3.20.0 ∫ (a + bx)(d + ex)7(a2 + 2abx + b2x2)5∕2 dx [1990]

\(\int (a+b x) (d+e x)^7 (a^2+2 a b x+b^2 x^2)^{5/2} \, dx\) [1990]

   Optimal result
   Rubi [A] (veriﬁed)
   Mathematica [A] (veriﬁed)
   Maple [B] (veriﬁed)
   Fricas [B] (veriﬁcation not implemented)
   Sympy [B] (veriﬁcation not implemented)
   Maxima [B] (veriﬁcation not implemented)
   Giac [B] (veriﬁcation not implemented)
   Mupad [F(-1)]

Optimal result

Integrand size = 33, antiderivative size = 362 \[ \int (a+b x) (d+e x)^7 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {(b d-a e)^6 (d+e x)^8 \sqrt {a^2+2 a b x+b^2 x^2}}{8 e^7 (a+b x)}-\frac {2 b (b d-a e)^5 (d+e x)^9 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x)}+\frac {3 b^2 (b d-a e)^4 (d+e x)^{10} \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^7 (a+b x)}-\frac {20 b^3 (b d-a e)^3 (d+e x)^{11} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x)}+\frac {5 b^4 (b d-a e)^2 (d+e x)^{12} \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^7 (a+b x)}-\frac {6 b^5 (b d-a e) (d+e x)^{13} \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^7 (a+b x)}+\frac {b^6 (d+e x)^{14} \sqrt {a^2+2 a b x+b^2 x^2}}{14 e^7 (a+b x)} \]

[Out]

1/8*(-a*e+b*d)^6*(e*x+d)^8*((b*x+a)^2)^(1/2)/e^7/(b*x+a)-2/3*b*(-a*e+b*d)^5*(e*x+d)^9*((b*x+a)^2)^(1/2)/e^7/(b

*x+a)+3/2*b^2*(-a*e+b*d)^4*(e*x+d)^10*((b*x+a)^2)^(1/2)/e^7/(b*x+a)-20/11*b^3*(-a*e+b*d)^3*(e*x+d)^11*((b*x+a)

^2)^(1/2)/e^7/(b*x+a)+5/4*b^4*(-a*e+b*d)^2*(e*x+d)^12*((b*x+a)^2)^(1/2)/e^7/(b*x+a)-6/13*b^5*(-a*e+b*d)*(e*x+d

)^13*((b*x+a)^2)^(1/2)/e^7/(b*x+a)+1/14*b^6*(e*x+d)^14*((b*x+a)^2)^(1/2)/e^7/(b*x+a)

Rubi [A] (veriﬁed)

Time = 0.34 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {784, 21, 45} \[ \int (a+b x) (d+e x)^7 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {3 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{10} (b d-a e)^4}{2 e^7 (a+b x)}-\frac {2 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^9 (b d-a e)^5}{3 e^7 (a+b x)}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^8 (b d-a e)^6}{8 e^7 (a+b x)}+\frac {b^6 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{14}}{14 e^7 (a+b x)}-\frac {6 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{13} (b d-a e)}{13 e^7 (a+b x)}+\frac {5 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{12} (b d-a e)^2}{4 e^7 (a+b x)}-\frac {20 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11} (b d-a e)^3}{11 e^7 (a+b x)} \]

[In]

Int[(a + b*x)*(d + e*x)^7*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

((b*d - a*e)^6*(d + e*x)^8*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(8*e^7*(a + b*x)) - (2*b*(b*d - a*e)^5*(d + e*x)^9*S

qrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^7*(a + b*x)) + (3*b^2*(b*d - a*e)^4*(d + e*x)^10*Sqrt[a^2 + 2*a*b*x + b^2*x

^2])/(2*e^7*(a + b*x)) - (20*b^3*(b*d - a*e)^3*(d + e*x)^11*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^7*(a + b*x))

+ (5*b^4*(b*d - a*e)^2*(d + e*x)^12*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*e^7*(a + b*x)) - (6*b^5*(b*d - a*e)*(d +

e*x)^13*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*e^7*(a + b*x)) + (b^6*(d + e*x)^14*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/

