3.1213 \(\int \frac{(a+b x+c x^2)^{3/2}}{(b d+2 c d x)^7} \, dx\)

Optimal. Leaf size=165 \[ \frac{\sqrt{a+b x+c x^2}}{128 c^2 d^7 \left (b^2-4 a c\right ) (b+2 c x)^2}+\frac{\tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{256 c^{5/2} d^7 \left (b^2-4 a c\right )^{3/2}}-\frac{\sqrt{a+b x+c x^2}}{64 c^2 d^7 (b+2 c x)^4}-\frac{\left (a+b x+c x^2\right )^{3/2}}{12 c d^7 (b+2 c x)^6} \]

[Out]

-Sqrt[a + b*x + c*x^2]/(64*c^2*d^7*(b + 2*c*x)^4) + Sqrt[a + b*x + c*x^2]/(128*c^2*(b^2 - 4*a*c)*d^7*(b + 2*c*
x)^2) - (a + b*x + c*x^2)^(3/2)/(12*c*d^7*(b + 2*c*x)^6) + ArcTan[(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 -
4*a*c]]/(256*c^(5/2)*(b^2 - 4*a*c)^(3/2)*d^7)

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Rubi [A]  time = 0.113958, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {684, 693, 688, 205} \[ \frac{\sqrt{a+b x+c x^2}}{128 c^2 d^7 \left (b^2-4 a c\right ) (b+2 c x)^2}+\frac{\tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{256 c^{5/2} d^7 \left (b^2-4 a c\right )^{3/2}}-\frac{\sqrt{a+b x+c x^2}}{64 c^2 d^7 (b+2 c x)^4}-\frac{\left (a+b x+c x^2\right )^{3/2}}{12 c d^7 (b+2 c x)^6} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^7,x]

[Out]

-Sqrt[a + b*x + c*x^2]/(64*c^2*d^7*(b + 2*c*x)^4) + Sqrt[a + b*x + c*x^2]/(128*c^2*(b^2 - 4*a*c)*d^7*(b + 2*c*
x)^2) - (a + b*x + c*x^2)^(3/2)/(12*c*d^7*(b + 2*c*x)^6) + ArcTan[(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 -
4*a*c]]/(256*c^(5/2)*(b^2 - 4*a*c)^(3/2)*d^7)

Rule 684

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(
a + b*x + c*x^2)^p)/(e*(m + 1)), x] - Dist[(b*p)/(d*e*(m + 1)), Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1
), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] &&
GtQ[p, 0] && LtQ[m, -1] &&  !(IntegerQ[m/2] && LtQ[m + 2*p + 3, 0]) && IntegerQ[2*p]

Rule 693

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-2*b*d*(d + e*x)^(m
+ 1)*(a + b*x + c*x^2)^(p + 1))/(d^2*(m + 1)*(b^2 - 4*a*c)), x] + Dist[(b^2*(m + 2*p + 3))/(d^2*(m + 1)*(b^2
- 4*a*c)), Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*
c, 0] && EqQ[2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] && LtQ[m, -1] && (IntegerQ[2*p] || (IntegerQ[m] && Rationa
lQ[p]) || IntegerQ[(m + 2*p + 3)/2])

Rule 688

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[4*c, Subst[Int[1/(b^2*e
- 4*a*c*e + 4*c*e*x^2), x], x, Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0]
&& EqQ[2*c*d - b*e, 0]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^7} \, dx &=-\frac{\left (a+b x+c x^2\right )^{3/2}}{12 c d^7 (b+2 c x)^6}+\frac{\int \frac{\sqrt{a+b x+c x^2}}{(b d+2 c d x)^5} \, dx}{8 c d^2}\\ &=-\frac{\sqrt{a+b x+c x^2}}{64 c^2 d^7 (b+2 c x)^4}-\frac{\left (a+b x+c x^2\right )^{3/2}}{12 c d^7 (b+2 c x)^6}+\frac{\int \frac{1}{(b d+2 c d x)^3 \sqrt{a+b x+c x^2}} \, dx}{128 c^2 d^4}\\ &=-\frac{\sqrt{a+b x+c x^2}}{64 c^2 d^7 (b+2 c x)^4}+\frac{\sqrt{a+b x+c x^2}}{128 c^2 \left (b^2-4 a c\right ) d^7 (b+2 c x)^2}-\frac{\left (a+b x+c x^2\right )^{3/2}}{12 c d^7 (b+2 c x)^6}+\frac{\int \frac{1}{(b d+2 c d x) \sqrt{a+b x+c x^2}} \, dx}{256 c^2 \left (b^2-4 a c\right ) d^6}\\ &=-\frac{\sqrt{a+b x+c x^2}}{64 c^2 d^7 (b+2 c x)^4}+\frac{\sqrt{a+b x+c x^2}}{128 c^2 \left (b^2-4 a c\right ) d^7 (b+2 c x)^2}-\frac{\left (a+b x+c x^2\right )^{3/2}}{12 c d^7 (b+2 c x)^6}+\frac{\operatorname{Subst}\left (\int \frac{1}{2 b^2 c d-8 a c^2 d+8 c^2 d x^2} \, dx,x,\sqrt{a+b x+c x^2}\right )}{64 c \left (b^2-4 a c\right ) d^6}\\ &=-\frac{\sqrt{a+b x+c x^2}}{64 c^2 d^7 (b+2 c x)^4}+\frac{\sqrt{a+b x+c x^2}}{128 c^2 \left (b^2-4 a c\right ) d^7 (b+2 c x)^2}-\frac{\left (a+b x+c x^2\right )^{3/2}}{12 c d^7 (b+2 c x)^6}+\frac{\tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{256 c^{5/2} \left (b^2-4 a c\right )^{3/2} d^7}\\ \end{align*}

