Optimal. Leaf size=266 \[ -\frac {c \sqrt {a+b x} \sqrt {c+d x}}{4 a x^4}+\frac {(7 b c-9 a d) \sqrt {a+b x} \sqrt {c+d x}}{24 a^2 x^3}-\frac {\left (35 b^2 c^2-46 a b c d+3 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{96 a^3 c x^2}+\frac {\left (105 b^3 c^3-145 a b^2 c^2 d+15 a^2 b c d^2+9 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{192 a^4 c^2 x}-\frac {(b c-a d)^2 \left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{9/2} c^{5/2}} \]
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Rubi [A]
time = 0.15, antiderivative size = 266, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {100, 156, 12,
95, 214} \begin {gather*} \frac {\sqrt {a+b x} \sqrt {c+d x} (7 b c-9 a d)}{24 a^2 x^3}-\frac {\left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right ) (b c-a d)^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{9/2} c^{5/2}}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (3 a^2 d^2-46 a b c d+35 b^2 c^2\right )}{96 a^3 c x^2}+\frac {\sqrt {a+b x} \sqrt {c+d x} \left (9 a^3 d^3+15 a^2 b c d^2-145 a b^2 c^2 d+105 b^3 c^3\right )}{192 a^4 c^2 x}-\frac {c \sqrt {a+b x} \sqrt {c+d x}}{4 a x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 95
Rule 100
Rule 156
Rule 214
Rubi steps
\begin {align*} \int \frac {(c+d x)^{3/2}}{x^5 \sqrt {a+b x}} \, dx &=-\frac {c \sqrt {a+b x} \sqrt {c+d x}}{4 a x^4}-\frac {\int \frac {\frac {1}{2} c (7 b c-9 a d)+d (3 b c-4 a d) x}{x^4 \sqrt {a+b x} \sqrt {c+d x}} \, dx}{4 a}\\ &=-\frac {c \sqrt {a+b x} \sqrt {c+d x}}{4 a x^4}+\frac {(7 b c-9 a d) \sqrt {a+b x} \sqrt {c+d x}}{24 a^2 x^3}+\frac {\int \frac {\frac {1}{4} c \left (35 b^2 c^2-46 a b c d+3 a^2 d^2\right )+b c d (7 b c-9 a d) x}{x^3 \sqrt {a+b x} \sqrt {c+d x}} \, dx}{12 a^2 c}\\ &=-\frac {c \sqrt {a+b x} \sqrt {c+d x}}{4 a x^4}+\frac {(7 b c-9 a d) \sqrt {a+b x} \sqrt {c+d x}}{24 a^2 x^3}-\frac {\left (35 b^2 c^2-46 a b c d+3 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{96 a^3 c x^2}-\frac {\int \frac {\frac {1}{8} c \left (105 b^3 c^3-145 a b^2 c^2 d+15 a^2 b c d^2+9 a^3 d^3\right )+\frac {1}{4} b c d \left (35 b^2 c^2-46 a b c d+3 a^2 d^2\right ) x}{x^2 \sqrt {a+b x} \sqrt {c+d x}} \, dx}{24 a^3 c^2}\\ &=-\frac {c \sqrt {a+b x} \sqrt {c+d x}}{4 a x^4}+\frac {(7 b c-9 a d) \sqrt {a+b x} \sqrt {c+d x}}{24 a^2 x^3}-\frac {\left (35 b^2 c^2-46 a b c d+3 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{96 a^3 c x^2}+\frac {\left (105 b^3 c^3-145 a b^2 c^2 d+15 a^2 b c d^2+9 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{192 a^4 c^2 x}+\frac {\int \frac {3 c (b c-a d)^2 \left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right )}{16 x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{24 a^4 c^3}\\ &=-\frac {c \sqrt {a+b x} \sqrt {c+d x}}{4 a x^4}+\frac {(7 b c-9 a d) \sqrt {a+b x} \sqrt {c+d x}}{24 a^2 x^3}-\frac {\left (35 b^2 c^2-46 a b c d+3 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{96 a^3 c x^2}+\frac {\left (105 b^3 c^3-145 a b^2 c^2 d+15 a^2 b c d^2+9 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{192 a^4 c^2 x}+\frac {\left ((b c-a d)^2 \left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right )\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{128 a^4 c^2}\\ &=-\frac {c \sqrt {a+b x} \sqrt {c+d x}}{4 a x^4}+\frac {(7 b c-9 a d) \sqrt {a+b x} \sqrt {c+d x}}{24 a^2 x^3}-\frac {\left (35 b^2 c^2-46 a b c d+3 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{96 a^3 c x^2}+\frac {\left (105 b^3 c^3-145 a b^2 c^2 d+15 a^2 b c d^2+9 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{192 a^4 c^2 x}+\frac {\left ((b c-a d)^2 \left (35 b^2 c^2+10 a b c d+3 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 )}{64 a^4 c^2}\\ &=-\frac {c \sqrt {a+b x} \sqrt {c+d x}}{4 a x^4}+\frac {(7 b c-9 a d) \sqrt {a+b x} \sqrt {c+d x}}{24 a^2 x^3}-\frac {\left (35 b^2 c^2-46 a b c d+3 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{96 