Optimal. Leaf size=294 \[ \frac {d \left (19 b^2 c^2+14 a b c d-a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 c^2}-\frac {\left (5 b^2 c^2+12 a b c d-a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{8 c^2 x}-\frac {(5 b c+3 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{12 c x^2}-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{3 x^3}-\frac {\left (5 b^3 c^3+45 a b^2 c^2 d+15 a^2 b c d^2-a^3 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 \sqrt {a} c^{3/2}}+b^{3/2} \sqrt {d} (3 b c+5 a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right ) \]
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Rubi [A]
time = 0.21, antiderivative size = 294, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 9, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {99, 154, 159,
163, 65, 223, 212, 95, 214} \begin {gather*} -\frac {\sqrt {a+b x} (c+d x)^{3/2} \left (-a^2 d^2+12 a b c d+5 b^2 c^2\right )}{8 c^2 x}+\frac {d \sqrt {a+b x} \sqrt {c+d x} \left (-a^2 d^2+14 a b c d+19 b^2 c^2\right )}{8 c^2}-\frac {\left (-a^3 d^3+15 a^2 b c d^2+45 a b^2 c^2 d+5 b^3 c^3\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 \sqrt {a} c^{3/2}}+b^{3/2} \sqrt {d} (5 a d+3 b c) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{3 x^3}-\frac {(a+b x)^{3/2} (c+d x)^{3/2} (3 a d+5 b c)}{12 c x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 95
Rule 99
Rule 154
Rule 159
Rule 163
Rule 212
Rule 214
Rule 223
Rubi steps
\begin {align*} \int \frac {(a+b x)^{5/2} (c+d x)^{3/2}}{x^4} \, dx &=-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{3 x^3}+\frac {1}{3} \int \frac {(a+b x)^{3/2} \sqrt {c+d x} \left (\frac {1}{2} (5 b c+3 a d)+4 b d x\right )}{x^3} \, dx\\ &=-\frac {(5 b c+3 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{12 c x^2}-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{3 x^3}+\frac {\int \frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {3}{4} \left (5 b^2 c^2+12 a b c d-a^2 d^2\right )+\frac {3}{2} b d (7 b c+a d) x\right )}{x^2} \, dx}{6 c}\\ &=-\frac {\left (5 b^2 c^2+12 a b c d-a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{8 c^2 x}-\frac {(5 b c+3 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{12 c x^2}-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{3 x^3}+\frac {\int \frac {\sqrt {c+d x} \left (\frac {3}{8} \left (5 b^3 c^3+45 a b^2 c^2 d+15 a^2 b c d^2-a^3 d^3\right )+\frac {3}{4} b d \left (19 b^2 c^2+14 a b c d-a^2 d^2\right ) x\right )}{x \sqrt {a+b x}} \, dx}{6 c^2}\\ &=\frac {d \left (19 b^2 c^2+14 a b c d-a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 c^2}-\frac {\left (5 b^2 c^2+12 a b c d-a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{8 c^2 x}-\frac {(5 b c+3 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{12 c x^2}-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{3 x^3}+\frac {\int \frac {\frac {3}{8} b c \left (5 b^3 c^3+45 a b^2 c^2 d+15 a^2 b c d^2-a^3 d^3\right )+3 b^3 c^2 d (3 b c+5 a d) x}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{6 b c^2}\\ &=\frac {d \left (19 b^2 c^2+14 a b c d-a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 c^2}-\frac {\left (5 b^2 c^2+12 a b c d-a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{8 c^2 x}-\frac {(5 b c+3 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{12 c x^2}-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{3 x^3}+\frac {1}{2} \left (b^2 d (3 b c+5 a d)\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx+\frac {\left (5 b^3 c^3+45 a b^2 c^2 d+15 a^2 b c d^2-a^3 d^3\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{16 c}\\ &=\frac {d \left (19 b^2 c^2+14 a b c d-a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 c^2}-\frac {\left (5 b^2 c^2+12 a b c d-a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{8 c^2 x}-\frac {(5 b c+3 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{12 c x^2}-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{3 x^3}+(b d (3 b c+5 a d)) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )+\frac {\left (5 b^3 c^3+45 a b^2 c^2 d+15 a^2 b c d^2-a^3 d^3\right ) \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{8 c}\\ &=\frac {d \left (19 b^2 c^2+14 a b c d-a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 c^2}-\frac {\left (5 b^2 c^2+12 a b c d-a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{8 c^2 x}-\frac {(5 b c+3 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{12 c x^2}-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{3 x^3}-\frac {\left (5 b^3 c^3+45 a b^2 c^2 d+15 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 \sqrt {a} c^{3/2}}+(b d (3 b c+5 a d)) \text {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )\\ &=\frac {d \left (19 b^2 c^2+14 a b c d-a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 c^2}-\frac {\left (5 b^2 c^2+12 a b c d-a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{8 c^2 x}-\frac {(5 b c+3 a d) (a+b x)^{3/2} (c+d x)^{3/2}}{12 c x^2}-\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{3 x^3}-\frac {\left (5 b^3 c^3+45 a b^2 c^2 d+15 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 \sqrt {a} c^{3/2}}+b^{3/2} \sqrt {d} (3 b c+5 a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.78, size = 214, normalized size = 0.73 \begin {gather*} -\frac {\sqrt {a+b x} \sqrt {c+d x} \left (3 b^2 c x^2 (11 c-8 d x)+2 a b c x (13 c+34 d x)+a^2 \left (8 c^2+14 c d x+3 d^2 x^2\right )\right )}{24 c x^3}+\frac {\left (-5 b^3 c^3-45 a b^2 c^2 d-15 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 \sqrt {a} c^{3/2}}+b^{3/2} \sqrt {d} (3 b c+5 a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right ) \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. \(607\) vs.
