Integrand size = 31, antiderivative size = 282 \[ \int \frac {(d+c x)^2}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\frac {2 \sqrt {b^2+a^2 x^2} \left (4 b^4 c d-4 b^4 c^2 x+25 a^2 b^2 d^2 x+30 a^2 b^2 c d x^2-3 a^2 b^2 c^2 x^3+60 a^4 d^2 x^3+40 a^4 c d x^4+12 a^4 c^2 x^5\right )}{15 a^2 \left (a x+\sqrt {b^2+a^2 x^2}\right )^{7/2}}+\frac {2 \left (-8 b^6 c^2+35 a^2 b^4 d^2+98 a^2 b^4 c d x-49 a^2 b^4 c^2 x^2+385 a^4 b^2 d^2 x^2+350 a^4 b^2 c d x^3+21 a^4 b^2 c^2 x^4+420 a^6 d^2 x^4+280 a^6 c d x^5+84 a^6 c^2 x^6\right )}{105 a^3 \left (a x+\sqrt {b^2+a^2 x^2}\right )^{7/2}} \]
2/15*(a^2*x^2+b^2)^(1/2)*(12*a^4*c^2*x^5+40*a^4*c*d*x^4+60*a^4*d^2*x^3-3*a ^2*b^2*c^2*x^3+30*a^2*b^2*c*d*x^2+25*a^2*b^2*d^2*x-4*b^4*c^2*x+4*b^4*c*d)/ a^2/(a*x+(a^2*x^2+b^2)^(1/2))^(7/2)+2/105*(84*a^6*c^2*x^6+280*a^6*c*d*x^5+ 420*a^6*d^2*x^4+21*a^4*b^2*c^2*x^4+350*a^4*b^2*c*d*x^3+385*a^4*b^2*d^2*x^2 -49*a^2*b^4*c^2*x^2+98*a^2*b^4*c*d*x+35*a^2*b^4*d^2-8*b^6*c^2)/a^3/(a*x+(a ^2*x^2+b^2)^(1/2))^(7/2)
Time = 0.30 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.72 \[ \int \frac {(d+c x)^2}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\frac {2 \left (-8 b^6 c^2+28 a^5 x^3 \left (15 d^2+10 c d x+3 c^2 x^2\right ) \left (a x+\sqrt {b^2+a^2 x^2}\right )+7 a b^4 \left (4 c (d-c x) \sqrt {b^2+a^2 x^2}+a \left (5 d^2+14 c d x-7 c^2 x^2\right )\right )+7 a^3 b^2 x \left (\sqrt {b^2+a^2 x^2} \left (25 d^2+30 c d x-3 c^2 x^2\right )+a x \left (55 d^2+50 c d x+3 c^2 x^2\right )\right )\right )}{105 a^3 \left (a x+\sqrt {b^2+a^2 x^2}\right )^{7/2}} \]
(2*(-8*b^6*c^2 + 28*a^5*x^3*(15*d^2 + 10*c*d*x + 3*c^2*x^2)*(a*x + Sqrt[b^ 2 + a^2*x^2]) + 7*a*b^4*(4*c*(d - c*x)*Sqrt[b^2 + a^2*x^2] + a*(5*d^2 + 14 *c*d*x - 7*c^2*x^2)) + 7*a^3*b^2*x*(Sqrt[b^2 + a^2*x^2]*(25*d^2 + 30*c*d*x - 3*c^2*x^2) + a*x*(55*d^2 + 50*c*d*x + 3*c^2*x^2))))/(105*a^3*(a*x + Sqr t[b^2 + a^2*x^2])^(7/2))
Time = 0.47 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.78, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {2544, 2035, 2195, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {(c x+d)^2}{\sqrt {\sqrt {a^2 x^2+b^2}+a x}} \, dx\) |
\(\Big \downarrow \) 2544 |
\(\displaystyle \frac {\int \frac {\left (b^2+\left (a x+\sqrt {b^2+a^2 x^2}\right )^2\right ) \left (c b^2-c \left (a x+\sqrt {b^2+a^2 x^2}\right )^2-2 a d \left (a x+\sqrt {b^2+a^2 x^2}\right )\right )^2}{\left (a x+\sqrt {b^2+a^2 x^2}\right )^{9/2}}d\left (a x+\sqrt {b^2+a^2 x^2}\right )}{8 a^3}\) |
\(\Big \downarrow \) 2035 |
\(\displaystyle \frac {\int \frac {\left (b^2+\left (a x+\sqrt {b^2+a^2 x^2}\right )^2\right ) \left (c b^2-c \left (a x+\sqrt {b^2+a^2 x^2}\right )^2-2 a d \left (a x+\sqrt {b^2+a^2 x^2}\right )\right )^2}{\left (a x+\sqrt {b^2+a^2 x^2}\right )^4}d\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{4 a^3}\) |
\(\Big \downarrow \) 2195 |
