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

Optimal. Leaf size=108 \[ \frac{8 (2 b d-a e) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{b^{3/2} \sqrt{e}}+\frac{6 d^2 \sqrt{a+b x}}{\sqrt{d+e x} (b d-a e)}+\frac{8 \sqrt{a+b x} \sqrt{d+e x}}{b} \]

[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])

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Rubi [A]  time = 0.119984, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.132, Rules used = {949, 80, 63, 217, 206} \[ \frac{8 (2 b d-a e) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{b^{3/2} \sqrt{e}}+\frac{6 d^2 \sqrt{a+b x}}{\sqrt{d+e x} (b d-a e)}+\frac{8 \sqrt{a+b x} \sqrt{d+e x}}{b} \]

Antiderivative was successfully verified.

[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 949

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]

Rule 80

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 63

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 217

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 206

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

Rubi steps

\begin{align*} \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{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)) \operatorname{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)) \operatorname{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]  time = 0.357536, size = 134, normalized size = 1.24 \[ \frac{2 \left (\frac{b \sqrt{a+b x} (b d (7 d+4 e x)-4 a e (d+e x))}{b d-a e}+\frac{4 \sqrt{b d-a e} (2 b d-a e) \sqrt{\frac{b (d+e x)}{b d-a e}} \sinh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b d-a e}}\right )}{\sqrt{e}}\right )}{b^2 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[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*((b*Sqrt[a + b*x]*(-4*a*e*(d + e*x) + b*d*(7*d + 4*e*x)))/(b*d - a*e) + (4*Sqrt[b*d - a*e]*(2*b*d - a*e)*Sq
rt[(b*(d + e*x))/(b*d - a*e)]*ArcSinh[(Sqrt[e]*Sqrt[a + b*x])/Sqrt[b*d - a*e]])/Sqrt[e]))/(b^2*Sqrt[d + e*x])

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Maple [B]  time = 0.388, size = 438, normalized size = 4.1 \begin{align*} -2\,{\frac{\sqrt{bx+a}}{b\sqrt{be} \left ( ae-bd \right ) \sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{ex+d}} \left ( 2\,\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ) x{a}^{2}{e}^{3}-6\,\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ) xabd{e}^{2}+4\,\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ) x{b}^{2}{d}^{2}e+2\,\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ){a}^{2}d{e}^{2}-6\,\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ) ab{d}^{2}e+4\,\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ){b}^{2}{d}^{3}-4\,xa{e}^{2}\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+4\,xbde\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}-4\,ade\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+7\,b{d}^{2}\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [B]  time = 2.35676, size = 1000, normalized size = 9.26 \begin{align*} \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 ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [B]  time = 1.21235, size = 261, normalized size = 2.42 \begin{align*} -\frac{8 \,{\left (2 \, b d - a e\right )} e^{\left (-\frac{1}{2}\right )} \log \left ({\left | -\sqrt{b x + a} \sqrt{b} e^{\frac{1}{2}} + \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e} \right |}\right )}{\sqrt{b}{\left | b \right |}} + \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{\left | b \right |} e^{2} - a b^{2}{\left | b \right |} e^{3}} + \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{\left | b \right |} e^{2} - a b^{2}{\left | b \right |} e^{3}}\right )}}{\sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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]

-8*(2*b*d - a*e)*e^(-1/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/(sqrt
(b)*abs(b)) + 2*sqrt(b*x + a)*(4*(b^3*d*e^3 - a*b^2*e^4)*(b*x + a)/(b^3*d*abs(b)*e^2 - a*b^2*abs(b)*e^3) + (7*
b^4*d^2*e^2 - 8*a*b^3*d*e^3 + 4*a^2*b^2*e^4)/(b^3*d*abs(b)*e^2 - a*b^2*abs(b)*e^3))/sqrt(b^2*d + (b*x + a)*b*e
- a*b*e)