Optimal. Leaf size=161 \[ -\frac {b c^2 \sqrt {a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{8 d^{3/2} (a+b x)}-\frac {b c x \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+d x^2}}{8 d (a+b x)}+\frac {(4 a+3 b x) \sqrt {a^2+2 a b x+b^2 x^2} \left (c+d x^2\right )^{3/2}}{12 d (a+b x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {1001, 780, 195, 217, 206} \[ -\frac {b c^2 \sqrt {a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{8 d^{3/2} (a+b x)}-\frac {b c x \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+d x^2}}{8 d (a+b x)}+\frac {(4 a+3 b x) \sqrt {a^2+2 a b x+b^2 x^2} \left (c+d x^2\right )^{3/2}}{12 d (a+b x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 195
Rule 206
Rule 217
Rule 780
Rule 1001
Rubi steps
\begin {align*} \int x \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+d x^2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int x \left (2 a b+2 b^2 x\right ) \sqrt {c+d x^2} \, dx}{2 a b+2 b^2 x}\\ &=\frac {(4 a+3 b x) \sqrt {a^2+2 a b x+b^2 x^2} \left (c+d x^2\right )^{3/2}}{12 d (a+b x)}-\frac {\left (b^2 c \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \sqrt {c+d x^2} \, dx}{2 d \left (2 a b+2 b^2 x\right )}\\ &=-\frac {b c x \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+d x^2}}{8 d (a+b x)}+\frac {(4 a+3 b x) \sqrt {a^2+2 a b x+b^2 x^2} \left (c+d x^2\right )^{3/2}}{12 d (a+b x)}-\frac {\left (b^2 c^2 \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \frac {1}{\sqrt {c+d x^2}} \, dx}{4 d \left (2 a b+2 b^2 x\right )}\\ &=-\frac {b c x \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+d x^2}}{8 d (a+b x)}+\frac {(4 a+3 b x) \sqrt {a^2+2 a b x+b^2 x^2} \left (c+d x^2\right )^{3/2}}{12 d (a+b x)}-\frac {\left (b^2 c^2 \sqrt {a^2+2 a b x+b^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{1-d x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{4 d \left (2 a b+2 b^2 x\right )}\\ &=-\frac {b c x \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+d x^2}}{8 d (a+b x)}+\frac {(4 a+3 b x) \sqrt {a^2+2 a b x+b^2 x^2} \left (c+d x^2\right )^{3/2}}{12 d (a+b x)}-\frac {b c^2 \sqrt {a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{8 d^{3/2} (a+b x)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.12, size = 117, normalized size = 0.73 \[ \frac {\sqrt {(a+b x)^2} \sqrt {c+d x^2} \left (\sqrt {d} \sqrt {\frac {d x^2}{c}+1} \left (8 a \left (c+d x^2\right )+3 b x \left (c+2 d x^2\right )\right )-3 b c^{3/2} \sinh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )\right )}{24 d^{3/2} (a+b x) \sqrt {\frac {d x^2}{c}+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.00, size = 157, normalized size = 0.98 \[ \left [\frac {3 \, b c^{2} \sqrt {d} \log \left (-2 \, d x^{2} + 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) + 2 \, {\left (6 \, b d^{2} x^{3} + 8 \, a d^{2} x^{2} + 3 \, b c d x + 8 \, a c d\right )} \sqrt {d x^{2} + c}}{48 \, d^{2}}, \frac {3 \, b c^{2} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) + {\left (6 \, b d^{2} x^{3} + 8 \, a d^{2} x^{2} + 3 \, b c d x + 8 \, a c d\right )} \sqrt {d x^{2} + c}}{24 \, d^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.21, size = 98, normalized size = 0.61 \[ \frac {b c^{2} \log \left ({\left | -\sqrt {d} x + \sqrt {d x^{2} + c} \right |}\right ) \mathrm {sgn}\left (b x + a\right )}{8 \, d^{\frac {3}{2}}} + \frac {1}{24} \, \sqrt {d x^{2} + c} {\left ({\left (2 \, {\left (3 \, b x \mathrm {sgn}\left (b x + a\right ) + 4 \, a \mathrm {sgn}\left (b x + a\right )\right )} x + \frac {3 \, b c \mathrm {sgn}\left (b x + a\right )}{d}\right )} x + \frac {8 \, a c \mathrm {sgn}\left (b x + a\right )}{d}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.01, size = 83, normalized size = 0.52 \[ \frac {\left (-3 b \,c^{2} \ln \left (\sqrt {d}\, x +\sqrt {d \,x^{2}+c}\right )-3 \sqrt {d \,x^{2}+c}\, b c \sqrt {d}\, x +6 \left (d \,x^{2}+c \right )^{\frac {3}{2}} b \sqrt {d}\, x +8 \left (d \,x^{2}+c \right )^{\frac {3}{2}} a \sqrt {d}\right ) \mathrm {csgn}\left (b x +a \right )}{24 d^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {d x^{2} + c} \sqrt {{\left (b x + a\right )}^{2}} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,\sqrt {{\left (a+b\,x\right )}^2}\,\sqrt {d\,x^2+c} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \sqrt {c + d x^{2}} \sqrt {\left (a + b x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________