(14*e^7*(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 )^5 (d+e x)^7 \, dx}{b^4 \left (a b+b^2 x\right )} \\ & = \frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int (a+b x)^6 (d+e x)^7 \, 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)^6 (d+e x)^7}{e^6}-\frac {6 b (b d-a e)^5 (d+e x)^8}{e^6}+\frac {15 b^2 (b d-a e)^4 (d+e x)^9}{e^6}-\frac {20 b^3 (b d-a e)^3 (d+e x)^{10}}{e^6}+\frac {15 b^4 (b d-a e)^2 (d+e x)^{11}}{e^6}-\frac {6 b^5 (b d-a e) (d+e x)^{12}}{e^6}+\frac {b^6 (d+e x)^{13}}{e^6}\right ) \, dx}{a b+b^2 x} \\ & = \frac {(b d-a e)^6 (d+e x)^8 \sqrt {a^2+2 a b x+b^2 x^2}}{8 e^7 (a+b x)}-\frac {2 b (b d-a e)^5 (d+e x)^9 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x)}+\frac {3 b^2 (b d-a e)^4 (d+e x)^{10} \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^7 (a+b x)}-\frac {20 b^3 (b d-a e)^3 (d+e x)^{11} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x)}+\frac {5 b^4 (b d-a e)^2 (d+e x)^{12} \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^7 (a+b x)}-\frac {6 b^5 (b d-a e) (d+e x)^{13} \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^7 (a+b x)}+\frac {b^6 (d+e x)^{14} \sqrt {a^2+2 a b x+b^2 x^2}}{14 e^7 (a+b x)} \\ \end{align*}

Mathematica [A] (veriﬁed)

Time = 1.13 (sec) , antiderivative size = 602, normalized size of antiderivative = 1.66 \[ \int (a+b x) (d+e x)^7 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {x \sqrt {(a+b x)^2} \left (3003 a^6 \left (8 d^7+28 d^6 e x+56 d^5 e^2 x^2+70 d^4 e^3 x^3+56 d^3 e^4 x^4+28 d^2 e^5 x^5+8 d e^6 x^6+e^7 x^7\right )+2002 a^5 b x \left (36 d^7+168 d^6 e x+378 d^5 e^2 x^2+504 d^4 e^3 x^3+420 d^3 e^4 x^4+216 d^2 e^5 x^5+63 d e^6 x^6+8 e^7 x^7\right )+1001 a^4 b^2 x^2 \left (120 d^7+630 d^6 e x+1512 d^5 e^2 x^2+2100 d^4 e^3 x^3+1800 d^3 e^4 x^4+945 d^2 e^5 x^5+280 d e^6 x^6+36 e^7 x^7\right )+364 a^3 b^3 x^3 \left (330 d^7+1848 d^6 e x+4620 d^5 e^2 x^2+6600 d^4 e^3 x^3+5775 d^3 e^4 x^4+3080 d^2 e^5 x^5+924 d e^6 x^6+120 e^7 x^7\right )+91 a^2 b^4 x^4 \left (792 d^7+4620 d^6 e x+11880 d^5 e^2 x^2+17325 d^4 e^3 x^3+15400 d^3 e^4 x^4+8316 d^2 e^5 x^5+2520 d e^6 x^6+330 e^7 x^7\right )+14 a b^5 x^5 \left (1716 d^7+10296 d^6 e x+27027 d^5 e^2 x^2+40040 d^4 e^3 x^3+36036 d^3 e^4 x^4+19656 d^2 e^5 x^5+6006 d e^6 x^6+792 e^7 x^7\right )+b^6 x^6 \left (3432 d^7+21021 d^6 e x+56056 d^5 e^2 x^2+84084 d^4 e^3 x^3+76440 d^3 e^4 x^4+42042 d^2 e^5 x^5+12936 d e^6 x^6+1716 e^7 x^7\right )\right )}{24024 (a+b x)} \]

[In]

Integrate[(a + b*x)*(d + e*x)^7*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(x*Sqrt[(a + b*x)^2]*(3003*a^6*(8*d^7 + 28*d^6*e*x + 56*d^5*e^2*x^2 + 70*d^4*e^3*x^3 + 56*d^3*e^4*x^4 + 28*d^2

*e^5*x^5 + 8*d*e^6*x^6 + e^7*x^7) + 2002*a^5*b*x*(36*d^7 + 168*d^6*e*x + 378*d^5*e^2*x^2 + 504*d^4*e^3*x^3 + 4

20*d^3*e^4*x^4 + 216*d^2*e^5*x^5 + 63*d*e^6*x^6 + 8*e^7*x^7) + 1001*a^4*b^2*x^2*(120*d^7 + 630*d^6*e*x + 1512*

d^5*e^2*x^2 + 2100*d^4*e^3*x^3 + 1800*d^3*e^4*x^4 + 945*d^2*e^5*x^5 + 280*d*e^6*x^6 + 36*e^7*x^7) + 364*a^3*b^

