Optimal. Leaf size=304 \[ -\frac {\left (5 b^3 c^3-55 a b^2 c^2 d-17 a^2 b c d^2+3 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b^2 d}+\frac {1}{96} \left (50 a c-\frac {5 b c^2}{d}+\frac {3 a^2 d}{b}\right ) \sqrt {a+b x} (c+d x)^{3/2}+\frac {(5 b c+3 a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 d}+\frac {1}{4} (a+b x)^{3/2} (c+d x)^{5/2}-2 a^{3/2} c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )-\frac {\left (5 b^4 c^4-60 a b^3 c^3 d-90 a^2 b^2 c^2 d^2+20 a^3 b c d^3-3 a^4 d^4\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{5/2} d^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.20, antiderivative size = 304, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {103, 159, 163,
65, 223, 212, 95, 214} \begin {gather*} -2 a^{3/2} c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )+\frac {1}{96} \sqrt {a+b x} (c+d x)^{3/2} \left (\frac {3 a^2 d}{b}+50 a c-\frac {5 b c^2}{d}\right )-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (3 a^3 d^3-17 a^2 b c d^2-55 a b^2 c^2 d+5 b^3 c^3\right )}{64 b^2 d}-\frac {\left (-3 a^4 d^4+20 a^3 b c d^3-90 a^2 b^2 c^2 d^2-60 a b^3 c^3 d+5 b^4 c^4\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{5/2} d^{3/2}}+\frac {1}{4} (a+b x)^{3/2} (c+d x)^{5/2}+\frac {\sqrt {a+b x} (c+d x)^{5/2} (3 a d+5 b c)}{24 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 65
Rule 95
Rule 103
Rule 159
Rule 163
Rule 212
Rule 214
Rule 223
Rubi steps
\begin {align*} \int \frac {(a+b x)^{3/2} (c+d x)^{5/2}}{x} \, dx &=\frac {1}{4} (a+b x)^{3/2} (c+d x)^{5/2}-\frac {1}{4} \int \frac {\sqrt {a+b x} (c+d x)^{3/2} \left (-4 a c+\frac {1}{2} (-5 b c-3 a d) x\right )}{x} \, dx\\ &=\frac {(5 b c+3 a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 d}+\frac {1}{4} (a+b x)^{3/2} (c+d x)^{5/2}-\frac {\int \frac {(c+d x)^{3/2} \left (-12 a^2 c d+\frac {1}{4} \left (5 b^2 c^2-50 a b c d-3 a^2 d^2\right ) x\right )}{x \sqrt {a+b x}} \, dx}{12 d}\\ &=\frac {1}{96} \left (50 a c-\frac {5 b c^2}{d}+\frac {3 a^2 d}{b}\right ) \sqrt {a+b x} (c+d x)^{3/2}+\frac {(5 b c+3 a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 d}+\frac {1}{4} (a+b x)^{3/2} (c+d x)^{5/2}-\frac {\int \frac {\sqrt {c+d x} \left (-24 a^2 b c^2 d+\frac {3}{8} \left (5 b^3 c^3-55 a b^2 c^2 d-17 a^2 b c d^2+3 a^3 d^3\right ) x\right )}{x \sqrt {a+b x}} \, dx}{24 b d}\\ &=-\frac {\left (5 b^3 c^3-55 a b^2 c^2 d-17 a^2 b c d^2+3 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b^2 d}+\frac {1}{96} \left (50 a c-\frac {5 b c^2}{d}+\frac {3 a^2 d}{b}\right ) \sqrt {a+b x} (c+d x)^{3/2}+\frac {(5 b c+3 a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 d}+\frac {1}{4} (a+b x)^{3/2} (c+d x)^{5/2}-\frac {\int \frac {-24 a^2 b^2 c^3 d+\frac {3}{16} \left (5 b^4 c^4-60 a b^3 c^3 d-90 a^2 b^2 c^2 d^2+20 a^3 b c d^3-3 a^4 d^4\right ) x}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{24 