\(\int (A+B x) (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2} \, dx\) [1835]

Optimal result

Integrand size = 35, antiderivative size = 164 \[ \int (A+B x) (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {2 (b d-a e) (B d-A e) (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^3 (a+b x)}-\frac {2 (2 b B d-A b e-a B e) (d+e x)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^3 (a+b x)}+\frac {2 b B (d+e x)^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^3 (a+b x)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {784, 78} \[ \int (A+B x) (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2} \, dx=-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (-a B e-A b e+2 b B d)}{11 e^3 (a+b x)}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e) (B d-A e)}{9 e^3 (a+b x)}+\frac {2 b B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{13/2}}{13 e^3 (a+b x)} \]

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

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

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 \left (a b+b^2 x\right ) (A+B x) (d+e x)^{7/2} \, dx}{a b+b^2 x} \\ & = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (-\frac {b (b d-a e) (-B d+A e) (d+e x)^{7/2}}{e^2}+\frac {b (-2 b B d+A b e+a B e) (d+e x)^{9/2}}{e^2}+\frac {b^2 B (d+e x)^{11/2}}{e^2}\right ) \, dx}{a b+b^2 x} \\ & = \frac {2 (b d-a e) (B d-A e) (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^3 (a+b x)}-\frac {2 (2 b B d-A b e-a B 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 B (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 = 0.04 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.54 \[ \int (A+B x) (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {2 \sqrt {(a+b x)^2} (d+e x)^{9/2} \left (13 A b e (-2 d+9 e x)+13 a e (-2 B d+11 A e+9 B e x)+b B \left (8 d^2-36 d e x+99 e^2 x^2\right )\right )}{1287 e^3 (a+b x)} \]

[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)*(13*A*b*e*(-2*d + 9*e*x) + 13*a*e*(-2*B*d + 11*A*e + 9*B*e*x) + b*B*(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 = 0.22 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.48

method	result	size
default	\(\frac {2 \,\operatorname {csgn}\left (b x +a \right ) \left (e x +d \right )^{\frac {9}{2}} \left (99 B b \,e^{2} x^{2}+117 A b \,e^{2} x +117 B a \,e^{2} x -36 B b d e x +143 A a \,e^{2}-26 A b d e -26 B a d e +8 B b \,d^{2}\right )}{1287 e^{3}}\)	\(79\)
gosper	\(\frac {2 \left (e x +d \right )^{\frac {9}{2}} \left (99 B b \,e^{2} x^{2}+117 A b \,e^{2} x +117 B a \,e^{2} x -36 B b d e x +143 A a \,e^{2}-26 A b d e -26 B a d e +8 B b \,d^{2}\right ) \sqrt {\left (b x +a \right )^{2}}}{1287 e^{3} \left (b x +a \right )}\)	\(89\)
risch	\(\frac {2 \sqrt {\left (b x +a \right )^{2}}\, \left (99 B \,e^{6} b \,x^{6}+117 A b \,e^{6} x^{5}+117 B a \,e^{6} x^{5}+360 B b d \,e^{5} x^{5}+143 A a \,e^{6} x^{4}+442 A b d \,e^{5} x^{4}+442 B a d \,e^{5} x^{4}+458 B b \,d^{2} e^{4} x^{4}+572 A a d \,e^{5} x^{3}+598 A b \,d^{2} e^{4} x^{3}+598 B a \,d^{2} e^{4} x^{3}+212 B b \,d^{3} e^{3} x^{3}+858 A a \,d^{2} e^{4} x^{2}+312 A b \,d^{3} e^{3} x^{2}+312 B a \,d^{3} e^{3} x^{2}+3 B b \,d^{4} e^{2} x^{2}+572 A a \,d^{3} e^{3} x +13 A b \,d^{4} e^{2} x +13 B a \,d^{4} e^{2} x -4 B b \,d^{5} e x +143 A a \,d^{4} e^{2}-26 A b \,d^{5} e -26 B a \,d^{5} e +8 B b \,d^{6}\right ) \sqrt {e x +d}}{1287 \left (b x +a \right ) e^{3}}\)	\(293\)

[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*b*e^2*x^2+117*A*b*e^2*x+117*B*a*e^2*x-36*B*b*d*e*x+143*A*a*e^2-26*A*b*d
*e-26*B*a*d*e+8*B*b*d^2)/e^3

