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

Optimal. Leaf size=128 \[ \frac {c \left (\sqrt {a^2 x^2+b^2}+a x\right )^{3/2}}{6 a^2}-\frac {b^2 d}{3 a \left (\sqrt {a^2 x^2+b^2}+a x\right )^{3/2}}+\frac {d \sqrt {\sqrt {a^2 x^2+b^2}+a x}}{a}+\frac {b^4 c}{10 a^2 \left (\sqrt {a^2 x^2+b^2}+a x\right )^{5/2}} \]

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Rubi [A]  time = 0.11, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2119, 1628} \begin {gather*} \frac {c \left (\sqrt {a^2 x^2+b^2}+a x\right )^{3/2}}{6 a^2}-\frac {b^2 d}{3 a \left (\sqrt {a^2 x^2+b^2}+a x\right )^{3/2}}+\frac {d \sqrt {\sqrt {a^2 x^2+b^2}+a x}}{a}+\frac {b^4 c}{10 a^2 \left (\sqrt {a^2 x^2+b^2}+a x\right )^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b^4*c)/(10*a^2*(a*x + Sqrt[b^2 + a^2*x^2])^(5/2)) - (b^2*d)/(3*a*(a*x + Sqrt[b^2 + a^2*x^2])^(3/2)) + (d*Sqrt
[a*x + Sqrt[b^2 + a^2*x^2]])/a + (c*(a*x + Sqrt[b^2 + a^2*x^2])^(3/2))/(6*a^2)

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}{\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 )}{x^{7/2}} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{4 a^2}\\ &=\frac {\operatorname {Subst}\left (\int \left (-\frac {b^4 c}{x^{7/2}}+\frac {2 a b^2 d}{x^{5/2}}+\frac {2 a d}{\sqrt {x}}+c \sqrt {x}\right ) \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{4 a^2}\\ &=\frac {b^4 c}{10 a^2 \left (a x+\sqrt {b^2+a^2 x^2}\right )^{5/2}}-\frac {b^2 d}{3 a \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}+\frac {d \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{a}+\frac {c \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{6 a^2}\\ \end {align*}

