\(\int (a+b x) (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2} \, dx\) [2092]

Optimal result

Integrand size = 35, antiderivative size = 152 \[ \int (a+b x) (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {2 (b d-a e)^2 (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^3 (a+b x)}-\frac {4 b (b d-a e) (d+e x)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^3 (a+b x)}+\frac {2 b^2 (d+e x)^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^3 (a+b x)} \]

[Out]

2/9*(-a*e+b*d)^2*(e*x+d)^(9/2)*((b*x+a)^2)^(1/2)/e^3/(b*x+a)-4/11*b*(-a*e+b*d)*(e*x+d)^(11/2)*((b*x+a)^2)^(1/2
)/e^3/(b*x+a)+2/13*b^2*(e*x+d)^(13/2)*((b*x+a)^2)^(1/2)/e^3/(b*x+a)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {784, 21, 45} \[ \int (a+b x) (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {2 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{13/2}}{13 e^3 (a+b x)}-\frac {4 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)}{11 e^3 (a+b x)}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^2}{9 e^3 (a+b x)} \]

[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 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 45

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])

Rule 784

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]

Rubi steps \begin{align*} \text {integral}& = \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] (verified)

Time = 0.03 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.52 \[ \int (a+b x) (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {2 \sqrt {(a+b x)^2} (d+e x)^{9/2} \left (143 a^2 e^2+26 a b e (-2 d+9 e x)+b^2 \left (8 d^2-36 d e x+99 e^2 x^2\right )\right )}{1287 e^3 (a+b x)} \]

[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))

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 2.

Time = 0.25 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.45

method	result	size
default	\(\frac {2 \,\operatorname {csgn}\left (b x +a \right ) \left (e x +d \right )^{\frac {9}{2}} \left (99 b^{2} e^{2} x^{2}+234 a b \,e^{2} x -36 b^{2} d e x +143 e^{2} a^{2}-52 a b d e +8 b^{2} d^{2}\right )}{1287 e^{3}}\)	\(69\)
gosper	\(\frac {2 \left (e x +d \right )^{\frac {9}{2}} \left (99 b^{2} e^{2} x^{2}+234 a b \,e^{2} x -36 b^{2} d e x +143 e^{2} a^{2}-52 a b d e +8 b^{2} d^{2}\right ) \sqrt {\left (b x +a \right )^{2}}}{1287 e^{3} \left (b x +a \right )}\)	\(79\)
risch	\(\frac {2 \sqrt {\left (b x +a \right )^{2}}\, \left (99 b^{2} e^{6} x^{6}+234 a b \,e^{6} x^{5}+360 b^{2} d \,e^{5} x^{5}+143 a^{2} e^{6} x^{4}+884 a b d \,e^{5} x^{4}+458 b^{2} d^{2} e^{4} x^{4}+572 a^{2} d \,e^{5} x^{3}+1196 a b \,d^{2} e^{4} x^{3}+212 b^{2} d^{3} e^{3} x^{3}+858 a^{2} d^{2} e^{4} x^{2}+624 a b \,d^{3} e^{3} x^{2}+3 b^{2} d^{4} e^{2} x^{2}+572 a^{2} d^{3} e^{3} x +26 a b \,d^{4} e^{2} x -4 b^{2} d^{5} e x +143 a^{2} d^{4} e^{2}-52 a b \,d^{5} e +8 b^{2} d^{6}\right ) \sqrt {e x +d}}{1287 \left (b x +a \right ) e^{3}}\)	\(239\)

[In]

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

[Out]

2/1287*csgn(b*x+a)*(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)/
e^3

Fricas [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.39 \[ \int (a+b x) (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {2 \, {\left (99 \, b^{2} e^{6} x^{6} + 8 \, b^{2} d^{6} - 52 \, a b d^{5} e + 143 \, a^{2} d^{4} e^{2} + 18 \, {\left (20 \, b^{2} d e^{5} + 13 \, a b e^{6}\right )} x^{5} + {\left (458 \, b^{2} d^{2} e^{4} + 884 \, a b d e^{5} + 143 \, a^{2} e^{6}\right )} x^{4} + 4 \, {\left (53 \, b^{2} d^{3} e^{3} + 299 \, a b d^{2} e^{4} + 143 \, a^{2} d e^{5}\right )} x^{3} + 3 \, {\left (b^{2} d^{4} e^{2} + 208 \, a b d^{3} e^{3} + 286 \, a^{2} d^{2} e^{4}\right )} x^{2} - 2 \, {\left (2 \, b^{2} d^{5} e - 13 \, a b d^{4} e^{2} - 286 \, a^{2} d^{3} e^{3}\right )} x\right )} \sqrt {e x + d}}{1287 \, e^{3}} \]

[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

Sympy [F(-1)]

Timed out. \[ \int (a+b x) (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 263 vs. \(2 (107) = 214\).

