Optimal. Leaf size=319 \[ \frac {5}{8} \left (b^2 c^2+10 a b c d+5 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}+\frac {5 \left (b^2 c^2+8 a b c d+3 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{12 c}+\frac {5 b (5 b c+3 a d) \sqrt {a+b x} (c+d x)^{5/2}}{12 c}-\frac {5 (b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{4 c x}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{2 x^2}-\frac {5}{4} \sqrt {a} \sqrt {c} (3 b c+a d) (b c+3 a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )+\frac {5 (b c+a d) \left (b^2 c^2+14 a b c d+a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 \sqrt {b} \sqrt {d}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.27, antiderivative size = 319, normalized size of antiderivative = 1.00, number of steps
used = 11, 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 {5 \sqrt {a+b x} (c+d x)^{3/2} \left (3 a^2 d^2+8 a b c d+b^2 c^2\right )}{12 c}+\frac {5}{8} \sqrt {a+b x} \sqrt {c+d x} \left (5 a^2 d^2+10 a b c d+b^2 c^2\right )+\frac {5 (a d+b c) \left (a^2 d^2+14 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 \sqrt {b} \sqrt {d}}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{2 x^2}-\frac {5 (a+b x)^{3/2} (c+d x)^{5/2} (a d+b c)}{4 c x}+\frac {5 b \sqrt {a+b x} (c+d x)^{5/2} (3 a d+5 b c)}{12 c}-\frac {5}{4} \sqrt {a} \sqrt {c} (a d+3 b c) (3 a d+b c) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
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)^{5/2}}{x^3} \, dx &=-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{2 x^2}+\frac {1}{2} \int \frac {(a+b x)^{3/2} (c+d x)^{3/2} \left (\frac {5}{2} (b c+a d)+5 b d x\right )}{x^2} \, dx\\ &=-\frac {5 (b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{4 c x}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{2 x^2}+\frac {\int \frac {\sqrt {a+b x} (c+d x)^{3/2} \left (\frac {5}{4} (3 b c+a d) (b c+3 a d)+\frac {5}{2} b d (5 b c+3 a d) x\right )}{x} \, dx}{2 c}\\ &=\frac {5 b (5 b c+3 a d) \sqrt {a+b x} (c+d x)^{5/2}}{12 c}-\frac {5 (b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{4 c x}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{2 x^2}+\frac {\int \frac {(c+d x)^{3/2} \left (\frac {15}{4} a d (3 b c+a d) (b c+3 a d)+5 b d \left (b^2 c^2+8 a b c d+3 a^2 d^2\right ) x\right )}{x \sqrt {a+b x}} \, dx}{6 c d}\\ &=\frac {5 \left (b^2 c^2+8 a b c d+3 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{12 c}+\frac {5 b (5 b c+3 a d) \sqrt {a+b x} (c+d x)^{5/2}}{12 c}-\frac {5 (b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{4 c x}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{2 x^2}+\frac {\int \frac {\sqrt {c+d x} \left (\frac {15}{2} a b c d (3 b c+a d) (b c+3 a d)+\frac {15}{2} b^2 c d \left (b^2 c^2+10 a b c d+5 a^2 d^2\right ) x\right )}{x \sqrt {a+b x}} \, dx}{12 b c d}\\ &=\frac {5}{8} \left (b^2 c^2+10 a b c d+5 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}+\frac {5 \left (b^2 c^2+8 a b c d+3 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{12 c}+\frac {5 b (5 b c+3 a d) \sqrt {a+b x} (c+d x)^{5/2}}{12 c}-\frac {5 (b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{4 c x}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{2 x^2}+\frac {\int \frac {\frac {15}{2} a b^2 c^2 d (3 b c+a d) (b c+3 a d)+\frac {15}{4} b^2 c d (b c+a d) \left (b^2 c^2+14 a b c d+a^2 d^2\right ) x}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{12 b^2 c d}\\ &=\frac {5}{8} \left (b^2 c^2+10 a b c d+5 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}+\frac {5 \left (b^2 c^2+8 a b c d+3 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{12 c}+\frac {5 b (5 b c+3 a d) \sqrt {a+b x} (c+d x)^{5/2}}{12 c}-\frac {5 (b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{4 c x}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{2 x^2}+\frac {1}{8} (5 a c (3 b c+a d) (b c+3 a d)) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx+\frac {1}{16} \left (5 (b c+a d) \left (b^2 c^2+14 a b c d+a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx\\ &=\frac {5}{8} \left (b^2 c^2+10 a b c d+5 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}+\frac {5 \left (b^2 c^2+8 a b c d+3 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{12 c}+\frac {5 b (5 b c+3 a d) \sqrt {a+b x} (c+d x)^{5/2}}{12 c}-\frac {5 (b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{4 c x}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{2 x^2}+\frac {1}{4} (5 a c (3 b c+a d) (b c+3 a d)) \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 c+a d) \left (b^2 c^2+14 a b c d+a^2 d^2\right )\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 )}{8 b}\\ &=\frac {5}{8} \left (b^2 c^2+10 a b c d+5 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}+\frac {5 \left (b^2 c^2+8 a b c d+3 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{12 c}+\frac {5 b (5 b c+3 a d) \sqrt {a+b x} (c+d x)^{5/2}}{12 c}-\frac {5 (b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{4 c x}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{2 x^2}-\frac {5}{4} \sqrt {a} \sqrt {c} (3 b c+a d) (b c+3 a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )+\frac {\left (5 (b c+a d) \left (b^2 c^2+14 a b c d+a^2 d^2\right )\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 )}{8 b}\\ &=\frac {5}{8} \left (b^2 c^2+10 a b c d+5 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}+\frac {5 \left (b^2 c^2+8 a b c d+3 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{12 c}+\frac {5 b (5 b c+3 a