3*x^3*(330*d^7 + 1848*d^6*e*x + 4620*d^5*e^2*x^2 + 6600*d^4*e^3*x^3 + 5775*d^3*e^4*x^4 + 3080*d^2*e^5*x^5 + 92

4*d*e^6*x^6 + 120*e^7*x^7) + 91*a^2*b^4*x^4*(792*d^7 + 4620*d^6*e*x + 11880*d^5*e^2*x^2 + 17325*d^4*e^3*x^3 +

15400*d^3*e^4*x^4 + 8316*d^2*e^5*x^5 + 2520*d*e^6*x^6 + 330*e^7*x^7) + 14*a*b^5*x^5*(1716*d^7 + 10296*d^6*e*x

+ 27027*d^5*e^2*x^2 + 40040*d^4*e^3*x^3 + 36036*d^3*e^4*x^4 + 19656*d^2*e^5*x^5 + 6006*d*e^6*x^6 + 792*e^7*x^7

) + b^6*x^6*(3432*d^7 + 21021*d^6*e*x + 56056*d^5*e^2*x^2 + 84084*d^4*e^3*x^3 + 76440*d^3*e^4*x^4 + 42042*d^2*

e^5*x^5 + 12936*d*e^6*x^6 + 1716*e^7*x^7)))/(24024*(a + b*x))

Maple [B] (veriﬁed)

Leaf count of result is larger than twice the leaf count of optimal. \(815\) vs. \(2(271)=542\).

Time = 1.32 (sec) , antiderivative size = 816, normalized size of antiderivative = 2.25


