3.29.22 \(\int \frac {(d+c x)^2}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx\)

Optimal. Leaf size=282 \[ \frac {2 \sqrt {a^2 x^2+b^2} \left (12 a^4 c^2 x^5+40 a^4 c d x^4+60 a^4 d^2 x^3-3 a^2 b^2 c^2 x^3+30 a^2 b^2 c d x^2+25 a^2 b^2 d^2 x-4 b^4 c^2 x+4 b^4 c d\right )}{15 a^2 \left (\sqrt {a^2 x^2+b^2}+a x\right )^{7/2}}+\frac {2 \left (84 a^6 c^2 x^6+280 a^6 c d x^5+420 a^6 d^2 x^4+21 a^4 b^2 c^2 x^4+350 a^4 b^2 c d x^3+385 a^4 b^2 d^2 x^2-49 a^2 b^4 c^2 x^2+98 a^2 b^4 c d x+35 a^2 b^4 d^2-8 b^6 c^2\right )}{105 a^3 \left (\sqrt {a^2 x^2+b^2}+a x\right )^{7/2}} \]

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Rubi [A]  time = 0.23, antiderivative size = 232, normalized size of antiderivative = 0.82, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {2119, 1628} \begin {gather*} \frac {c d \left (\sqrt {a^2 x^2+b^2}+a x\right )^{3/2}}{3 a^2}+\frac {b^4 c d}{5 a^2 \left (\sqrt {a^2 x^2+b^2}+a x\right )^{5/2}}+\frac {b^2 \left (b^2 c^2-4 a^2 d^2\right )}{12 a^3 \left (\sqrt {a^2 x^2+b^2}+a x\right )^{3/2}}-\frac {\sqrt {\sqrt {a^2 x^2+b^2}+a x} \left (b^2 c^2-4 a^2 d^2\right )}{4 a^3}+\frac {c^2 \left (\sqrt {a^2 x^2+b^2}+a x\right )^{5/2}}{20 a^3}-\frac {b^6 c^2}{28 a^3 \left (\sqrt {a^2 x^2+b^2}+a x\right )^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + c*x)^2/Sqrt[a*x + Sqrt[b^2 + a^2*x^2]],x]

[Out]

-1/28*(b^6*c^2)/(a^3*(a*x + Sqrt[b^2 + a^2*x^2])^(7/2)) + (b^4*c*d)/(5*a^2*(a*x + Sqrt[b^2 + a^2*x^2])^(5/2))
+ (b^2*(b^2*c^2 - 4*a^2*d^2))/(12*a^3*(a*x + Sqrt[b^2 + a^2*x^2])^(3/2)) - ((b^2*c^2 - 4*a^2*d^2)*Sqrt[a*x + S
qrt[b^2 + a^2*x^2]])/(4*a^3) + (c*d*(a*x + Sqrt[b^2 + a^2*x^2])^(3/2))/(3*a^2) + (c^2*(a*x + Sqrt[b^2 + a^2*x^
2])^(5/2))/(20*a^3)

Rule 1628

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 2119

Int[((g_.) + (h_.)*(x_))^(m_.)*((e_.)*(x_) + (f_.)*Sqrt[(a_.) + (c_.)*(x_)^2])^(n_.), x_Symbol] :> Dist[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] && IntegerQ[m]

Rubi steps

\begin {align*} \int \frac {(d+c x)^2}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (b^2+x^2\right ) \left (-b^2 c+2 a d x+c x^2\right )^2}{x^{9/2}} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{8 a^3}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {b^6 c^2}{x^{9/2}}-\frac {4 a b^4 c d}{x^{7/2}}-\frac {b^2 \left (b^2 c^2-4 a^2 d^2\right )}{x^{5/2}}+\frac {-b^2 c^2+4 a^2 d^2}{\sqrt {x}}+4 a c d \sqrt {x}+c^2 x^{3/2}\right ) \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{8 a^3}\\ &=-\frac {b^6 c^2}{28 a^3 \left (a x+\sqrt {b^2+a^2 x^2}\right )^{7/2}}+\frac {b^4 c d}{5 a^2 \left (a x+\sqrt {b^2+a^2 x^2}\right )^{5/2}}+\frac {b^2 \left (b^2 c^2-4 a^2 d^2\right )}{12 a^3 \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}-\frac {\left (b^2 c^2-4 a^2 d^2\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{4 a^3}+\frac {c d \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{3 a^2}+\frac {c^2 \left (a x+\sqrt {b^2+a^2 x^2}\right )^{5/2}}{20 a^3}\\ \end {align*}

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Mathematica [B]  time = 7.65, size = 1949, normalized size = 6.91

result too large to display

Antiderivative was successfully verified.