d) \sqrt {a+b x} (c+d x)^{5/2}}{12 c}-\frac {5 (b c+a d) (a+b x)^{3/2} (c+d x)^{5/2}}{4 c x}-\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{2 x^2}-\frac {5}{4} \sqrt {a} \sqrt {c} (3 b c+a d) (b c+3 a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )+\frac {5 (b c+a d) \left (b^2 c^2+14 a b c d+a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 \sqrt {b} \sqrt {d}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.83, size = 244, normalized size = 0.76 \begin {gather*} \frac {1}{24} \left (\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-3 a^2 \left (4 c^2+18 c d x-11 d^2 x^2\right )+b^2 x^2 \left (33 c^2+26 c d x+8 d^2 x^2\right )+2 a b x \left (-27 c^2+61 c d x+13 d^2 x^2\right )\right )}{x^2}-30 \sqrt {a} \sqrt {c} \left (3 b^2 c^2+10 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )+\frac {15 \left (b^3 c^3+15 a b^2 c^2 d+15 a^2 b c d^2+a^3 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{\sqrt {b} \sqrt {d}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(730\) vs.
\(2(261)=522\).
time = 0.08, size = 731, normalized size = 2.29
method | result | size |
default | \(\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (16 b^{2} d^{2} x^{4} \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b 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}\, a^{3} d^{3} x^{2}+225 \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 c \,d^{2} x^{2}+225 \sqrt {a c}\, \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^{2} d \,x^{2}+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^{3} c^{3} x^{2}-90 \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^{3} c \,d^{2} x^{2}-300 \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 \,c^{2} d \,x^{2}-90 \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 \,b^{2} c^{3} x^{2}+52 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a b \,d^{2} x^{3}+52 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, b^{2} c d \,x^{3}+66 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} d^{2} x^{2}+244 \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}-108 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} c d x -108 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a b \,c^{2} x -24 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, a^{2} c^{2}\right )}{48 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}\, x^{2} \sqrt {a c}}\) | \(731\) |
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 = 7.69, size = 1469, normalized size = 4.61 \begin {gather*} \left [\frac {15 \, {\left (b^{3} c^{3} + 15 \, a b^{2} c^{2} d + 15 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \sqrt {b d} x^{2} \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 ) + 30 \, {\left (3 \, b^{3} c^{2} d + 10 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3}\right )} \sqrt {a c} x^{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 ) + 4 \, {\left (8 \, b^{3} d^{3} x^{4} - 12 \, a^{2} b c^{2} d + 26 \, {\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{3} + {\left (33 \, b^{3} c^{2} d + 122 \, a b^{2} c d^{2} + 33 \, a^{2} b d^{3}\right )} x^{2} - 54 \, {\left (a b^{2} c^{2} d + a^{2} b c d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{96 \, b d x^{2}}, -\frac {15 \, {\left (b^{3} c^{3} + 15 \, a b^{2} c^{2} d + 15 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \sqrt {-b d} x^{2} \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 ) - 15 \, {\left (3 \, b^{3} c^{2} d + 10 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3}\right )} \sqrt {a c} x^{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 ) - 2 \, {\left (8 \, b^{3} d^{3} x^{4} - 12 \, a^{2} b c^{2} d + 26 \, {\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{3} + {\left (33 \, b^{3} c^{2} d + 122 \, a b^{2} c d^{2} + 33 \, a^{2} b d^{3}\right )} x^{2} - 54 \, {\left (a b^{2} c^{2} d + a^{2} b c d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, b d x^{2}}, \frac {60 \, {\left (3 \, b^{3} c^{2} d + 10 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3}\right )} \sqrt {-a c} x^{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 ) + 15 \, {\left (b^{3} c^{3} + 15 \, a b^{2} c^{2} d + 15 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \sqrt {b d} x^{2} \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 (8 \, b^{3} d^{3} x^{4} - 12 \, a^{2} b c^{2} d + 26 \, {\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{3} + {\left (33 \, b^{3} c^{2} d + 122 \, a b^{2} c d^{2} + 33 \, a^{2} b d^{3}\right )} x^{2} - 54 \, {\left (a b^{2} c^{2} d + a^{2} b c d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{96 \, b d x^{2}}, \frac {30 \, {\left (3 \, b^{3} c^{2} d + 10 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3}\right )} \sqrt {-a c} x^{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 ) - 15 \, {\left (b^{3} c^{3} + 15 \, a b^{2} c^{2} d + 15 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \sqrt {-b d} x^{2} \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 (8 \, b^{3} d^{3} x^{4} - 12 \, a^{2} b c^{2} d + 26 \, {\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{3} + {\left (33 \, b^{3} c^{2} d + 122 \, a b^{2} c d^{2} + 33 \, a^{2} b d^{3}\right )} x^{2} - 54 \, {\left (a b^{2} c^{2} d + a^{2} b c d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, b d x^{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 {5}{2}} \left (c + d x\right )^{\frac {5}{2}}}{x^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1336 vs.