b^2 d}\\ &=-\frac {\left (5 b^3 c^3-55 a b^2 c^2 d-17 a^2 b c d^2+3 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b^2 d}+\frac {1}{96} \left (50 a c-\frac {5 b c^2}{d}+\frac {3 a^2 d}{b}\right ) \sqrt {a+b x} (c+d x)^{3/2}+\frac {(5 b c+3 a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 d}+\frac {1}{4} (a+b x)^{3/2} (c+d x)^{5/2}+\left (a^2 c^3\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx-\frac {\left (5 b^4 c^4-60 a b^3 c^3 d-90 a^2 b^2 c^2 d^2+20 a^3 b c d^3-3 a^4 d^4\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{128 b^2 d}\\ &=-\frac {\left (5 b^3 c^3-55 a b^2 c^2 d-17 a^2 b c d^2+3 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b^2 d}+\frac {1}{96} \left (50 a c-\frac {5 b c^2}{d}+\frac {3 a^2 d}{b}\right ) \sqrt {a+b x} (c+d x)^{3/2}+\frac {(5 b c+3 a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 d}+\frac {1}{4} (a+b x)^{3/2} (c+d x)^{5/2}+\left (2 a^2 c^3\right ) \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )-\frac {\left (5 b^4 c^4-60 a b^3 c^3 d-90 a^2 b^2 c^2 d^2+20 a^3 b c d^3-3 a^4 d^4\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{64 b^3 d}\\ &=-\frac {\left (5 b^3 c^3-55 a b^2 c^2 d-17 a^2 b c d^2+3 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b^2 d}+\frac {1}{96} \left (50 a c-\frac {5 b c^2}{d}+\frac {3 a^2 d}{b}\right ) \sqrt {a+b x} (c+d x)^{3/2}+\frac {(5 b c+3 a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 d}+\frac {1}{4} (a+b x)^{3/2} (c+d x)^{5/2}-2 a^{3/2} c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )-\frac {\left (5 b^4 c^4-60 a b^3 c^3 d-90 a^2 b^2 c^2 d^2+20 a^3 b c d^3-3 a^4 d^4\right ) \text {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{64 b^3 d}\\ &=-\frac {\left (5 b^3 c^3-55 a b^2 c^2 d-17 a^2 b c d^2+3 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b^2 d}+\frac {1}{96} \left (50 a c-\frac {5 b c^2}{d}+\frac {3 a^2 d}{b}\right ) \sqrt {a+b x} (c+d x)^{3/2}+\frac {(5 b c+3 a d) \sqrt {a+b x} (c+d x)^{5/2}}{24 d}+\frac {1}{4} (a+b x)^{3/2} (c+d x)^{5/2}-2 a^{3/2} c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )-\frac {\left (5 b^4 c^4-60 a b^3 c^3 d-90 a^2 b^2 c^2 d^2+20 a^3 b c d^3-3 a^4 d^4\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{5/2} d^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.69, size = 251, normalized size = 0.83 \begin {gather*} \frac {\sqrt {a+b x} \sqrt {c+d x} \left (-9 a^3 d^3+3 a^2 b d^2 (19 c+2 d x)+a b^2 d \left (337 c^2+244 c d x+72 d^2 x^2\right )+b^3 \left (15 c^3+118 c^2 d x+136 c d^2 x^2+48 d^3 x^3\right )\right )}{192 b^2 d}-2 a^{3/2} c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )-\frac {\left (5 b^4 c^4-60 a b^3 c^3 d-90 a^2 b^2 c^2 d^2+20 a^3 b c d^3-3 a^4 d^4\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{64 b^{5/2} d^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(708\) vs.
\(2(254)=508\).
