Optimal. Leaf size=209 \[ -\frac {3 \left (a^2 d^2+6 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 \sqrt {a} \sqrt {c}}-\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{2 x^2}-\frac {3 \sqrt {a+b x} (c+d x)^{3/2} (a d+b c)}{4 c x}+\frac {3 d \sqrt {a+b x} \sqrt {c+d x} (a d+3 b c)}{4 c}+3 \sqrt {b} \sqrt {d} (a d+b c) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right ) \]
________________________________________________________________________________________
Rubi [A] time = 0.18, antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {97, 149, 154, 157, 63, 217, 206, 93, 208} \begin {gather*} -\frac {3 \left (a^2 d^2+6 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 \sqrt {a} \sqrt {c}}-\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{2 x^2}-\frac {3 \sqrt {a+b x} (c+d x)^{3/2} (a d+b c)}{4 c x}+\frac {3 d \sqrt {a+b x} \sqrt {c+d x} (a d+3 b c)}{4 c}+3 \sqrt {b} \sqrt {d} (a d+b c) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 63
Rule 93
Rule 97
Rule 149
Rule 154
Rule 157
Rule 206
Rule 208
Rule 217
Rubi steps
\begin {align*} \int \frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x^3} \, dx &=-\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{2 x^2}+\frac {1}{2} \int \frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {3}{2} (b c+a d)+3 b d x\right )}{x^2} \, dx\\ &=-\frac {3 (b c+a d) \sqrt {a+b x} (c+d x)^{3/2}}{4 c x}-\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{2 x^2}+\frac {\int \frac {\sqrt {c+d x} \left (\frac {3}{4} \left (b^2 c^2+6 a b c d+a^2 d^2\right )+\frac {3}{2} b d (3 b c+a d) x\right )}{x \sqrt {a+b x}} \, dx}{2 c}\\ &=\frac {3 d (3 b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{4 c}-\frac {3 (b c+a d) \sqrt {a+b x} (c+d x)^{3/2}}{4 c x}-\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{2 x^2}+\frac {\int \frac {\frac {3}{4} b c \left (b^2 c^2+6 a b c d+a^2 d^2\right )+3 b^2 c d (b c+a d) x}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{2 b c}\\ &=\frac {3 d (3 b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{4 c}-\frac {3 (b c+a d) \sqrt {a+b x} (c+d x)^{3/2}}{4 c x}-\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{2 x^2}+\frac {1}{2} (3 b d (b c+a d)) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx+\frac {1}{8} \left (3 \left (b^2 c^2+6 a b c d+a^2 d^2\right )\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx\\ &=\frac {3 d (3 b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{4 c}-\frac {3 (b c+a d) \sqrt {a+b x} (c+d x)^{3/2}}{4 c x}-\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{2 x^2}+(3 d (b c+a d)) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )+\frac {1}{4} \left (3 \left (b^2 c^2+6 a b c d+a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )\\ &=\frac {3 d (3 b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{4 c}-\frac {3 (b c+a d) \sqrt {a+b x} (c+d x)^{3/2}}{4 c x}-\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{2 x^2}-\frac {3 \left (b^2 c^2+6 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 )}{4 \sqrt {a} \sqrt {c}}+(3 d (b c+a d)) \operatorname {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )\\ &=\frac {3 d (3 b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{4 c}-\frac {3 (b c+a d) \sqrt {a+b x} (c+d x)^{3/2}}{4 c x}-\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{2 x^2}-\frac {3 \left (b^2 c^2+6 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 )}{4 \sqrt {a} \sqrt {c}}+3 \sqrt {b} \sqrt {d} (b c+a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 1.36, size = 197, normalized size = 0.94 \begin {gather*} -\frac {3 \left (a^2 d^2+6 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 \sqrt {a} \sqrt {c}}-\frac {\sqrt {a+b x} \sqrt {c+d x} (a (2 c+5 d x)+b x (5 c-4 d x))}{4 x^2}+\frac {3 \sqrt {d} \sqrt {b c-a d} (a d+b c) \sqrt {\frac {b (c+d x)}{b c-a d}} \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{\sqrt {c+d x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.58, size = 367, normalized size = 1.76 \begin {gather*} -\frac {3 \left (a^2 d^2+6 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{4 \sqrt {a} \sqrt {c}}+\frac {\sqrt {c+d x} \left (\frac {5 a^3 d^3 (c+d x)}{a+b x}-\frac {9 a^3 b d^2 (c+d x)^2}{(a+b x)^2}+\frac {6 a^2 b^2 c d (c+d x)^2}{(a+b x)^2}+\frac {9 a^2 b c d^2 (c+d x)}{a+b x}-3 a^2 c d^3-\frac {5 b^3 c^3 (c+d x)}{a+b x}+\frac {3 a b^3 c^2 (c+d x)^2}{(a+b x)^2}-\frac {9 a b^2 c^2 d (c+d x)}{a+b x}-6 a b c^2 d^2+9 b^2 c^3 d\right )}{4 \sqrt {a+b x} \left (c-\frac {a (c+d x)}{a+b x}\right )^2 \left (\frac {b (c+d x)}{a+b x}-d\right )}+3 \left (a \sqrt {b} d^{3/2}+b^{3/2} c \sqrt {d}\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 4.32, size = 1089, normalized size = 5.21 \begin {gather*} \left [\frac {12 \, {\left (a b c^{2} + a^{2} c d\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 ) + 3 \, {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\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 (4 \, a b c d x^{2} - 2 \, a^{2} c^{2} - 5 \, {\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{16 \, a c x^{2}}, -\frac {24 \, {\left (a b c^{2} + a^{2} c d\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 ) - 3 \, {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\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 (4 \, a b c d x^{2} - 2 \, a^{2} c^{2} - 5 \, {\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{16 \, a c x^{2}}, \frac {3 \, {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\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 ) + 6 \, {\left (a b c^{2} + a^{2} c d\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 ) + 2 \, {\left (4 \, a b c d x^{2} - 2 \, a^{2} c^{2} - 5 \, {\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{8 \, a c x^{2}}, \frac {3 \, {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\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 ) - 12 \, {\left (a b c^{2} + a^{2} c d\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 (4 \, a b c d x^{2} - 2 \, a^{2} c^{2} - 5 \, {\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{8 \, a c x^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 16.64, size = 1168, normalized size = 5.59
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.02, size = 497, normalized size = 2.38 \begin {gather*} \frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (-3 \sqrt {b d}\, a^{2} d^{2} x^{2} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )-18 \sqrt {b d}\, a b c d \,x^{2} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )+12 \sqrt {a c}\, a b \,d^{2} x^{2} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-3 \sqrt {b d}\, b^{2} c^{2} x^{2} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}}{x}\right )+12 \sqrt {a c}\, b^{2} c d \,x^{2} \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+8 \sqrt {b d}\, \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, b d \,x^{2}-10 \sqrt {b d}\, \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a d x -10 \sqrt {b d}\, \sqrt {a c}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, b c x -4 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, a c \right )}{8 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, \sqrt {a c}\, x^{2}} \end {gather*}
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]
________________________________________________________________________________________
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 )}^{3/2}}{x^3} \,d x \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 {3}{2}}}{x^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________