\(\int \frac {15 d^2+20 d e x+8 e^2 x^2}{\sqrt {a+b x} (d+e x)^{3/2}} \, dx\) [846]

Optimal result

Integrand size = 38, antiderivative size = 108 \[ \int \frac {15 d^2+20 d e x+8 e^2 x^2}{\sqrt {a+b x} (d+e x)^{3/2}} \, dx=\frac {6 d^2 \sqrt {a+b x}}{(b d-a e) \sqrt {d+e x}}+\frac {8 \sqrt {a+b x} \sqrt {d+e x}}{b}+\frac {8 (2 b d-a e) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{b^{3/2} \sqrt {e}} \]

[Out]

8*(-a*e+2*b*d)*arctanh(e^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(e*x+d)^(1/2))/b^(3/2)/e^(1/2)+6*d^2*(b*x+a)^(1/2)/(-a*e+
b*d)/(e*x+d)^(1/2)+8*(b*x+a)^(1/2)*(e*x+d)^(1/2)/b

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.132, Rules used = {963, 81, 65, 223, 212} \[ \int \frac {15 d^2+20 d e x+8 e^2 x^2}{\sqrt {a+b x} (d+e x)^{3/2}} \, dx=\frac {8 (2 b d-a e) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{b^{3/2} \sqrt {e}}+\frac {6 d^2 \sqrt {a+b x}}{\sqrt {d+e x} (b d-a e)}+\frac {8 \sqrt {a+b x} \sqrt {d+e x}}{b} \]

[In]

Int[(15*d^2 + 20*d*e*x + 8*e^2*x^2)/(Sqrt[a + b*x]*(d + e*x)^(3/2)),x]

[Out]

(6*d^2*Sqrt[a + b*x])/((b*d - a*e)*Sqrt[d + e*x]) + (8*Sqrt[a + b*x]*Sqrt[d + e*x])/b + (8*(2*b*d - a*e)*ArcTa
nh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(b^(3/2)*Sqrt[e])

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 81

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(c + d*x)^
(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 963

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> With[{Qx = PolynomialQuotient[(a + b*x + c*x^2)^p, d + e*x, x], R = PolynomialRemainder[(a + b*x + c*x^2)^p,
d + e*x, x]}, Simp[R*(d + e*x)^(m + 1)*((f + g*x)^(n + 1)/((m + 1)*(e*f - d*g))), x] + Dist[1/((m + 1)*(e*f -
d*g)), Int[(d + e*x)^(m + 1)*(f + g*x)^n*ExpandToSum[(m + 1)*(e*f - d*g)*Qx - g*R*(m + n + 2), x], x], x]] /;
FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&& IGtQ[p, 0] && LtQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {6 d^2 \sqrt {a+b x}}{(b d-a e) \sqrt {d+e x}}+\frac {2 \int \frac {6 d (b d-a e)+4 e (b d-a e) x}{\sqrt {a+b x} \sqrt {d+e x}} \, dx}{b d-a e} \\ & = \frac {6 d^2 \sqrt {a+b x}}{(b d-a e) \sqrt {d+e x}}+\frac {8 \sqrt {a+b x} \sqrt {d+e x}}{b}+\frac {(4 (2 b d-a e)) \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}} \, dx}{b} \\ & = \frac {6 d^2 \sqrt {a+b x}}{(b d-a e) \sqrt {d+e x}}+\frac {8 \sqrt {a+b x} \sqrt {d+e x}}{b}+\frac {(8 (2 b d-a e)) \text {Subst}\left (\int \frac {1}{\sqrt {d-\frac {a e}{b}+\frac {e x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{b^2} \\ & = \frac {6 d^2 \sqrt {a+b x}}{(b d-a e) \sqrt {d+e x}}+\frac {8 \sqrt {a+b x} \sqrt {d+e x}}{b}+\frac {(8 (2 b d-a e)) \text {Subst}\left (\int \frac {1}{1-\frac {e x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {d+e x}}\right )}{b^2} \\ & = \frac {6 d^2 \sqrt {a+b x}}{(b d-a e) \sqrt {d+e x}}+\frac {8 \sqrt {a+b x} \sqrt {d+e x}}{b}+\frac {8 (2 b d-a e) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{b^{3/2} \sqrt {e}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.13 \[ \int \frac {15 d^2+20 d e x+8 e^2 x^2}{\sqrt {a+b x} (d+e x)^{3/2}} \, dx=\frac {2 \left (\frac {\sqrt {b} \sqrt {a+b x} (-4 a e (d+e x)+b d (7 d+4 e x))}{\sqrt {d+e x}}+\frac {4 \left (2 b^2 d^2-3 a b d e+a^2 e^2\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{\sqrt {e}}\right )}{b^{3/2} (b d-a e)} \]

[In]

Integrate[(15*d^2 + 20*d*e*x + 8*e^2*x^2)/(Sqrt[a + b*x]*(d + e*x)^(3/2)),x]

[Out]

(2*((Sqrt[b]*Sqrt[a + b*x]*(-4*a*e*(d + e*x) + b*d*(7*d + 4*e*x)))/Sqrt[d + e*x] + (4*(2*b^2*d^2 - 3*a*b*d*e +
a^2*e^2)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/Sqrt[e]))/(b^(3/2)*(b*d - a*e))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(437\) vs. \(2(88)=176\).