\(\displaystyle \frac {\int \left (\frac {c^2 b^6}{\left (a x+\sqrt {b^2+a^2 x^2}\right )^4}-\frac {4 a c d b^4}{\left (a x+\sqrt {b^2+a^2 x^2}\right )^3}-c^2 \left (1-\frac {4 a^2 d^2}{b^2 c^2}\right ) b^2+c^2 \left (a x+\sqrt {b^2+a^2 x^2}\right )^2+4 a c d \left (a x+\sqrt {b^2+a^2 x^2}\right )+\frac {4 a^2 b^2 d^2-b^4 c^2}{\left (a x+\sqrt {b^2+a^2 x^2}\right )^2}\right )d\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{4 a^3}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {b^2 \left (b^2 c^2-4 a^2 d^2\right )}{3 \left (\sqrt {a^2 x^2+b^2}+a x\right )^{3/2}}-\sqrt {\sqrt {a^2 x^2+b^2}+a x} \left (b^2 c^2-4 a^2 d^2\right )+\frac {1}{5} c^2 \left (\sqrt {a^2 x^2+b^2}+a x\right )^{5/2}+\frac {4}{3} a c d \left (\sqrt {a^2 x^2+b^2}+a x\right )^{3/2}-\frac {b^6 c^2}{7 \left (\sqrt {a^2 x^2+b^2}+a x\right )^{7/2}}+\frac {4 a b^4 c d}{5 \left (\sqrt {a^2 x^2+b^2}+a x\right )^{5/2}}}{4 a^3}\) |
(-1/7*(b^6*c^2)/(a*x + Sqrt[b^2 + a^2*x^2])^(7/2) + (4*a*b^4*c*d)/(5*(a*x + Sqrt[b^2 + a^2*x^2])^(5/2)) + (b^2*(b^2*c^2 - 4*a^2*d^2))/(3*(a*x + Sqrt [b^2 + a^2*x^2])^(3/2)) - (b^2*c^2 - 4*a^2*d^2)*Sqrt[a*x + Sqrt[b^2 + a^2* x^2]] + (4*a*c*d*(a*x + Sqrt[b^2 + a^2*x^2])^(3/2))/3 + (c^2*(a*x + Sqrt[b ^2 + a^2*x^2])^(5/2))/5)/(4*a^3)
3.29.22.3.1 Defintions of rubi rules used
Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k Subst [Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti onQ[m] && AlgebraicFunctionQ[Fx, x]
Int[(Pq_)*((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_ Symbol] :> Int[ExpandIntegrand[(d*x)^m*Pq*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && PolyQ[Pq, x^2] && IGtQ[p, -2]
Int[((g_.) + (h_.)*(x_))^(m_.)*((e_.)*(x_) + (f_.)*Sqrt[(a_.) + (c_.)*(x_)^ 2])^(n_.), x_Symbol] :> Simp[1/(2^(m + 1)*e^(m + 1)) Subst[Int[x^(n - m - 2)*(a*f^2 + x^2)*((-a)*f^2*h + 2*e*g*x + h*x^2)^m, x], x, e*x + f*Sqrt[a + c*x^2]], x] /; FreeQ[{a, c, e, f, g, h, n}, x] && EqQ[e^2 - c*f^2, 0] && I ntegerQ[m]
\[\int \frac {\left (c x +d \right )^{2}}{\sqrt {a x +\sqrt {a^{2} x^{2}+b^{2}}}}d x\]
Time = 0.32 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.59 \[ \int \frac {(d+c x)^2}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=-\frac {2 \, {\left (15 \, a^{4} c^{2} x^{4} + 42 \, a^{4} c d x^{3} + 14 \, a^{2} b^{2} c d x + 8 \, b^{4} c^{2} - 35 \, a^{2} b^{2} d^{2} + {\left (a^{2} b^{2} c^{2} + 35 \, a^{4} d^{2}\right )} x^{2} - {\left (15 \, a^{3} c^{2} x^{3} + 42 \, a^{3} c d x^{2} + 28 \, a b^{2} c d + {\left (4 \, a b^{2} c^{2} + 35 \, a^{3} d^{2}\right )} x\right )} \sqrt {a^{2} x^{2} + b^{2}}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}{105 \, a^{3} b^{2}} \]
-2/105*(15*a^4*c^2*x^4 + 42*a^4*c*d*x^3 + 14*a^2*b^2*c*d*x + 8*b^4*c^2 - 3 5*a^2*b^2*d^2 + (a^2*b^2*c^2 + 35*a^4*d^2)*x^2 - (15*a^3*c^2*x^3 + 42*a^3* c*d*x^2 + 28*a*b^2*c*d + (4*a*b^2*c^2 + 35*a^3*d^2)*x)*sqrt(a^2*x^2 + b^2) )*sqrt(a*x + sqrt(a^2*x^2 + b^2))/(a^3*b^2)
\[ \int \frac {(d+c x)^2}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\int \frac {\left (c x + d\right )^{2}}{\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}\, dx \]
\[ \int \frac {(d+c x)^2}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\int { \frac {{\left (c x + d\right )}^{2}}{\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}} \,d x } \]
\[ \int \frac {(d+c x)^2}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\int { \frac {{\left (c x + d\right )}^{2}}{\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}} \,d x } \]
Timed out. \[ \int \frac {(d+c x)^2}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\int \frac {{\left (d+c\,x\right )}^2}{\sqrt {a\,x+\sqrt {a^2\,x^2+b^2}}} \,d x \]