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Mathematica [B]  time = 5.09, size = 1053, normalized size = 8.23 \begin {gather*} \frac {2 \left (b^2+a^2 x^2\right )^{3/2} \left (2 c b^{26}+a \left (\sqrt {b^2+a^2 x^2} (5 d+49 c x)+2 a x (65 d+302 c x)\right ) b^{24}+40 a^3 x^2 \left (2 \sqrt {b^2+a^2 x^2} (21 d+62 c x)+a x (361 d+773 c x)\right ) b^{22}+2288 a^5 x^4 \left (2 \sqrt {b^2+a^2 x^2} (20 d+33 c x)+a x (205 d+277 c x)\right ) b^{20}+9152 a^7 x^6 \left (2 \sqrt {b^2+a^2 x^2} (105 d+118 c x)+a x (765 d+749 c x)\right ) b^{18}+18304 a^9 x^8 \left (5 \sqrt {b^2+a^2 x^2} (225 d+191 c x)+a x (3175 d+2441 c x)\right ) b^{16}+106496 a^{11} x^{10} \left (2 \sqrt {b^2+a^2 x^2} (605 d+414 c x)+a x (2785 d+1773 c x)\right ) b^{14}+1392640 a^{13} x^{12} \left (14 \sqrt {b^2+a^2 x^2} (26 d+15 c x)+a x (707 d+387 c x)\right ) b^{12}+3342336 a^{15} x^{14} \left (2 \sqrt {b^2+a^2 x^2} (195 d+98 c x)+a x (655 d+317 c x)\right ) b^{10}+1245184 a^{17} x^{16} \left (3 \sqrt {b^2+a^2 x^2} (595 d+267 c x)+2 a x (1320 d+577 c x)\right ) b^8+524288 a^{19} x^{18} \left (10 \sqrt {b^2+a^2 x^2} (475 d+194 c x)+a x (6275 d+2519 c x)\right ) b^6+1048576 a^{21} x^{20} \left (2 \sqrt {b^2+a^2 x^2} (840 d+317 c x)+a x (2005 d+749 c x)\right ) b^4+20971520 a^{23} x^{22} \left (2 \sqrt {b^2+a^2 x^2} (17 d+6 c x)+a x (37 d+13 c x)\right ) b^2+41943040 a^{25} x^{24} (3 d+c x) \left (a x+\sqrt {b^2+a^2 x^2}\right )\right )}{15 a^2 \left (a x+\sqrt {b^2+a^2 x^2}\right )^{9/2} \left (512 a^{11} x^{11}+512 a^{10} \sqrt {b^2+a^2 x^2} x^{10}+1536 a^9 b^2 x^9+1280 a^8 b^2 \sqrt {b^2+a^2 x^2} x^8+1696 a^7 b^4 x^7+1120 a^6 b^4 \sqrt {b^2+a^2 x^2} x^6+832 a^5 b^6 x^5+400 a^4 b^6 \sqrt {b^2+a^2 x^2} x^4+170 a^3 b^8 x^3+50 a^2 b^8 \sqrt {b^2+a^2 x^2} x^2+10 a b^{10} x+b^{10} \sqrt {b^2+a^2 x^2}\right ) \left (b^2+a x \left (a x+\sqrt {b^2+a^2 x^2}\right )\right ) \left (b^{10}+a x \left (41 a x+9 \sqrt {b^2+a^2 x^2}\right ) b^8+40 a^3 x^3 \left (7 a x+3 \sqrt {b^2+a^2 x^2}\right ) b^6+16 a^5 x^5 \left (43 a x+27 \sqrt {b^2+a^2 x^2}\right ) b^4+64 a^7 x^7 \left (11 a x+9 \sqrt {b^2+a^2 x^2}\right ) b^2+256 a^9 x^9 \left (a x+\sqrt {b^2+a^2 x^2}\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(b^2 + a^2*x^2)^(3/2)*(2*b^26*c + 41943040*a^25*x^24*(3*d + c*x)*(a*x + Sqrt[b^2 + a^2*x^2]) + 20971520*a^2
3*b^2*x^22*(a*x*(37*d + 13*c*x) + 2*(17*d + 6*c*x)*Sqrt[b^2 + a^2*x^2]) + 1392640*a^13*b^12*x^12*(a*x*(707*d +
387*c*x) + 14*(26*d + 15*c*x)*Sqrt[b^2 + a^2*x^2]) + 2288*a^5*b^20*x^4*(a*x*(205*d + 277*c*x) + 2*(20*d + 33*
c*x)*Sqrt[b^2 + a^2*x^2]) + a*b^24*(2*a*x*(65*d + 302*c*x) + (5*d + 49*c*x)*Sqrt[b^2 + a^2*x^2]) + 40*a^3*b^22
*x^2*(a*x*(361*d + 773*c*x) + 2*(21*d + 62*c*x)*Sqrt[b^2 + a^2*x^2]) + 3342336*a^15*b^10*x^14*(a*x*(655*d + 31
7*c*x) + 2*(195*d + 98*c*x)*Sqrt[b^2 + a^2*x^2]) + 9152*a^7*b^18*x^6*(a*x*(765*d + 749*c*x) + 2*(105*d + 118*c
*x)*Sqrt[b^2 + a^2*x^2]) + 18304*a^9*b^16*x^8*(a*x*(3175*d + 2441*c*x) + 5*(225*d + 191*c*x)*Sqrt[b^2 + a^2*x^
2]) + 524288*a^19*b^6*x^18*(a*x*(6275*d + 2519*c*x) + 10*(475*d + 194*c*x)*Sqrt[b^2 + a^2*x^2]) + 1245184*a^17
*b^8*x^16*(2*a*x*(1320*d + 577*c*x) + 3*(595*d + 267*c*x)*Sqrt[b^2 + a^2*x^2]) + 1048576*a^21*b^4*x^20*(a*x*(2
005*d + 749*c*x) + 2*(840*d + 317*c*x)*Sqrt[b^2 + a^2*x^2]) + 106496*a^11*b^14*x^10*(a*x*(2785*d + 1773*c*x) +
2*(605*d + 414*c*x)*Sqrt[b^2 + a^2*x^2])))/(15*a^2*(a*x + Sqrt[b^2 + a^2*x^2])^(9/2)*(10*a*b^10*x + 170*a^3*b
^8*x^3 + 832*a^5*b^6*x^5 + 1696*a^7*b^4*x^7 + 1536*a^9*b^2*x^9 + 512*a^11*x^11 + b^10*Sqrt[b^2 + a^2*x^2] + 50
*a^2*b^8*x^2*Sqrt[b^2 + a^2*x^2] + 400*a^4*b^6*x^4*Sqrt[b^2 + a^2*x^2] + 1120*a^6*b^4*x^6*Sqrt[b^2 + a^2*x^2]
+ 1280*a^8*b^2*x^8*Sqrt[b^2 + a^2*x^2] + 512*a^10*x^10*Sqrt[b^2 + a^2*x^2])*(b^2 + a*x*(a*x + Sqrt[b^2 + a^2*x
^2]))*(b^10 + 256*a^9*x^9*(a*x + Sqrt[b^2 + a^2*x^2]) + 40*a^3*b^6*x^3*(7*a*x + 3*Sqrt[b^2 + a^2*x^2]) + 64*a^
7*b^2*x^7*(11*a*x + 9*Sqrt[b^2 + a^2*x^2]) + a*b^8*x*(41*a*x + 9*Sqrt[b^2 + a^2*x^2]) + 16*a^5*b^4*x^5*(43*a*x
+ 27*Sqrt[b^2 + a^2*x^2])))