[In]

Integrate[(d + c*x)^2/Sqrt[a*x + Sqrt[b^2 + a^2*x^2]],x]

[Out]

(2*d^2*Sqrt[b^2 + a^2*x^2]*(2*a^2*x*Sqrt[b^2 + a^2*x^2] + a*(b^2 + 2*a^2*x^2))*(6*a^2*x*Sqrt[b^2 + a^2*x^2]*(3
*b^2 + a^2*x^2) + a*(7*b^4 + 21*a^2*b^2*x^2 + 6*a^4*x^4))*(Sqrt[a^2]*b^2 + 2*Sqrt[a^2]*x*(a^2*x + a*Sqrt[b^2 +
a^2*x^2]))^3)/(15*a^4*Sqrt[a^2]*b^2*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]*(a^2*x*Sqrt[b^2 + a^2*x^2]*(9*b^8 + 120*a
^2*b^6*x^2 + 432*a^4*b^4*x^4 + 576*a^6*b^2*x^6 + 256*a^8*x^8) + a*(b^10 + 41*a^2*b^8*x^2 + 280*a^4*b^6*x^4 + 6
88*a^6*b^4*x^6 + 704*a^8*b^2*x^8 + 256*a^10*x^10))) + (4*c*d*Sqrt[b^2 + a^2*x^2]*(2*a^2*x*Sqrt[b^2 + a^2*x^2]
+ a*(b^2 + 2*a^2*x^2))*(5*a^2*x*Sqrt[b^2 + a^2*x^2]*(11*b^4 + 19*a^2*b^2*x^2 + 12*a^4*x^4) + a*(22*b^6 + 95*a^
2*b^4*x^2 + 125*a^4*b^2*x^4 + 60*a^6*x^6))*(Sqrt[a^2]*b^2 + 2*Sqrt[a^2]*x*(a^2*x + a*Sqrt[b^2 + a^2*x^2]))^3)/
(105*a^6*b^2*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]*(Sqrt[a^2]*Sqrt[b^2 + a^2*x^2]*(b^10 + 50*a^2*b^8*x^2 + 400*a^4*b
^6*x^4 + 1120*a^6*b^4*x^6 + 1280*a^8*b^2*x^8 + 512*a^10*x^10) + 2*a*(5*Sqrt[a^2]*b^10*x + 85*(a^2)^(3/2)*b^8*x
^3 + 416*(a^2)^(5/2)*b^6*x^5 + 848*(a^2)^(7/2)*b^4*x^7 + 768*(a^2)^(9/2)*b^2*x^9 + 256*(a^2)^(11/2)*x^11))) +
(2*c^2*Sqrt[b^2 + a^2*x^2]*(2*a^2*x*Sqrt[b^2 + a^2*x^2] + a*(b^2 + 2*a^2*x^2))*(28*a^2*x*Sqrt[b^2 + a^2*x^2]*(
-b^6 - a^2*b^4*x^2 + 3*a^4*b^2*x^4 + 2*a^6*x^6) + a*(-8*b^8 - 49*a^2*b^6*x^2 + 7*a^4*b^4*x^4 + 112*a^6*b^2*x^6
+ 56*a^8*x^8))*(Sqrt[a^2]*b^2 + 2*Sqrt[a^2]*x*(a^2*x + a*Sqrt[b^2 + a^2*x^2]))^3)/(63*a^6*Sqrt[a^2]*b^2*Sqrt[
a*x + Sqrt[b^2 + a^2*x^2]]*(a^2*x*Sqrt[b^2 + a^2*x^2]*(11*b^10 + 220*a^2*b^8*x^2 + 1232*a^4*b^6*x^4 + 2816*a^6
*b^4*x^6 + 2816*a^8*b^2*x^8 + 1024*a^10*x^10) + a*(b^12 + 61*a^2*b^10*x^2 + 620*a^4*b^8*x^4 + 2352*a^6*b^6*x^6
+ 4096*a^8*b^4*x^8 + 3328*a^10*b^2*x^10 + 1024*a^12*x^12))) - (Sqrt[a^2]*d^2*x*(a*x + Sqrt[b^2 + a^2*x^2])*((
2*b^4)/(3*(a*x + Sqrt[b^2 + a^2*x^2])^(3/2)) + (2*(a*x + Sqrt[b^2 + a^2*x^2])^(5/2))/5))/(2*b^2*(a/Sqrt[a^2] +
(Sqrt[a^2]*x)/Sqrt[b^2 + a^2*x^2])*(-(Sqrt[a^2]*b^2) + Sqrt[a^2]*(a*x + Sqrt[b^2 + a^2*x^2])^2)*(Sqrt[a^2]/a
- (-(Sqrt[a^2]*b^2) + Sqrt[a^2]*(a*x + Sqrt[b^2 + a^2*x^2])^2)/(2*a*(a*x + Sqrt[b^2 + a^2*x^2])^2))) - (2*a^2*
c*d*x^2*(-21*b^6 + 105*b^4*(a*x + Sqrt[b^2 + a^2*x^2])^2 - 35*b^2*(a*x + Sqrt[b^2 + a^2*x^2])^4 + 15*(a*x + Sq
rt[b^2 + a^2*x^2])^6))/(105*b^2*(a/Sqrt[a^2] + (Sqrt[a^2]*x)/Sqrt[b^2 + a^2*x^2])*Sqrt[a*x + Sqrt[b^2 + a^2*x^
2]]*(-(Sqrt[a^2]*b^2) + Sqrt[a^2]*(a*x + Sqrt[b^2 + a^2*x^2])^2)^2*(Sqrt[a^2]/a - (-(Sqrt[a^2]*b^2) + Sqrt[a^2
]*(a*x + Sqrt[b^2 + a^2*x^2])^2)/(2*a*(a*x + Sqrt[b^2 + a^2*x^2])^2))) - ((a^2)^(3/2)*c^2*x^3*(45*b^8 - 210*b^
6*(a*x + Sqrt[b^2 + a^2*x^2])^2 - 126*b^2*(a*x + Sqrt[b^2 + a^2*x^2])^6 + 35*(a*x + Sqrt[b^2 + a^2*x^2])^8))/(
315*b^2*(a/Sqrt[a^2] + (Sqrt[a^2]*x)/Sqrt[b^2 + a^2*x^2])*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]*(-(Sqrt[a^2]*b^2) +
Sqrt[a^2]*(a*x + Sqrt[b^2 + a^2*x^2])^2)^3*(Sqrt[a^2]/a - (-(Sqrt[a^2]*b^2) + Sqrt[a^2]*(a*x + Sqrt[b^2 + a^2*
x^2])^2)/(2*a*(a*x + Sqrt[b^2 + a^2*x^2])^2)))