method	result	size

gosper	\(\frac {x \left (1716 b^{6} e^{7} x^{13}+11088 x^{12} b^{5} a \,e^{7}+12936 x^{12} b^{6} d \,e^{6}+30030 x^{11} b^{4} a^{2} e^{7}+84084 x^{11} b^{5} a d \,e^{6}+42042 x^{11} b^{6} d^{2} e^{5}+43680 x^{10} a^{3} b^{3} e^{7}+229320 x^{10} b^{4} a^{2} d \,e^{6}+275184 x^{10} b^{5} a \,d^{2} e^{5}+76440 x^{10} b^{6} d^{3} e^{4}+36036 x^{9} a^{4} b^{2} e^{7}+336336 x^{9} a^{3} b^{3} d \,e^{6}+756756 x^{9} b^{4} a^{2} d^{2} e^{5}+504504 x^{9} b^{5} a \,d^{3} e^{4}+84084 x^{9} b^{6} d^{4} e^{3}+16016 x^{8} a^{5} b \,e^{7}+280280 x^{8} a^{4} b^{2} d \,e^{6}+1121120 x^{8} a^{3} b^{3} d^{2} e^{5}+1401400 x^{8} b^{4} a^{2} d^{3} e^{4}+560560 x^{8} b^{5} a \,d^{4} e^{3}+56056 x^{8} b^{6} d^{5} e^{2}+3003 x^{7} a^{6} e^{7}+126126 x^{7} a^{5} b d \,e^{6}+945945 x^{7} a^{4} b^{2} d^{2} e^{5}+2102100 x^{7} a^{3} b^{3} d^{3} e^{4}+1576575 x^{7} b^{4} a^{2} d^{4} e^{3}+378378 x^{7} b^{5} a \,d^{5} e^{2}+21021 x^{7} b^{6} d^{6} e +24024 x^{6} a^{6} d \,e^{6}+432432 x^{6} a^{5} b \,d^{2} e^{5}+1801800 x^{6} a^{4} b^{2} d^{3} e^{4}+2402400 x^{6} a^{3} b^{3} d^{4} e^{3}+1081080 x^{6} b^{4} a^{2} d^{5} e^{2}+144144 x^{6} b^{5} a \,d^{6} e +3432 x^{6} b^{6} d^{7}+84084 x^{5} a^{6} d^{2} e^{5}+840840 x^{5} a^{5} b \,d^{3} e^{4}+2102100 x^{5} a^{4} b^{2} d^{4} e^{3}+1681680 x^{5} a^{3} b^{3} d^{5} e^{2}+420420 x^{5} b^{4} a^{2} d^{6} e +24024 x^{5} b^{5} a \,d^{7}+168168 a^{6} d^{3} e^{4} x^{4}+1009008 a^{5} b \,d^{4} e^{3} x^{4}+1513512 a^{4} b^{2} d^{5} e^{2} x^{4}+672672 a^{3} b^{3} d^{6} e \,x^{4}+72072 a^{2} b^{4} d^{7} x^{4}+210210 x^{3} a^{6} d^{4} e^{3}+756756 x^{3} a^{5} b \,d^{5} e^{2}+630630 x^{3} a^{4} b^{2} d^{6} e +120120 x^{3} a^{3} b^{3} d^{7}+168168 a^{6} d^{5} e^{2} x^{2}+336336 a^{5} b \,d^{6} e \,x^{2}+120120 a^{4} b^{2} d^{7} x^{2}+84084 x \,a^{6} d^{6} e +72072 x \,a^{5} b \,d^{7}+24024 a^{6} d^{7}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{24024 \left (b x +a \right )^{5}}\)	\(816\)
default	\(\frac {x \left (1716 b^{6} e^{7} x^{13}+11088 x^{12} b^{5} a \,e^{7}+12936 x^{12} b^{6} d \,e^{6}+30030 x^{11} b^{4} a^{2} e^{7}+84084 x^{11} b^{5} a d \,e^{6}+42042 x^{11} b^{6} d^{2} e^{5}+43680 x^{10} a^{3} b^{3} e^{7}+229320 x^{10} b^{4} a^{2} d \,e^{6}+275184 x^{10} b^{5} a \,d^{2} e^{5}+76440 x^{10} b^{6} d^{3} e^{4}+36036 x^{9} a^{4} b^{2} e^{7}+336336 x^{9} a^{3} b^{3} d \,e^{6}+756756 x^{9} b^{4} a^{2} d^{2} e^{5}+504504 x^{9} b^{5} a \,d^{3} e^{4}+84084 x^{9} b^{6} d^{4} e^{3}+16016 x^{8} a^{5} b \,e^{7}+280280 x^{8} a^{4} b^{2} d \,e^{6}+1121120 x^{8} a^{3} b^{3} d^{2} e^{5}+1401400 x^{8} b^{4} a^{2} d^{3} e^{4}+560560 x^{8} b^{5} a \,d^{4} e^{3}+56056 x^{8} b^{6} d^{5} e^{2}+3003 x^{7} a^{6} e^{7}+126126 x^{7} a^{5} b d \,e^{6}+945945 x^{7} a^{4} b^{2} d^{2} e^{5}+2102100 x^{7} a^{3} b^{3} d^{3} e^{4}+1576575 x^{7} b^{4} a^{2} d^{4} e^{3}+378378 x^{7} b^{5} a \,d^{5} e^{2}+21021 x^{7} b^{6} d^{6} e +24024 x^{6} a^{6} d \,e^{6}+432432 x^{6} a^{5} b \,d^{2} e^{5}+1801800 x^{6} a^{4} b^{2} d^{3} e^{4}+2402400 x^{6} a^{3} b^{3} d^{4} e^{3}+1081080 x^{6} b^{4} a^{2} d^{5} e^{2}+144144 x^{6} b^{5} a \,d^{6} e +3432 x^{6} b^{6} d^{7}+84084 