Fricas [A] (verification not implemented)

none

Time = 0.38 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.40 \[ \int (A+B x) (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {2 \, {\left (99 \, B b e^{6} x^{6} + 8 \, B b d^{6} + 143 \, A a d^{4} e^{2} - 26 \, {\left (B a + A b\right )} d^{5} e + 9 \, {\left (40 \, B b d e^{5} + 13 \, {\left (B a + A b\right )} e^{6}\right )} x^{5} + {\left (458 \, B b d^{2} e^{4} + 143 \, A a e^{6} + 442 \, {\left (B a + A b\right )} d e^{5}\right )} x^{4} + 2 \, {\left (106 \, B b d^{3} e^{3} + 286 \, A a d e^{5} + 299 \, {\left (B a + A b\right )} d^{2} e^{4}\right )} x^{3} + 3 \, {\left (B b d^{4} e^{2} + 286 \, A a d^{2} e^{4} + 104 \, {\left (B a + A b\right )} d^{3} e^{3}\right )} x^{2} - {\left (4 \, B b d^{5} e - 572 \, A a d^{3} e^{3} - 13 \, {\left (B a + A b\right )} d^{4} e^{2}\right )} x\right )} \sqrt {e x + d}}{1287 \, e^{3}} \]

[In]

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

[Out]

2/1287*(99*B*b*e^6*x^6 + 8*B*b*d^6 + 143*A*a*d^4*e^2 - 26*(B*a + A*b)*d^5*e + 9*(40*B*b*d*e^5 + 13*(B*a + A*b)
*e^6)*x^5 + (458*B*b*d^2*e^4 + 143*A*a*e^6 + 442*(B*a + A*b)*d*e^5)*x^4 + 2*(106*B*b*d^3*e^3 + 286*A*a*d*e^5 +
299*(B*a + A*b)*d^2*e^4)*x^3 + 3*(B*b*d^4*e^2 + 286*A*a*d^2*e^4 + 104*(B*a + A*b)*d^3*e^3)*x^2 - (4*B*b*d^5*e
- 572*A*a*d^3*e^3 - 13*(B*a + A*b)*d^4*e^2)*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. \(2 (119) = 238\).

Time = 0.22 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.60 \[ \int (A+B x) (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\frac {2 \, {\left (9 \, b e^{5} x^{5} - 2 \, b d^{5} + 11 \, a d^{4} e + {\left (34 \, b d e^{4} + 11 \, a e^{5}\right )} x^{4} + 2 \, {\left (23 \, b d^{2} e^{3} + 22 \, a d e^{4}\right )} x^{3} + 6 \, {\left (4 \, b d^{3} e^{2} + 11 \, a d^{2} e^{3}\right )} x^{2} + {\left (b d^{4} e + 44 \, a d^{3} e^{2}\right )} x\right )} \sqrt {e x + d} A}{99 \, e^{2}} + \frac {2 \, {\left (99 \, b e^{6} x^{6} + 8 \, b d^{6} - 26 \, a d^{5} e + 9 \, {\left (40 \, b d e^{5} + 13 \, a e^{6}\right )} x^{5} + 2 \, {\left (229 \, b d^{2} e^{4} + 221 \, a d e^{5}\right )} x^{4} + 2 \, {\left (106 \, b d^{3} e^{3} + 299 \, a d^{2} e^{4}\right )} x^{3} + 3 \, {\left (b d^{4} e^{2} + 104 \, a d^{3} e^{3}\right )} x^{2} - {\left (4 \, b d^{5} e - 13 \, a d^{4} e^{2}\right )} x\right )} \sqrt {e x + d} B}{1287 \, e^{3}} \]

[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. 1164 vs. \(2 (119) = 238\).

Time = 0.32 (sec) , antiderivative size = 1164, normalized size of antiderivative = 7.10 \[ \int (A+B x) (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2} \, dx=\text {Too large to display} \]

[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*a*d^4*sgn(b*x + a) + 60060*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*A*a*d^3*sgn(b*
x + a) + 15015*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*B*a*d^4*sgn(b*x + a)/e + 15015*((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
*a*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*b*d^4*sgn(b*x +
a)/e^2 + 12012*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*B*a*d^3*sgn(b*x + a)/e + 120
12*(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*a*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*b*d^3*sgn(b*x + a)/e^
2 + 7722*(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*a*d^2*sg
n(b*x + a)/e + 7722*(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*a*sgn(b*x + a) + 858*(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)*B*b*d^2*sgn(b*x + a)/e^2 + 572*(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)*B*a*d*sgn(b*x + a)/e + 572*(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*b*d*sgn(b*x + a)/e^2 + 65*(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*a*sgn(b*x + a)/e + 65*(63*(e*x + d)^(1
1/2) - 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*b*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)