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IntegrateAlgebraic [A]  time = 0.20, size = 128, normalized size = 1.00 \begin {gather*} \frac {b^4 c}{10 a^2 \left (a x+\sqrt {b^2+a^2 x^2}\right )^{5/2}}-\frac {b^2 d}{3 a \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}+\frac {d \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{a}+\frac {c \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{6 a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b^4*c)/(10*a^2*(a*x + Sqrt[b^2 + a^2*x^2])^(5/2)) - (b^2*d)/(3*a*(a*x + Sqrt[b^2 + a^2*x^2])^(3/2)) + (d*Sqrt
[a*x + Sqrt[b^2 + a^2*x^2]])/a + (c*(a*x + Sqrt[b^2 + a^2*x^2])^(3/2))/(6*a^2)

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fricas [A]  time = 0.45, size = 98, normalized size = 0.77 \begin {gather*} -\frac {2 \, {\left (3 \, a^{3} c x^{3} + 5 \, a^{3} d x^{2} + a b^{2} c x - 5 \, a b^{2} d - {\left (3 \, a^{2} c x^{2} + 5 \, a^{2} d x + 2 \, b^{2} c\right )} \sqrt {a^{2} x^{2} + b^{2}}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}{15 \, a^{2} b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/15*(3*a^3*c*x^3 + 5*a^3*d*x^2 + a*b^2*c*x - 5*a*b^2*d - (3*a^2*c*x^2 + 5*a^2*d*x + 2*b^2*c)*sqrt(a^2*x^2 +
b^2))*sqrt(a*x + sqrt(a^2*x^2 + b^2))/(a^2*b^2)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {c x + d}{\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)/(a*x+(a^2*x^2+b^2)^(1/2))^(1/2),x, algorithm="giac")

[Out]

integrate((c*x + d)/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 \[\int \frac {c x +d}{\sqrt {a x +\sqrt {a^{2} x^{2}+b^{2}}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((c*x+d)/(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 {c x + d}{\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)/(a*x+(a^2*x^2+b^2)^(1/2))^(1/2),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {d+c\,x}{\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)/(a*x + (b^2 + a^2*x^2)^(1/2))^(1/2),x)

[Out]

int((d + c*x)/(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 {c x + d}{\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)/(a*x+(a**2*x**2+b**2)**(1/2))**(1/2),x)

[Out]

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

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