Optimal. Leaf size=315 \[ -\frac {\left (7 a^2 d^2+6 a b c d+3 b^2 c^2\right ) (b c-a d)^3 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{128 b^{9/2} d^{7/2}}+\frac {(a+b x)^{3/2} \sqrt {c+d x} \left (7 a^2 d^2+6 a b c d+3 b^2 c^2\right ) (b c-a d)}{64 b^4 d^2}+\frac {\sqrt {a+b x} \sqrt {c+d x} \left (7 a^2 d^2+6 a b c d+3 b^2 c^2\right ) (b c-a d)^2}{128 b^4 d^3}+\frac {(a+b x)^{3/2} (c+d x)^{3/2} \left (7 a^2 d^2+6 a b c d+3 b^2 c^2\right )}{48 b^3 d^2}-\frac {(a+b x)^{3/2} (c+d x)^{5/2} (7 a d+5 b c)}{40 b^2 d^2}+\frac {x (a+b x)^{3/2} (c+d x)^{5/2}}{5 b d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.29, antiderivative size = 315, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {90, 80, 50, 63, 217, 206} \[ \frac {\sqrt {a+b x} \sqrt {c+d x} \left (7 a^2 d^2+6 a b c d+3 b^2 c^2\right ) (b c-a d)^2}{128 b^4 d^3}+\frac {(a+b x)^{3/2} \sqrt {c+d x} \left (7 a^2 d^2+6 a b c d+3 b^2 c^2\right ) (b c-a d)}{64 b^4 d^2}+\frac {(a+b x)^{3/2} (c+d x)^{3/2} \left (7 a^2 d^2+6 a b c d+3 b^2 c^2\right )}{48 b^3 d^2}-\frac {\left (7 a^2 d^2+6 a b c d+3 b^2 c^2\right ) (b c-a d)^3 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{128 b^{9/2} d^{7/2}}-\frac {(a+b x)^{3/2} (c+d x)^{5/2} (7 a d+5 b c)}{40 b^2 d^2}+\frac {x (a+b x)^{3/2} (c+d x)^{5/2}}{5 b d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 50
Rule 63
Rule 80
Rule 90
Rule 206
Rule 217
Rubi steps
\begin {align*} \int x^2 \sqrt {a+b x} (c+d x)^{3/2} \, dx &=\frac {x (a+b x)^{3/2} (c+d x)^{5/2}}{5 b d}+\frac {\int \sqrt {a+b x} (c+d x)^{3/2} \left (-a c-\frac {1}{2} (5 b c+7 a d) x\right ) \, dx}{5 b d}\\ &=-\frac {(5 b c+7 a d) (a+b x)^{3/2} (c+d x)^{5/2}}{40 b^2 d^2}+\frac {x (a+b x)^{3/2} (c+d x)^{5/2}}{5 b d}+\frac {\left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right ) \int \sqrt {a+b x} (c+d x)^{3/2} \, dx}{16 b^2 d^2}\\ &=\frac {\left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right ) (a+b x)^{3/2} (c+d x)^{3/2}}{48 b^3 d^2}-\frac {(5 b c+7 a d) (a+b x)^{3/2} (c+d x)^{5/2}}{40 b^2 d^2}+\frac {x (a+b x)^{3/2} (c+d x)^{5/2}}{5 b d}+\frac {\left ((b c-a d) \left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right )\right ) \int \sqrt {a+b x} \sqrt {c+d x} \, dx}{32 b^3 d^2}\\ &=\frac {(b c-a d) \left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{64 b^4 d^2}+\frac {\left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right ) (a+b x)^{3/2} (c+d x)^{3/2}}{48 b^3 d^2}-\frac {(5 b c+7 a d) (a+b x)^{3/2} (c+d x)^{5/2}}{40 b^2 d^2}+\frac {x (a+b x)^{3/2} (c+d x)^{5/2}}{5 b d}+\frac {\left ((b c-a d)^2 \left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right )\right ) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{128 b^4 d^2}\\ &=\frac {(b c-a d)^2 \left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{128 b^4 d^3}+\frac {(b c-a d) \left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{64 b^4 d^2}+\frac {\left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right ) (a+b x)^{3/2} (c+d x)^{3/2}}{48 b^3 d^2}-\frac {(5 b c+7 a d) (a+b x)^{3/2} (c+d x)^{5/2}}{40 b^2 d^2}+\frac {x (a+b x)^{3/2} (c+d x)^{5/2}}{5 b d}-\frac {\left ((b c-a d)^3 \left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{256 b^4 d^3}\\ &=\frac {(b c-a d)^2 \left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{128 b^4 d^3}+\frac {(b c-a d) \left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{64 b^4 d^2}+\frac {\left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right ) (a+b x)^{3/2} (c+d x)^{3/2}}{48 b^3 d^2}-\frac {(5 b c+7 a d) (a+b x)^{3/2} (c+d x)^{5/2}}{40 b^2 d^2}+\frac {x (a+b x)^{3/2} (c+d x)^{5/2}}{5 b d}-\frac {\left ((b c-a d)^3 \left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{128 b^5 d^3}\\ &=\frac {(b c-a d)^2 \left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{128 b^4 d^3}+\frac {(b c-a d) \left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{64 b^4 d^2}+\frac {\left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right ) (a+b x)^{3/2} (c+d x)^{3/2}}{48 b^3 d^2}-\frac {(5 b c+7 a d) (a+b x)^{3/2} (c+d x)^{5/2}}{40 b^2 d^2}+\frac {x (a+b x)^{3/2} (c+d x)^{5/2}}{5 b d}-\frac {\left ((b c-a d)^3 \left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{128 b^5 d^3}\\ &=\frac {(b c-a d)^2 \left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{128 b^4 d^3}+\frac {(b c-a d) \left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{64 b^4 d^2}+\frac {\left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right ) (a+b x)^{3/2} (c+d