 \, {\left (8 \, a^{4} c^{3} + {\left (105 \, a b^{3} c^{3} - 100 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2}\right )} x^{3} + {\left (35 \, a^{2} b^{2} c^{3} - 38 \, a^{3} b c^{2} d + 3 \, a^{4} c d^{2}\right )} x^{2} - 14 \, {\left (a^{3} b c^{3} - a^{4} c^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, {\left (a^{5} b c^{2} x^{4} + a^{6} c^{2} x^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
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. 2310 vs.
\(2 (203) = 406\).
time = 11.28, size = 2310, normalized size = 9.59 \begin {gather*} \text {Too large to display} \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 (c+d\,x\right )}^{3/2}}{x^4\,{\left (a+b\,x\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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