3.2092 \(\int (a+b x) (d+e x)^{7/2} \sqrt{a^2+2 a b x+b^2 x^2} \, dx\)

Optimal. Leaf size=152 \[ \frac{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13/2}}{13 e^3 (a+b x)}-\frac{4 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)}{11 e^3 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^2}{9 e^3 (a+b x)} \]

[Out]

(2*(b*d - a*e)^2*(d + e*x)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^3*(a + b*x)) - (4*b*(b*d - a*e)*(d + e*x)
^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^3*(a + b*x)) + (2*b^2*(d + e*x)^(13/2)*Sqrt[a^2 + 2*a*b*x + b^2*x
^2])/(13*e^3*(a + b*x))

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Rubi [A]  time = 0.0736086, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.086, Rules used = {770, 21, 43} \[ \frac{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13/2}}{13 e^3 (a+b x)}-\frac{4 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)}{11 e^3 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^2}{9 e^3 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(b*d - a*e)^2*(d + e*x)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^3*(a + b*x)) - (4*b*(b*d - a*e)*(d + e*x)
^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^3*(a + b*x)) + (2*b^2*(d + e*x)^(13/2)*Sqrt[a^2 + 2*a*b*x + b^2*x
^2])/(13*e^3*(a + b*x))

Rule 770

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dis
t[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(f + g*x)*(b/2 + c
*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int (a+b x) (d+e x)^{7/2} \sqrt{a^2+2 a b x+b^2 x^2} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int (a+b x) \left (a b+b^2 x\right ) (d+e x)^{7/2} \, dx}{a b+b^2 x}\\ &=\frac{\left (b \sqrt{a^2+2 a b x+b^2 x^2}\right ) \int (a+b x)^2 (d+e x)^{7/2} \, dx}{a b+b^2 x}\\ &=\frac{\left (b \sqrt{a^2+2 a b x+b^2 x^2}\right ) \int \left (\frac{(-b d+a e)^2 (d+e x)^{7/2}}{e^2}-\frac{2 b (b d-a e) (d+e x)^{9/2}}{e^2}+\frac{b^2 (d+e x)^{11/2}}{e^2}\right ) \, dx}{a b+b^2 x}\\ &=\frac{2 (b d-a e)^2 (d+e x)^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}}{9 e^3 (a+b x)}-\frac{4 b (b d-a e) (d+e x)^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}}{11 e^3 (a+b x)}+\frac{2 b^2 (d+e x)^{13/2} \sqrt{a^2+2 a b x+b^2 x^2}}{13 e^3 (a+b x)}\\ \end{align*}

Mathematica [A]  time = 0.0692384, size = 79, normalized size = 0.52 \[ \frac{2 \sqrt{(a+b x)^2} (d+e x)^{9/2} \left (143 a^2 e^2+26 a b e (9 e x-2 d)+b^2 \left (8 d^2-36 d e x+99 e^2 x^2\right )\right )}{1287 e^3 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[(a + b*x)^2]*(d + e*x)^(9/2)*(143*a^2*e^2 + 26*a*b*e*(-2*d + 9*e*x) + b^2*(8*d^2 - 36*d*e*x + 99*e^2*x
^2)))/(1287*e^3*(a + b*x))

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Maple [A]  time = 0.006, size = 79, normalized size = 0.5 \begin{align*}{\frac{198\,{x}^{2}{b}^{2}{e}^{2}+468\,xab{e}^{2}-72\,x{b}^{2}de+286\,{a}^{2}{e}^{2}-104\,abde+16\,{b}^{2}{d}^{2}}{1287\,{e}^{3} \left ( bx+a \right ) } \left ( ex+d \right ) ^{{\frac{9}{2}}}\sqrt{ \left ( bx+a \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(e*x+d)^(7/2)*((b*x+a)^2)^(1/2),x)

[Out]

2/1287*(e*x+d)^(9/2)*(99*b^2*e^2*x^2+234*a*b*e^2*x-36*b^2*d*e*x+143*a^2*e^2-52*a*b*d*e+8*b^2*d^2)*((b*x+a)^2)^
(1/2)/e^3/(b*x+a)

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Maxima [B]  time = 1.10388, size = 355, normalized size = 2.34 \begin{align*} \frac{2 \,{\left (9 \, b e^{5} x^{5} - 2 \, b d^{5} + 11 \, a d^{4} e +{\left (34 \, b d e^{4} + 11 \, a e^{5}\right )} x^{4} + 2 \,{\left (23 \, b d^{2} e^{3} + 22 \, a d e^{4}\right )} x^{3} + 6 \,{\left (4 \, b d^{3} e^{2} + 11 \, a d^{2} e^{3}\right )} x^{2} +{\left (b d^{4} e + 44 \, a d^{3} e^{2}\right )} x\right )} \sqrt{e x + d} a}{99 \, e^{2}} + \frac{2 \,{\left (99 \, b e^{6} x^{6} + 8 \, b d^{6} - 26 \, a d^{5} e + 9 \,{\left (40 \, b d e^{5} + 13 \, a e^{6}\right )} x^{5} + 2 \,{\left (229 \, b d^{2} e^{4} + 221 \, a d e^{5}\right )} x^{4} + 2 \,{\left (106 \, b d^{3} e^{3} + 299 \, a d^{2} e^{4}\right )} x^{3} + 3 \,{\left (b d^{4} e^{2} + 104 \, a d^{3} e^{3}\right )} x^{2} -{\left (4 \, b d^{5} e - 13 \, a d^{4} e^{2}\right )} x\right )} \sqrt{e x + d} b}{1287 \, e^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^(7/2)*((b*x+a)^2)^(1/2),x, algorithm="maxima")