x)^{3/2}}{48 b^3 d^2}-\frac {(5 b c+7 a d) (a+b x)^{3/2} (c+d x)^{5/2}}{40 b^2 d^2}+\frac {x (a+b x)^{3/2} (c+d x)^{5/2}}{5 b d}-\frac {(b c-a d)^3 \left (3 b^2 c^2+6 a b c d+7 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{128 b^{9/2} d^{7/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.77, size = 268, normalized size = 0.85 \[ \frac {b \sqrt {d} \sqrt {a+b x} (c+d x) \left (-105 a^4 d^4+10 a^3 b d^3 (19 c+7 d x)-2 a^2 b^2 d^2 \left (18 c^2+61 c d x+28 d^2 x^2\right )+6 a b^3 d \left (-5 c^3+3 c^2 d x+16 c d^2 x^2+8 d^3 x^3\right )+3 b^4 \left (15 c^4-10 c^3 d x+8 c^2 d^2 x^2+176 c d^3 x^3+128 d^4 x^4\right )\right )-15 (b c-a d)^{7/2} \left (7 a^2 d^2+6 a b c d+3 b^2 c^2\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{1920 b^5 d^{7/2} \sqrt {c+d x}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.88, size = 704, normalized size = 2.23 \[ \left [-\frac {15 \, {\left (3 \, b^{5} c^{5} - 3 \, a b^{4} c^{4} d - 2 \, a^{2} b^{3} c^{3} d^{2} - 6 \, a^{3} b^{2} c^{2} d^{3} + 15 \, a^{4} b c d^{4} - 7 \, a^{5} d^{5}\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \, {\left (384 \, b^{5} d^{5} x^{4} + 45 \, b^{5} c^{4} d - 30 \, a b^{4} c^{3} d^{2} - 36 \, a^{2} b^{3} c^{2} d^{3} + 190 \, a^{3} b^{2} c d^{4} - 105 \, a^{4} b d^{5} + 48 \, {\left (11 \, b^{5} c d^{4} + a b^{4} d^{5}\right )} x^{3} + 8 \, {\left (3 \, b^{5} c^{2} d^{3} + 12 \, a b^{4} c d^{4} - 7 \, a^{2} b^{3} d^{5}\right )} x^{2} - 2 \, {\left (15 \, b^{5} c^{3} d^{2} - 9 \, a b^{4} c^{2} d^{3} + 61 \, a^{2} b^{3} c d^{4} - 35 \, a^{3} b^{2} d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{7680 \, b^{5} d^{4}}, \frac {15 \, {\left (3 \, b^{5} c^{5} - 3 \, a b^{4} c^{4} d - 2 \, a^{2} b^{3} c^{3} d^{2} - 6 \, a^{3} b^{2} c^{2} d^{3} + 15 \, a^{4} b c d^{4} - 7 \, a^{5} d^{5}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) + 2 \, {\left (384 \, b^{5} d^{5} x^{4} + 45 \, b^{5} c^{4} d - 30 \, a b^{4} c^{3} d^{2} - 36 \, a^{2} b^{3} c^{2} d^{3} + 190 \, a^{3} b^{2} c d^{4} - 105 \, a^{4} b d^{5} + 48 \, {\left (11 \, b^{5} c d^{4} + a b^{4} d^{5}\right )} x^{3} + 8 \, {\left (3 \, b^{5} c^{2} d^{3} + 12 \, a b^{4} c d^{4} - 7 \, a^{2} b^{3} d^{5}\right )} x^{2} - 2 \, {\left (15 \, b^{5} c^{3} d^{2} - 9 \, a b^{4} c^{2} d^{3} + 61 \, a^{2} b^{3} c d^{4} - 35 \, a^{3} b^{2} d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3840 \, b^{5} d^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 2.96, size = 1170, normalized size = 3.71 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.02, size = 942, normalized size = 2.99 \[ \frac {\sqrt {d x +c}\, \sqrt {b x +a}\, \left (768 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, b^{4} d^{4} x^{4}+105 a^{5} d^{5} \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 )-225 a^{4} b c \,d^{4} \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 )+90 a^{3} b^{2} c^{2} d^{3} \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 )+30 a^{2} b^{3} c^{3} d^{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 )+45 a \,b^{4} c^{4} d \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 )+96 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, a \,b^{3} d^{4} x^{3}-45 b^{5} c^{5} \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 )+1056 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, b^{4} c \,d^{3} x^{3}-112 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, a^{2} b^{2} d^{4} x^{2}+192 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, a \,b^{3} c \,d^{3} x^{2}+48 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, b^{4} c^{2} d^{2} x^{2}+140 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} b \,d^{4} x -244 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{2} b^{2} c \,d^{3} x +36 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a \,b^{3} c^{2} d^{2} x -60 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, b^{4} c^{3} d x -210 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{4} d^{4}+380 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{3} b c \,d^{3}-72 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a^{2} b^{2} c^{2} d^{2}-60 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, a \,b^{3} c^{3} d +90 \sqrt {b d}\, \sqrt {b d \,x^{2}+a d x +b c x +a c}\, b^{4} c^{4}\right )}{3840 \sqrt {b d \,x^{2}+a d x +b c x +a c}\, \sqrt {b d}\, b^{4} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^2\,\sqrt {a+b\,x}\,{\left (c+d\,x\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________