Mathematica [C]  time = 0.0319288, size = 62, normalized size = 0.38 \[ \frac{2 (a+x (b+c x))^{5/2} \, _2F_1\left (\frac{5}{2},4;\frac{7}{2};\frac{4 c (a+x (b+c x))}{4 a c-b^2}\right )}{5 d^7 \left (b^2-4 a c\right )^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^7,x]

[Out]

(2*(a + x*(b + c*x))^(5/2)*Hypergeometric2F1[5/2, 4, 7/2, (4*c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])/(5*(b^2 - 4
*a*c)^4*d^7)

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Maple [B]  time = 0.23, size = 682, normalized size = 4.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x+a)^(3/2)/(2*c*d*x+b*d)^7,x)

[Out]

-1/192/d^7/c^6/(4*a*c-b^2)/(x+1/2*b/c)^6*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(5/2)+1/192/d^7/c^4/(4*a*c-b^2)^2
/(x+1/2*b/c)^4*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(5/2)+1/96/d^7/c^2/(4*a*c-b^2)^3/(x+1/2*b/c)^2*((x+1/2*b/c)
^2*c+1/4*(4*a*c-b^2)/c)^(5/2)-1/96/d^7/c/(4*a*c-b^2)^3*((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(3/2)-1/64/d^7/c/(4
*a*c-b^2)^3*(4*(x+1/2*b/c)^2*c+(4*a*c-b^2)/c)^(1/2)*a+1/256/d^7/c^2/(4*a*c-b^2)^3*(4*(x+1/2*b/c)^2*c+(4*a*c-b^
2)/c)^(1/2)*b^2+1/16/d^7/c/(4*a*c-b^2)^3/((4*a*c-b^2)/c)^(1/2)*ln((1/2*(4*a*c-b^2)/c+1/2*((4*a*c-b^2)/c)^(1/2)
*(4*(x+1/2*b/c)^2*c+(4*a*c-b^2)/c)^(1/2))/(x+1/2*b/c))*a^2-1/32/d^7/c^2/(4*a*c-b^2)^3/((4*a*c-b^2)/c)^(1/2)*ln
((1/2*(4*a*c-b^2)/c+1/2*((4*a*c-b^2)/c)^(1/2)*(4*(x+1/2*b/c)^2*c+(4*a*c-b^2)/c)^(1/2))/(x+1/2*b/c))*a*b^2+1/25
6/d^7/c^3/(4*a*c-b^2)^3/((4*a*c-b^2)/c)^(1/2)*ln((1/2*(4*a*c-b^2)/c+1/2*((4*a*c-b^2)/c)^(1/2)*(4*(x+1/2*b/c)^2
*c+(4*a*c-b^2)/c)^(1/2))/(x+1/2*b/c))*b^4

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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((c*x^2+b*x+a)^(3/2)/(2*c*d*x+b*d)^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 48.1422, size = 2276, normalized size = 13.79 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^(3/2)/(2*c*d*x+b*d)^7,x, algorithm="fricas")

[Out]