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IntegrateAlgebraic [A]  time = 0.33, size = 282, normalized size = 1.00 \begin {gather*} \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}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(d + c*x)^2/Sqrt[a*x + Sqrt[b^2 + a^2*x^2]],x]

[Out]

(2*Sqrt[b^2 + a^2*x^2]*(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))/(15*a^2*(a*x + Sqrt[b^2 + a^2*x^2])^(7/2)) + (2*(-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))/(105*a^3*(a*x + Sqrt[b^2 + a^2*x^2])^(7/2)
)

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fricas [A]  time = 1.13, size = 167, normalized size = 0.59 \begin {gather*} -\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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x+d)^2/(a*x+(a^2*x^2+b^2)^(1/2))^(1/2),x, algorithm="fricas")

[Out]

-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 - 35*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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (c x + d\right )}^{2}}{\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x+d)^2/(a*x+(a^2*x^2+b^2)^(1/2))^(1/2),x, algorithm="giac")

[Out]

integrate((c*x + d)^2/sqrt(a*x + sqrt(a^2*x^2 + b^2)), x)

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maple [F]  time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (c x +d \right )^{2}}{\sqrt {a x +\sqrt {a^{2} x^{2}+b^{2}}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x+d)^2/(a*x+(a^2*x^2+b^2)^(1/2))^(1/2),x)

[Out]

int((c*x+d)^2/(a*x+(a^2*x^2+b^2)^(1/2))^(1/2),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (c x + d\right )}^{2}}{\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x+d)^2/(a*x+(a^2*x^2+b^2)^(1/2))^(1/2),x, algorithm="maxima")

[Out]

integrate((c*x + d)^2/sqrt(a*x + sqrt(a^2*x^2 + b^2)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (d+c\,x\right )}^2}{\sqrt {a\,x+\sqrt {a^2\,x^2+b^2}}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + c*x)^2/(a*x + (b^2 + a^2*x^2)^(1/2))^(1/2),x)

[Out]

int((d + c*x)^2/(a*x + (b^2 + a^2*x^2)^(1/2))^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c x + d\right )^{2}}{\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x+d)**2/(a*x+(a**2*x**2+b**2)**(1/2))**(1/2),x)

[Out]

Integral((c*x + d)**2/sqrt(a*x + sqrt(a**2*x**2 + b**2)), x)

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