[Out]

2/99*(9*b*e^5*x^5 - 2*b*d^5 + 11*a*d^4*e + (34*b*d*e^4 + 11*a*e^5)*x^4 + 2*(23*b*d^2*e^3 + 22*a*d*e^4)*x^3 + 6
*(4*b*d^3*e^2 + 11*a*d^2*e^3)*x^2 + (b*d^4*e + 44*a*d^3*e^2)*x)*sqrt(e*x + d)*a/e^2 + 2/1287*(99*b*e^6*x^6 + 8
*b*d^6 - 26*a*d^5*e + 9*(40*b*d*e^5 + 13*a*e^6)*x^5 + 2*(229*b*d^2*e^4 + 221*a*d*e^5)*x^4 + 2*(106*b*d^3*e^3 +
299*a*d^2*e^4)*x^3 + 3*(b*d^4*e^2 + 104*a*d^3*e^3)*x^2 - (4*b*d^5*e - 13*a*d^4*e^2)*x)*sqrt(e*x + d)*b/e^3

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Fricas [A]  time = 0.992556, size = 474, normalized size = 3.12 \begin{align*} \frac{2 \,{\left (99 \, b^{2} e^{6} x^{6} + 8 \, b^{2} d^{6} - 52 \, a b d^{5} e + 143 \, a^{2} d^{4} e^{2} + 18 \,{\left (20 \, b^{2} d e^{5} + 13 \, a b e^{6}\right )} x^{5} +{\left (458 \, b^{2} d^{2} e^{4} + 884 \, a b d e^{5} + 143 \, a^{2} e^{6}\right )} x^{4} + 4 \,{\left (53 \, b^{2} d^{3} e^{3} + 299 \, a b d^{2} e^{4} + 143 \, a^{2} d e^{5}\right )} x^{3} + 3 \,{\left (b^{2} d^{4} e^{2} + 208 \, a b d^{3} e^{3} + 286 \, a^{2} d^{2} e^{4}\right )} x^{2} - 2 \,{\left (2 \, b^{2} d^{5} e - 13 \, a b d^{4} e^{2} - 286 \, a^{2} d^{3} e^{3}\right )} x\right )} \sqrt{e x + d}}{1287 \, e^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^(7/2)*((b*x+a)^2)^(1/2),x, algorithm="fricas")

[Out]

2/1287*(99*b^2*e^6*x^6 + 8*b^2*d^6 - 52*a*b*d^5*e + 143*a^2*d^4*e^2 + 18*(20*b^2*d*e^5 + 13*a*b*e^6)*x^5 + (45
8*b^2*d^2*e^4 + 884*a*b*d*e^5 + 143*a^2*e^6)*x^4 + 4*(53*b^2*d^3*e^3 + 299*a*b*d^2*e^4 + 143*a^2*d*e^5)*x^3 +
3*(b^2*d^4*e^2 + 208*a*b*d^3*e^3 + 286*a^2*d^2*e^4)*x^2 - 2*(2*b^2*d^5*e - 13*a*b*d^4*e^2 - 286*a^2*d^3*e^3)*x
)*sqrt(e*x + d)/e^3

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)**(7/2)*((b*x+a)**2)**(1/2),x)

[Out]

Timed out

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Giac [B]  time = 1.23846, size = 900, normalized size = 5.92 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^(7/2)*((b*x+a)^2)^(1/2),x, algorithm="giac")

[Out]

2/45045*(6006*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a*b*d^3*e^(-1)*sgn(b*x + a) + 429*(15*(x*e + d)^(7/2)
- 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*b^2*d^3*e^(-2)*sgn(b*x + a) + 15015*(x*e + d)^(3/2)*a^2*d^3*s
gn(b*x + a) + 2574*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a*b*d^2*e^(-1)*sgn(b*x
+ a) + 429*(35*(x*e + d)^(9/2) - 135*(x*e + d)^(7/2)*d + 189*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*b
^2*d^2*e^(-2)*sgn(b*x + a) + 9009*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a^2*d^2*sgn(b*x + a) + 858*(35*(x*
e + d)^(9/2) - 135*(x*e + d)^(7/2)*d + 189*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*a*b*d*e^(-1)*sgn(b*x
+ a) + 39*(315*(x*e + d)^(11/2) - 1540*(x*e + d)^(9/2)*d + 2970*(x*e + d)^(7/2)*d^2 - 2772*(x*e + d)^(5/2)*d^
3 + 1155*(x*e + d)^(3/2)*d^4)*b^2*d*e^(-2)*sgn(b*x + a) + 1287*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35
*(x*e + d)^(3/2)*d^2)*a^2*d*sgn(b*x + a) + 26*(315*(x*e + d)^(11/2) - 1540*(x*e + d)^(9/2)*d + 2970*(x*e + d)^
(7/2)*d^2 - 2772*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4)*a*b*e^(-1)*sgn(b*x + a) + 5*(693*(x*e + d)^(1
3/2) - 4095*(x*e + d)^(11/2)*d + 10010*(x*e + d)^(9/2)*d^2 - 12870*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*
d^4 - 3003*(x*e + d)^(3/2)*d^5)*b^2*e^(-2)*sgn(b*x + a) + 143*(35*(x*e + d)^(9/2) - 135*(x*e + d)^(7/2)*d + 18
9*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*a^2*sgn(b*x + a))*e^(-1)