a^3 c x^2}+\frac {\left (105 b^3 c^3-145 a b^2 c^2 d+15 a^2 b c d^2+9 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{192 a^4 c^2 x}-\frac {(b c-a d)^2 \left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{9/2} c^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.48, size = 202, normalized size = 0.76 \begin {gather*} \frac {\sqrt {a+b x} \sqrt {c+d x} \left (105 b^3 c^3 x^3-5 a b^2 c^2 x^2 (14 c+29 d x)+a^2 b c x \left (56 c^2+92 c d x+15 d^2 x^2\right )-3 a^3 \left (16 c^3+24 c^2 d x+2 c d^2 x^2-3 d^3 x^3\right )\right )}{192 a^4 c^2 x^4}-\frac {(b c-a d)^2 \left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{9/2} c^{5/2}} \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. \(592\) vs.
\(2(228)=456\).
time = 0.06, size = 593, normalized size = 2.23
method | result | size |
default | \(-\frac {\sqrt {d x +c}\, \sqrt {b x +a}\, \left (9 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{4} d^{4} x^{4}+12 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{3} b c \,d^{3} x^{4}+54 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{2} b^{2} c^{2} d^{2} x^{4}-180 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a \,b^{3} c^{3} d \,x^{4}+105 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) b^{4} c^{4} x^{4}-18 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{3} d^{3} x^{3}-30 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} b c \,d^{2} x^{3}+290 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a \,b^{2} c^{2} d \,x^{3}-210 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, b^{3} c^{3} x^{3}+12 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{3} c \,d^{2} x^{2}-184 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} b \,c^{2} d \,x^{2}+140 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a \,b^{2} c^{3} x^{2}+144 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{3} c^{2} d x -112 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} b \,c^{3} x +96 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{3} c^{3} \sqrt {a c}\right )}{384 a^{4} c^{2} \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, x^{4} \sqrt {a c}}\) | \(593\) |
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 = 3.88, size = 574, normalized size = 2.16 \begin {gather*} \left [\frac {3 \, {\left (35 \, b^{4} c^{4} - 60 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a^{3} b c d^{3} + 3 \, a^{4} d^{4}\right )} \sqrt {a c} x^{4} \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 (48 \, a^{4} c^{4} - {\left (105 \, a b^{3} c^{4} - 145 \, a^{2} b^{2} c^{3} d + 15 \, a^{3} b c^{2} d^{2} + 9 \, a^{4} c d^{3}\right )} x^{3} + 2 \, {\left (35 \, a^{2} b^{2} c^{4} - 46 \, a^{3} b c^{3} d + 3 \, a^{4} c^{2} d^{2}\right )} x^{2} - 8 \, {\left (7 \, a^{3} b c^{4} - 9 \, a^{4} c^{3} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{768 \, a^{5} c^{3} x^{4}}, \frac {3 \, {\left (35 \, b^{4} c^{4} - 60 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a^{3} b c d^{3} + 3 \, a^{4} d^{4}\right )} \sqrt {-a c} x^{4} \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 (48 \, a^{4} c^{4} - {\left (105 \, a b^{3} c^{4} - 145 \, a^{2} b^{2} c^{3} d + 15 \, a^{3} b c^{2} d^{2} + 9 \, a^{4} c d^{3}\right )} x^{3} + 2 \, {\left (35 \, a^{2} b^{2} c^{4} - 46 \, a^{3} b c^{3} d + 3 \, a^{4} c^{2} d^{2}\right )} x^{2} - 8 \, {\left (7 \, a^{3} b c^{4} - 9 \, a^{4} c^{3} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{384 \, a^{5} c^{3} x^{4}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c + d x\right )^{\frac {3}{2}}}{x^{5} \sqrt {a + b x}}\, dx \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. 3834 vs.
\(2 (228) = 456\).
time = 3.73, size = 3834, normalized size = 14.41 \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^5\,\sqrt {a+b\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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