Time = 0.48 (sec) , antiderivative size = 438, normalized size of antiderivative = 4.06

method	result	size
default	\(-\frac {2 \sqrt {b x +a}\, \left (2 \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a^{2} e^{3} x -6 \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a b d \,e^{2} x +4 \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) b^{2} d^{2} e x +2 \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a^{2} d \,e^{2}-6 \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a b \,d^{2} e +4 \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) b^{2} d^{3}-4 a \,e^{2} x \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+4 b d e x \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}-4 a d e \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+7 b \,d^{2} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\right )}{b \sqrt {b e}\, \left (a e -b d \right ) \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {e x +d}}\)	\(438\)

[In]

int((8*e^2*x^2+20*d*e*x+15*d^2)/(e*x+d)^(3/2)/(b*x+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-2*(b*x+a)^(1/2)*(2*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^2*e^3*x-6*ln
(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a*b*d*e^2*x+4*ln(1/2*(2*b*e*x+2*((b*
x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*b^2*d^2*e*x+2*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*
(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^2*d*e^2-6*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(
b*e)^(1/2))*a*b*d^2*e+4*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*b^2*d^3-4*
a*e^2*x*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+4*b*d*e*x*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-4*a*d*e*((b*x+a)*(e*
x+d))^(1/2)*(b*e)^(1/2)+7*b*d^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2))/b/(b*e)^(1/2)/(a*e-b*d)/((b*x+a)*(e*x+d))
^(1/2)/(e*x+d)^(1/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 225 vs. \(2 (88) = 176\).

Time = 0.39 (sec) , antiderivative size = 463, normalized size of antiderivative = 4.29 \[ \int \frac {15 d^2+20 d e x+8 e^2 x^2}{\sqrt {a+b x} (d+e x)^{3/2}} \, dx=\left [-\frac {2 \, {\left ({\left (2 \, b^{2} d^{3} - 3 \, a b d^{2} e + a^{2} d e^{2} + {\left (2 \, b^{2} d^{2} e - 3 \, a b d e^{2} + a^{2} e^{3}\right )} x\right )} \sqrt {b e} \log \left (8 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 6 \, a b d e + a^{2} e^{2} - 4 \, {\left (2 \, b e x + b d + a e\right )} \sqrt {b e} \sqrt {b x + a} \sqrt {e x + d} + 8 \, {\left (b^{2} d e + a b e^{2}\right )} x\right ) - {\left (7 \, b^{2} d^{2} e - 4 \, a b d e^{2} + 4 \, {\left (b^{2} d e^{2} - a b e^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}\right )}}{b^{3} d^{2} e - a b^{2} d e^{2} + {\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x}, -\frac {2 \, {\left (2 \, {\left (2 \, b^{2} d^{3} - 3 \, a b d^{2} e + a^{2} d e^{2} + {\left (2 \, b^{2} d^{2} e - 3 \, a b d e^{2} + a^{2} e^{3}\right )} x\right )} \sqrt {-b e} \arctan \left (\frac {{\left (2 \, b e x + b d + a e\right )} \sqrt {-b e} \sqrt {b x + a} \sqrt {e x + d}}{2 \, {\left (b^{2} e^{2} x^{2} + a b d e + {\left (b^{2} d e + a b e^{2}\right )} x\right )}}\right ) - {\left (7 \, b^{2} d^{2} e - 4 \, a b d e^{2} + 4 \, {\left (b^{2} d e^{2} - a b e^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}\right )}}{b^{3} d^{2} e - a b^{2} d e^{2} + {\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x}\right ] \]

[In]

integrate((8*e^2*x^2+20*d*e*x+15*d^2)/(e*x+d)^(3/2)/(b*x+a)^(1/2),x, algorithm="fricas")

[Out]