[1/1536*(3*(64*c^6*x^6 + 192*b*c^5*x^5 + 240*b^2*c^4*x^4 + 160*b^3*c^3*x^3 + 60*b^4*c^2*x^2 + 12*b^5*c*x + b^6
)*sqrt(-b^2*c + 4*a*c^2)*log(-(4*c^2*x^2 + 4*b*c*x - b^2 + 8*a*c + 4*sqrt(-b^2*c + 4*a*c^2)*sqrt(c*x^2 + b*x +
a))/(4*c^2*x^2 + 4*b*c*x + b^2)) - 4*(3*b^6*c - 4*a*b^4*c^2 - 160*a^2*b^2*c^3 + 512*a^3*c^4 - 48*(b^2*c^5 - 4
*a*c^6)*x^4 - 96*(b^3*c^4 - 4*a*b*c^5)*x^3 - 16*(b^4*c^3 + 10*a*b^2*c^4 - 56*a^2*c^5)*x^2 + 32*(b^5*c^2 - 11*a
*b^3*c^3 + 28*a^2*b*c^4)*x)*sqrt(c*x^2 + b*x + a))/(64*(b^4*c^9 - 8*a*b^2*c^10 + 16*a^2*c^11)*d^7*x^6 + 192*(b
^5*c^8 - 8*a*b^3*c^9 + 16*a^2*b*c^10)*d^7*x^5 + 240*(b^6*c^7 - 8*a*b^4*c^8 + 16*a^2*b^2*c^9)*d^7*x^4 + 160*(b^
7*c^6 - 8*a*b^5*c^7 + 16*a^2*b^3*c^8)*d^7*x^3 + 60*(b^8*c^5 - 8*a*b^6*c^6 + 16*a^2*b^4*c^7)*d^7*x^2 + 12*(b^9*
c^4 - 8*a*b^7*c^5 + 16*a^2*b^5*c^6)*d^7*x + (b^10*c^3 - 8*a*b^8*c^4 + 16*a^2*b^6*c^5)*d^7), -1/768*(3*(64*c^6*
x^6 + 192*b*c^5*x^5 + 240*b^2*c^4*x^4 + 160*b^3*c^3*x^3 + 60*b^4*c^2*x^2 + 12*b^5*c*x + b^6)*sqrt(b^2*c - 4*a*
c^2)*arctan(1/2*sqrt(b^2*c - 4*a*c^2)*sqrt(c*x^2 + b*x + a)/(c^2*x^2 + b*c*x + a*c)) + 2*(3*b^6*c - 4*a*b^4*c^
2 - 160*a^2*b^2*c^3 + 512*a^3*c^4 - 48*(b^2*c^5 - 4*a*c^6)*x^4 - 96*(b^3*c^4 - 4*a*b*c^5)*x^3 - 16*(b^4*c^3 +
10*a*b^2*c^4 - 56*a^2*c^5)*x^2 + 32*(b^5*c^2 - 11*a*b^3*c^3 + 28*a^2*b*c^4)*x)*sqrt(c*x^2 + b*x + a))/(64*(b^4
*c^9 - 8*a*b^2*c^10 + 16*a^2*c^11)*d^7*x^6 + 192*(b^5*c^8 - 8*a*b^3*c^9 + 16*a^2*b*c^10)*d^7*x^5 + 240*(b^6*c^
7 - 8*a*b^4*c^8 + 16*a^2*b^2*c^9)*d^7*x^4 + 160*(b^7*c^6 - 8*a*b^5*c^7 + 16*a^2*b^3*c^8)*d^7*x^3 + 60*(b^8*c^5
- 8*a*b^6*c^6 + 16*a^2*b^4*c^7)*d^7*x^2 + 12*(b^9*c^4 - 8*a*b^7*c^5 + 16*a^2*b^5*c^6)*d^7*x + (b^10*c^3 - 8*a
*b^8*c^4 + 16*a^2*b^6*c^5)*d^7)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{a \sqrt{a + b x + c x^{2}}}{b^{7} + 14 b^{6} c x + 84 b^{5} c^{2} x^{2} + 280 b^{4} c^{3} x^{3} + 560 b^{3} c^{4} x^{4} + 672 b^{2} c^{5} x^{5} + 448 b c^{6} x^{6} + 128 c^{7} x^{7}}\, dx + \int \frac{b x \sqrt{a + b x + c x^{2}}}{b^{7} + 14 b^{6} c x + 84 b^{5} c^{2} x^{2} + 280 b^{4} c^{3} x^{3} + 560 b^{3} c^{4} x^{4} + 672 b^{2} c^{5} x^{5} + 448 b c^{6} x^{6} + 128 c^{7} x^{7}}\, dx + \int \frac{c x^{2} \sqrt{a + b x + c x^{2}}}{b^{7} + 14 b^{6} c x + 84 b^{5} c^{2} x^{2} + 280 b^{4} c^{3} x^{3} + 560 b^{3} c^{4} x^{4} + 672 b^{2} c^{5} x^{5} + 448 b c^{6} x^{6} + 128 c^{7} x^{7}}\, dx}{d^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)**(3/2)/(2*c*d*x+b*d)**7,x)

[Out]

(Integral(a*sqrt(a + b*x + c*x**2)/(b**7 + 14*b**6*c*x + 84*b**5*c**2*x**2 + 280*b**4*c**3*x**3 + 560*b**3*c**
4*x**4 + 672*b**2*c**5*x**5 + 448*b*c**6*x**6 + 128*c**7*x**7), x) + Integral(b*x*sqrt(a + b*x + c*x**2)/(b**7
+ 14*b**6*c*x + 84*b**5*c**2*x**2 + 280*b**4*c**3*x**3 + 560*b**3*c**4*x**4 + 672*b**2*c**5*x**5 + 448*b*c**6
*x**6 + 128*c**7*x**7), x) + Integral(c*x**2*sqrt(a + b*x + c*x**2)/(b**7 + 14*b**6*c*x + 84*b**5*c**2*x**2 +
280*b**4*c**3*x**3 + 560*b**3*c**4*x**4 + 672*b**2*c**5*x**5 + 448*b*c**6*x**6 + 128*c**7*x**7), x))/d**7

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^(3/2)/(2*c*d*x+b*d)^7,x, algorithm="giac")

[Out]

Exception raised: TypeError