x^{5} a^{6} d^{2} e^{5}+840840 x^{5} a^{5} b \,d^{3} e^{4}+2102100 x^{5} a^{4} b^{2} d^{4} e^{3}+1681680 x^{5} a^{3} b^{3} d^{5} e^{2}+420420 x^{5} b^{4} a^{2} d^{6} e +24024 x^{5} b^{5} a \,d^{7}+168168 a^{6} d^{3} e^{4} x^{4}+1009008 a^{5} b \,d^{4} e^{3} x^{4}+1513512 a^{4} b^{2} d^{5} e^{2} x^{4}+672672 a^{3} b^{3} d^{6} e \,x^{4}+72072 a^{2} b^{4} d^{7} x^{4}+210210 x^{3} a^{6} d^{4} e^{3}+756756 x^{3} a^{5} b \,d^{5} e^{2}+630630 x^{3} a^{4} b^{2} d^{6} e +120120 x^{3} a^{3} b^{3} d^{7}+168168 a^{6} d^{5} e^{2} x^{2}+336336 a^{5} b \,d^{6} e \,x^{2}+120120 a^{4} b^{2} d^{7} x^{2}+84084 x \,a^{6} d^{6} e +72072 x \,a^{5} b \,d^{7}+24024 a^{6} d^{7}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{24024 \left (b x +a \right )^{5}}\)	\(816\)
risch	\(\frac {\sqrt {\left (b x +a \right )^{2}}\, b^{6} e^{7} x^{14}}{14 b x +14 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (6 b^{5} a \,e^{7}+7 b^{6} d \,e^{6}\right ) x^{13}}{13 b x +13 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (15 b^{4} a^{2} e^{7}+42 b^{5} a d \,e^{6}+21 b^{6} d^{2} e^{5}\right ) x^{12}}{12 b x +12 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (20 a^{3} b^{3} e^{7}+105 b^{4} a^{2} d \,e^{6}+126 b^{5} a \,d^{2} e^{5}+35 b^{6} d^{3} e^{4}\right ) x^{11}}{11 b x +11 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (15 a^{4} b^{2} e^{7}+140 a^{3} b^{3} d \,e^{6}+315 b^{4} a^{2} d^{2} e^{5}+210 b^{5} a \,d^{3} e^{4}+35 b^{6} d^{4} e^{3}\right ) x^{10}}{10 b x +10 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (6 a^{5} b \,e^{7}+105 a^{4} b^{2} d \,e^{6}+420 a^{3} b^{3} d^{2} e^{5}+525 b^{4} a^{2} d^{3} e^{4}+210 b^{5} a \,d^{4} e^{3}+21 b^{6} d^{5} e^{2}\right ) x^{9}}{9 b x +9 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (a^{6} e^{7}+42 a^{5} b d \,e^{6}+315 a^{4} b^{2} d^{2} e^{5}+700 a^{3} b^{3} d^{3} e^{4}+525 b^{4} a^{2} d^{4} e^{3}+126 b^{5} a \,d^{5} e^{2}+7 b^{6} d^{6} e \right ) x^{8}}{8 b x +8 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (7 a^{6} d \,e^{6}+126 a^{5} b \,d^{2} e^{5}+525 a^{4} b^{2} d^{3} e^{4}+700 a^{3} b^{3} d^{4} e^{3}+315 b^{4} a^{2} d^{5} e^{2}+42 b^{5} a \,d^{6} e +b^{6} d^{7}\right ) x^{7}}{7 b x +7 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (21 a^{6} d^{2} e^{5}+210 a^{5} b \,d^{3} e^{4}+525 a^{4} b^{2} d^{4} e^{3}+420 a^{3} b^{3} d^{5} e^{2}+105 b^{4} a^{2} d^{6} e +6 b^{5} a \,d^{7}\right ) x^{6}}{6 b x +6 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (35 a^{6} d^{3} e^{4}+210 a^{5} b \,d^{4} e^{3}+315 a^{4} b^{2} d^{5} e^{2}+140 a^{3} b^{3} d^{6} e +15 b^{4} a^{2} d^{7}\right ) x^{5}}{5 b x +5 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (35 a^{6} d^{4} e^{3}+126 a^{5} b \,d^{5} e^{2}+105 a^{4} b^{2} d^{6} e +20 a^{3} b^{3} d^{7}\right ) x^{4}}{4 b x +4 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (21 a^{6} d^{5} e^{2}+42 a^{5} b \,d^{6} e +15 a^{4} b^{2} d^{7}\right ) x^{3}}{3 b x +3 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (7 a^{6} d^{6} e +6 a^{5} b \,d^{7}\right ) x^{2}}{2 b x +2 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, a^{6} d^{7} x}{b x +a}\)	\(933\)