[-2*((2*b^2*d^3 - 3*a*b*d^2*e + a^2*d*e^2 + (2*b^2*d^2*e - 3*a*b*d*e^2 + a^2*e^3)*x)*sqrt(b*e)*log(8*b^2*e^2*x
^2 + b^2*d^2 + 6*a*b*d*e + a^2*e^2 - 4*(2*b*e*x + b*d + a*e)*sqrt(b*e)*sqrt(b*x + a)*sqrt(e*x + d) + 8*(b^2*d*
e + a*b*e^2)*x) - (7*b^2*d^2*e - 4*a*b*d*e^2 + 4*(b^2*d*e^2 - a*b*e^3)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(b^3*d^
2*e - a*b^2*d*e^2 + (b^3*d*e^2 - a*b^2*e^3)*x), -2*(2*(2*b^2*d^3 - 3*a*b*d^2*e + a^2*d*e^2 + (2*b^2*d^2*e - 3*
a*b*d*e^2 + a^2*e^3)*x)*sqrt(-b*e)*arctan(1/2*(2*b*e*x + b*d + a*e)*sqrt(-b*e)*sqrt(b*x + a)*sqrt(e*x + d)/(b^
2*e^2*x^2 + a*b*d*e + (b^2*d*e + a*b*e^2)*x)) - (7*b^2*d^2*e - 4*a*b*d*e^2 + 4*(b^2*d*e^2 - a*b*e^3)*x)*sqrt(b
*x + a)*sqrt(e*x + d))/(b^3*d^2*e - a*b^2*d*e^2 + (b^3*d*e^2 - a*b^2*e^3)*x)]

Sympy [F]

\[ \int \frac {15 d^2+20 d e x+8 e^2 x^2}{\sqrt {a+b x} (d+e x)^{3/2}} \, dx=\int \frac {15 d^{2} + 20 d e x + 8 e^{2} x^{2}}{\sqrt {a + b x} \left (d + e x\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate((8*e**2*x**2+20*d*e*x+15*d**2)/(e*x+d)**(3/2)/(b*x+a)**(1/2),x)

[Out]

Integral((15*d**2 + 20*d*e*x + 8*e**2*x**2)/(sqrt(a + b*x)*(d + e*x)**(3/2)), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {15 d^2+20 d e x+8 e^2 x^2}{\sqrt {a+b x} (d+e x)^{3/2}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((8*e^2*x^2+20*d*e*x+15*d^2)/(e*x+d)^(3/2)/(b*x+a)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 197 vs. \(2 (88) = 176\).

Time = 0.32 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.82 \[ \int \frac {15 d^2+20 d e x+8 e^2 x^2}{\sqrt {a+b x} (d+e x)^{3/2}} \, dx=\frac {2 \, \sqrt {b x + a} {\left (\frac {4 \, {\left (b^{3} d e^{3} - a b^{2} e^{4}\right )} {\left (b x + a\right )}}{b^{3} d e^{2} {\left | b \right |} - a b^{2} e^{3} {\left | b \right |}} + \frac {7 \, b^{4} d^{2} e^{2} - 8 \, a b^{3} d e^{3} + 4 \, a^{2} b^{2} e^{4}}{b^{3} d e^{2} {\left | b \right |} - a b^{2} e^{3} {\left | b \right |}}\right )}}{\sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}} - \frac {8 \, {\left (2 \, b d - a e\right )} \log \left ({\left | -\sqrt {b e} \sqrt {b x + a} + \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \right |}\right )}{\sqrt {b e} {\left | b \right |}} \]

[In]

integrate((8*e^2*x^2+20*d*e*x+15*d^2)/(e*x+d)^(3/2)/(b*x+a)^(1/2),x, algorithm="giac")

[Out]

2*sqrt(b*x + a)*(4*(b^3*d*e^3 - a*b^2*e^4)*(b*x + a)/(b^3*d*e^2*abs(b) - a*b^2*e^3*abs(b)) + (7*b^4*d^2*e^2 -
8*a*b^3*d*e^3 + 4*a^2*b^2*e^4)/(b^3*d*e^2*abs(b) - a*b^2*e^3*abs(b)))/sqrt(b^2*d + (b*x + a)*b*e - a*b*e) - 8*
(2*b*d - a*e)*log(abs(-sqrt(b*e)*sqrt(b*x + a) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/(sqrt(b*e)*abs(b))

Mupad [F(-1)]

Timed out. \[ \int \frac {15 d^2+20 d e x+8 e^2 x^2}{\sqrt {a+b x} (d+e x)^{3/2}} \, dx=\int \frac {15\,d^2+20\,d\,e\,x+8\,e^2\,x^2}{\sqrt {a+b\,x}\,{\left (d+e\,x\right )}^{3/2}} \,d x \]

[In]

int((15*d^2 + 8*e^2*x^2 + 20*d*e*x)/((a + b*x)^(1/2)*(d + e*x)^(3/2)),x)

[Out]

int((15*d^2 + 8*e^2*x^2 + 20*d*e*x)/((a + b*x)^(1/2)*(d + e*x)^(3/2)), x)