Time = 0.20 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.73 \[ \int (a+b x) (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {2 \, {\left (9 \, b e^{5} x^{5} - 2 \, b d^{5} + 11 \, a d^{4} e + {\left (34 \, b d e^{4} + 11 \, a e^{5}\right )} x^{4} + 2 \, {\left (23 \, b d^{2} e^{3} + 22 \, a d e^{4}\right )} x^{3} + 6 \, {\left (4 \, b d^{3} e^{2} + 11 \, a d^{2} e^{3}\right )} x^{2} + {\left (b d^{4} e + 44 \, a d^{3} e^{2}\right )} x\right )} \sqrt {e x + d} a}{99 \, e^{2}} + \frac {2 \, {\left (99 \, b e^{6} x^{6} + 8 \, b d^{6} - 26 \, a d^{5} e + 9 \, {\left (40 \, b d e^{5} + 13 \, a e^{6}\right )} x^{5} + 2 \, {\left (229 \, b d^{2} e^{4} + 221 \, a d e^{5}\right )} x^{4} + 2 \, {\left (106 \, b d^{3} e^{3} + 299 \, a d^{2} e^{4}\right )} x^{3} + 3 \, {\left (b d^{4} e^{2} + 104 \, a d^{3} e^{3}\right )} x^{2} - {\left (4 \, b d^{5} e - 13 \, a d^{4} e^{2}\right )} x\right )} \sqrt {e x + d} b}{1287 \, e^{3}} \]

[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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 881 vs. \(2 (107) = 214\).

Time = 0.28 (sec) , antiderivative size = 881, normalized size of antiderivative = 5.80 \[ \int (a+b x) (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

2/45045*(45045*sqrt(e*x + d)*a^2*d^4*sgn(b*x + a) + 60060*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*a^2*d^3*sgn(b*
x + a) + 30030*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*a*b*d^4*sgn(b*x + a)/e + 18018*(3*(e*x + d)^(5/2) - 10*(e
*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a^2*d^2*sgn(b*x + a) + 3003*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d
+ 15*sqrt(e*x + d)*d^2)*b^2*d^4*sgn(b*x + a)/e^2 + 24024*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e
*x + d)*d^2)*a*b*d^3*sgn(b*x + a)/e + 5148*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2
- 35*sqrt(e*x + d)*d^3)*a^2*d*sgn(b*x + a) + 5148*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/
2)*d^2 - 35*sqrt(e*x + d)*d^3)*b^2*d^3*sgn(b*x + a)/e^2 + 15444*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35
*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*a*b*d^2*sgn(b*x + a)/e + 143*(35*(e*x + d)^(9/2) - 180*(e*x + d)^
(7/2)*d + 378*(e*x + d)^(5/2)*d^2 - 420*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*a^2*sgn(b*x + a) + 858*(3
5*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 378*(e*x + d)^(5/2)*d^2 - 420*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x +
d)*d^4)*b^2*d^2*sgn(b*x + a)/e^2 + 1144*(35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 378*(e*x + d)^(5/2)*d^2
- 420*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*a*b*d*sgn(b*x + a)/e + 260*(63*(e*x + d)^(11/2) - 385*(e*x
+ d)^(9/2)*d + 990*(e*x + d)^(7/2)*d^2 - 1386*(e*x + d)^(5/2)*d^3 + 1155*(e*x + d)^(3/2)*d^4 - 693*sqrt(e*x +
d)*d^5)*b^2*d*sgn(b*x + a)/e^2 + 130*(63*(e*x + d)^(11/2) - 385*(e*x + d)^(9/2)*d + 990*(e*x + d)^(7/2)*d^2 -
1386*(e*x + d)^(5/2)*d^3 + 1155*(e*x + d)^(3/2)*d^4 - 693*sqrt(e*x + d)*d^5)*a*b*sgn(b*x + a)/e + 15*(231*(e*
x + d)^(13/2) - 1638*(e*x + d)^(11/2)*d + 5005*(e*x + d)^(9/2)*d^2 - 8580*(e*x + d)^(7/2)*d^3 + 9009*(e*x + d)
^(5/2)*d^4 - 6006*(e*x + d)^(3/2)*d^5 + 3003*sqrt(e*x + d)*d^6)*b^2*sgn(b*x + a)/e^2)/e

Mupad [F(-1)]

Timed out. \[ \int (a+b x) (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\int \sqrt {{\left (a+b\,x\right )}^2}\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^{7/2} \,d x \]

[In]

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

[Out]

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