time = 0.07, size = 709, normalized size = 2.33
method | result | size |
default | \(-\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (-96 b^{3} d^{3} x^{3} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}-144 a \,b^{2} d^{3} x^{2} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}-272 b^{3} c \,d^{2} x^{2} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+384 \sqrt {b d}\, \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^{3} d -9 \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 ) \sqrt {a c}\, a^{4} d^{4}+60 \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 ) \sqrt {a c}\, a^{3} b c \,d^{3}-270 \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 ) \sqrt {a c}\, a^{2} b^{2} c^{2} d^{2}-180 \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 ) \sqrt {a c}\, a \,b^{3} c^{3} d +15 \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 ) \sqrt {a c}\, b^{4} c^{4}-12 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} b \,d^{3} x -488 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a \,b^{2} c \,d^{2} x -236 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, b^{3} c^{2} d x +18 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{3} d^{3}-114 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} b c \,d^{2}-674 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a \,b^{2} c^{2} d -30 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, b^{3} c^{3}\right )}{384 b^{2} d \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}\, \sqrt {a c}}\) | \(709\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 14.22, size = 1481, normalized size = 4.87 \begin {gather*} \left [\frac {384 \, \sqrt {a c} a b^{3} c^{2} d^{2} \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 ) - 3 \, {\left (5 \, b^{4} c^{4} - 60 \, a b^{3} c^{3} d - 90 \, a^{2} b^{2} c^{2} d^{2} + 20 \, a^{3} b c d^{3} - 3 \, a^{4} d^{4}\right )} \sqrt {b d} \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 ) + 4 \, {\left (48 \, b^{4} d^{4} x^{3} + 15 \, b^{4} c^{3} d + 337 \, a b^{3} c^{2} d^{2} + 57 \, a^{2} b^{2} c d^{3} - 9 \, a^{3} b d^{4} + 8 \, {\left (17 \, b^{4} c d^{3} + 9 \, a b^{3} d^{4}\right )} x^{2} + 2 \, {\left (59 \, b^{4} c^{2} d^{2} + 122 \, a b^{3} c d^{3} + 3 \, a^{2} b^{2} d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{768 \, b^{3} d^{2}}, \frac {192 \, \sqrt {a c} a b^{3} c^{2} d^{2} \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 ) + 3 \, {\left (5 \, b^{4} c^{4} - 60 \, a b^{3} c^{3} d - 90 \, a^{2} b^{2} c^{2} d^{2} + 20 \, a^{3} b c d^{3} - 3 \, a^{4} d^{4}\right )} \sqrt {-b d} \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 (48 \, b^{4} d^{4} x^{3} + 15 \, b^{4} c^{3} d + 337 \, a b^{3} c^{2} d^{2} + 57 \, a^{2} b^{2} c d^{3} - 9 \, a^{3} b d^{4} + 8 \, {\left (17 \, b^{4} c d^{3} + 9 \, a b^{3} d^{4}\right )} x^{2} + 2 \, {\left (59 \, b^{4} c^{2} d^{2} + 122 \, a b^{3} c d^{3} + 3 \, a^{2} b^{2} d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{384 \, b^{3} d^{2}}, \frac {768 \, \sqrt {-a c} a b^{3} c^{2} d^{2} \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 ) - 3 \, {\left (5 \, b^{4} c^{4} - 60 \, a b^{3} c^{3} d - 90 \, a^{2} b^{2} c^{2} d^{2} + 20 \, a^{3} b c d^{3} - 3 \, a^{4} d^{4}\right )} \sqrt {b d} \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 ) + 4 \, {\left (48 \, b^{4} d^{4} x^{3} + 15 \, b^{4} c^{3} d + 337 \, a b^{3} c^{2} d^{2} + 57 \, a^{2} b^{2} c d^{3} - 9 \, a^{3} b d^{4} + 8 \, {\left (17 \, b^{4} c d^{3} + 9 \, a b^{3} d^{4}\right )} x^{2} + 2 \, {\left (59 \, b^{4} c^{2} d^{2} + 122 \, a b^{3} c d^{3} + 3 \, a^{2} b^{2} d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{768 \, b^{3} d^{2}}, \frac {384 \, \sqrt {-a c} a b^{3} c^{2} d^{2} \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 ) + 3 \, {\left (5 \, b^{4} c^{4} - 60 \, a b^{3} c^{3} d - 90 \, a^{2} b^{2} c^{2} d^{2} + 20 \, a^{3} b c d^{3} - 3 \, a^{4} d^{4}\right )} \sqrt {-b d} \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 (48 \, b^{4} d^{4} x^{3} + 15 \, b^{4} c^{3} d + 337 \, a b^{3} c^{2} d^{2} + 57 \, a^{2} b^{2} c d^{3} - 9 \, a^{3} b d^{4} + 8 \, {\left (17 \, b^{4} c d^{3} + 9 \, a b^{3} d^{4}\right )} x^{2} + 2 \, {\left (59 \, b^{4} c^{2} d^{2} + 122 \, a b^{3} c d^{3} + 3 \, a^{2} b^{2} d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{384 \, b^{3} d^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b x\right )^{\frac {3}{2}} \left (c + d x\right )^{\frac {5}{2}}}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^{3/2}\,{\left (c+d\,x\right )}^{5/2}}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________