\(2(244)=488\).
time = 0.08, size = 608, normalized size = 2.07
method | result | size |
default | \(\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (120 \ln \left (\frac {2 b d x +2 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a \,b^{2} c \,d^{2} x^{3} \sqrt {a c}+72 \ln \left (\frac {2 b d x +2 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{3} c^{2} d \,x^{3} \sqrt {a c}+3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{3} d^{3} x^{3} \sqrt {b d}-45 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a^{2} b c \,d^{2} x^{3} \sqrt {b d}-135 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) a \,b^{2} c^{2} d \,x^{3} \sqrt {b d}-15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}+2 a c}{x}\right ) b^{3} c^{3} x^{3} \sqrt {b d}+48 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, b^{2} c d \,x^{3}-6 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} d^{2} x^{2}-136 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a b c d \,x^{2}-66 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, b^{2} c^{2} x^{2}-28 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} c d x -52 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a b \,c^{2} x -16 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} c^{2}\right )}{48 c \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, x^{3} \sqrt {b d}\, \sqrt {a c}}\) | \(608\) |
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 = 4.01, size = 1357, normalized size = 4.62 \begin {gather*} \left [\frac {24 \, {\left (3 \, a b^{2} c^{3} + 5 \, a^{2} b c^{2} d\right )} \sqrt {b d} x^{3} \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 x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) - 3 \, {\left (5 \, b^{3} c^{3} + 45 \, a b^{2} c^{2} d + 15 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {a c} x^{3} \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 (24 \, a b^{2} c^{2} d x^{3} - 8 \, a^{3} c^{3} - {\left (33 \, a b^{2} c^{3} + 68 \, a^{2} b c^{2} d + 3 \, a^{3} c d^{2}\right )} x^{2} - 2 \, {\left (13 \, a^{2} b c^{3} + 7 \, a^{3} c^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{96 \, a c^{2} x^{3}}, -\frac {48 \, {\left (3 \, a b^{2} c^{3} + 5 \, a^{2} b c^{2} d\right )} \sqrt {-b d} x^{3} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) + 3 \, {\left (5 \, b^{3} c^{3} + 45 \, a b^{2} c^{2} d + 15 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {a c} x^{3} \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 (24 \, a b^{2} c^{2} d x^{3} - 8 \, a^{3} c^{3} - {\left (33 \, a b^{2} c^{3} + 68 \, a^{2} b c^{2} d + 3 \, a^{3} c d^{2}\right )} x^{2} - 2 \, {\left (13 \, a^{2} b c^{3} + 7 \, a^{3} c^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{96 \, a c^{2} x^{3}}, \frac {3 \, {\left (5 \, b^{3} c^{3} + 45 \, a b^{2} c^{2} d + 15 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {-a c} x^{3} \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 ) + 12 \, {\left (3 \, a b^{2} c^{3} + 5 \, a^{2} b c^{2} d\right )} \sqrt {b d} x^{3} \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 x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + 2 \, {\left (24 \, a b^{2} c^{2} d x^{3} - 8 \, a^{3} c^{3} - {\left (33 \, a b^{2} c^{3} + 68 \, a^{2} b c^{2} d + 3 \, a^{3} c d^{2}\right )} x^{2} - 2 \, {\left (13 \, a^{2} b c^{3} + 7 \, a^{3} c^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, a c^{2} x^{3}}, \frac {3 \, {\left (5 \, b^{3} c^{3} + 45 \, a b^{2} c^{2} d + 15 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {-a c} x^{3} \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 ) - 24 \, {\left (3 \, a b^{2} c^{3} + 5 \, a^{2} b c^{2} d\right )} \sqrt {-b d} x^{3} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) + 2 \, {\left (24 \, a b^{2} c^{2} d x^{3} - 8 \, a^{3} c^{3} - {\left (33 \, a b^{2} c^{3} + 68 \, a^{2} b c^{2} d + 3 \, a^{3} c d^{2}\right )} x^{2} - 2 \, {\left (13 \, a^{2} b c^{3} + 7 \, a^{3} c^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, a c^{2} x^{3}}\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 (a + b x\right )^{\frac {5}{2}} \left (c + d x\right )^{\frac {3}{2}}}{x^{4}}\, 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. 2316 vs.
\(2 (244) = 488\).
time = 13.81, size = 2316, normalized size = 7.88 \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 (a+b\,x\right )}^{5/2}\,{\left (c+d\,x\right )}^{3/2}}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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