\(2 (261) = 522\).
time = 4.30, size = 1336, normalized size = 4.19 \begin {gather*} \frac {2 \, \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} {\left (2 \, {\left (\frac {4 \, {\left (b x + a\right )} d^{2} {\left | b \right |}}{b} + \frac {13 \, b c d^{5} {\left | b \right |} + 5 \, a d^{6} {\left | b \right |}}{b d^{4}}\right )} {\left (b x + a\right )} + \frac {3 \, {\left (11 \, b^{2} c^{2} d^{4} {\left | b \right |} + 32 \, a b c d^{5} {\left | b \right |} + 5 \, a^{2} d^{6} {\left | b \right |}\right )}}{b d^{4}}\right )} \sqrt {b x + a} - \frac {60 \, {\left (3 \, \sqrt {b d} a b^{3} c^{3} {\left | b \right |} + 10 \, \sqrt {b d} a^{2} b^{2} c^{2} d {\left | b \right |} + 3 \, \sqrt {b d} a^{3} b c d^{2} {\left | b \right |}\right )} \arctan \left (-\frac {b^{2} c + a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt {-a b c d} b}\right )}{\sqrt {-a b c d} b} - \frac {15 \, {\left (\sqrt {b d} b^{3} c^{3} {\left | b \right |} + 15 \, \sqrt {b d} a b^{2} c^{2} d {\left | b \right |} + 15 \, \sqrt {b d} a^{2} b c d^{2} {\left | b \right |} + \sqrt {b d} a^{3} d^{3} {\left | b \right |}\right )} \log \left ({\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{b d} - \frac {24 \, {\left (9 \, \sqrt {b d} a b^{9} c^{6} {\left | b \right |} - 27 \, \sqrt {b d} a^{2} b^{8} c^{5} d {\left | b \right |} + 18 \, \sqrt {b d} a^{3} b^{7} c^{4} d^{2} {\left | b \right |} + 18 \, \sqrt {b d} a^{4} b^{6} c^{3} d^{3} {\left | b \right |} - 27 \, \sqrt {b d} a^{5} b^{5} c^{2} d^{4} {\left | b \right |} + 9 \, \sqrt {b d} a^{6} b^{4} c d^{5} {\left | b \right |} - 27 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b^{7} c^{5} {\left | b \right |} + 4 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{2} b^{6} c^{4} d {\left | b \right |} + 46 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{3} b^{5} c^{3} d^{2} {\left | b \right |} + 4 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{4} b^{4} c^{2} d^{3} {\left | b \right |} - 27 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{5} b^{3} c d^{4} {\left | b \right |} + 27 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a b^{5} c^{4} {\left | b \right |} + 45 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a^{2} b^{4} c^{3} d {\left | b \right |} + 45 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a^{3} b^{3} c^{2} d^{2} {\left | b \right |} + 27 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a^{4} b^{2} c d^{3} {\left | b \right |} - 9 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{6} a b^{3} c^{3} {\left | b \right |} - 22 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{6} a^{2} b^{2} c^{2} d {\left | b \right |} - 9 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{6} a^{3} b c d^{2} {\left | b \right |}\right )}}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2} - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{2} c - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b d + {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4}\right )}^{2}}}{48 \, b} \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 )}^{5/2}\,{\left (c+d\,x\right )}^{5/2}}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________