[In]

int((b*x+a)*(e*x+d)^7*(b^2*x^2+2*a*b*x+a^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/24024*x*(1716*b^6*e^7*x^13+11088*a*b^5*e^7*x^12+12936*b^6*d*e^6*x^12+30030*a^2*b^4*e^7*x^11+84084*a*b^5*d*e^

6*x^11+42042*b^6*d^2*e^5*x^11+43680*a^3*b^3*e^7*x^10+229320*a^2*b^4*d*e^6*x^10+275184*a*b^5*d^2*e^5*x^10+76440

*b^6*d^3*e^4*x^10+36036*a^4*b^2*e^7*x^9+336336*a^3*b^3*d*e^6*x^9+756756*a^2*b^4*d^2*e^5*x^9+504504*a*b^5*d^3*e

^4*x^9+84084*b^6*d^4*e^3*x^9+16016*a^5*b*e^7*x^8+280280*a^4*b^2*d*e^6*x^8+1121120*a^3*b^3*d^2*e^5*x^8+1401400*

a^2*b^4*d^3*e^4*x^8+560560*a*b^5*d^4*e^3*x^8+56056*b^6*d^5*e^2*x^8+3003*a^6*e^7*x^7+126126*a^5*b*d*e^6*x^7+945

945*a^4*b^2*d^2*e^5*x^7+2102100*a^3*b^3*d^3*e^4*x^7+1576575*a^2*b^4*d^4*e^3*x^7+378378*a*b^5*d^5*e^2*x^7+21021

*b^6*d^6*e*x^7+24024*a^6*d*e^6*x^6+432432*a^5*b*d^2*e^5*x^6+1801800*a^4*b^2*d^3*e^4*x^6+2402400*a^3*b^3*d^4*e^

3*x^6+1081080*a^2*b^4*d^5*e^2*x^6+144144*a*b^5*d^6*e*x^6+3432*b^6*d^7*x^6+84084*a^6*d^2*e^5*x^5+840840*a^5*b*d

^3*e^4*x^5+2102100*a^4*b^2*d^4*e^3*x^5+1681680*a^3*b^3*d^5*e^2*x^5+420420*a^2*b^4*d^6*e*x^5+24024*a*b^5*d^7*x^

5+168168*a^6*d^3*e^4*x^4+1009008*a^5*b*d^4*e^3*x^4+1513512*a^4*b^2*d^5*e^2*x^4+672672*a^3*b^3*d^6*e*x^4+72072*

a^2*b^4*d^7*x^4+210210*a^6*d^4*e^3*x^3+756756*a^5*b*d^5*e^2*x^3+630630*a^4*b^2*d^6*e*x^3+120120*a^3*b^3*d^7*x^

3+168168*a^6*d^5*e^2*x^2+336336*a^5*b*d^6*e*x^2+120120*a^4*b^2*d^7*x^2+84084*a^6*d^6*e*x+72072*a^5*b*d^7*x+240

24*a^6*d^7)*((b*x+a)^2)^(5/2)/(b*x+a)^5

Fricas [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 706 vs. \(2 (271) = 542\).

Time = 0.39 (sec) , antiderivative size = 706, normalized size of antiderivative = 1.95 \[ \int (a+b x) (d+e x)^7 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {1}{14} \, b^{6} e^{7} x^{14} + a^{6} d^{7} x + \frac {1}{13} \, {\left (7 \, b^{6} d e^{6} + 6 \, a b^{5} e^{7}\right )} x^{13} + \frac {1}{4} \, {\left (7 \, b^{6} d^{2} e^{5} + 14 \, a b^{5} d e^{6} + 5 \, a^{2} b^{4} e^{7}\right )} x^{12} + \frac {1}{11} \, {\left (35 \, b^{6} d^{3} e^{4} + 126 \, a b^{5} d^{2} e^{5} + 105 \, a^{2} b^{4} d e^{6} + 20 \, a^{3} b^{3} e^{7}\right )} x^{11} + \frac {1}{2} \, {\left (7 \, b^{6} d^{4} e^{3} + 42 \, a b^{5} d^{3} e^{4} + 63 \, a^{2} b^{4} d^{2} e^{5} + 28 \, a^{3} b^{3} d e^{6} + 3 \, a^{4} b^{2} e^{7}\right )} x^{10} + \frac {1}{3} \, {\left (7 \, b^{6} d^{5} e^{2} + 70 \, a b^{5} d^{4} e^{3} + 175 \, a^{2} b^{4} d^{3} e^{4} + 140 \, a^{3} b^{3} d^{2} e^{5} + 35 \, a^{4} b^{2} d e^{6} + 2 \, a^{5} b e^{7}\right )} x^{9} + \frac {1}{8} \, {\left (7 \, b^{6} d^{6} e + 126 \, a b^{5} d^{5} e^{2} + 525 \, a^{2} b^{4} d^{4} e^{3} + 700 \, a^{3} b^{3} d^{3} e^{4} + 315 \, a^{4} b^{2} d^{2} e^{5} + 42 \, a^{5} b d e^{6} + a^{6} e^{7}\right )} x^{8} + \frac {1}{7} \, {\left (b^{6} d^{7} + 42 \, a b^{5} d^{6} e + 315 \, a^{2} b^{4} d^{5} e^{2} + 700 \, a^{3} b^{3} d^{4} e^{3} + 525 \, a^{4} b^{2} d^{3} e^{4} + 126 \, a^{5} b d^{2} e^{5} + 7 \, a^{6} d e^{6}\right )} x^{7} + \frac {1}{2} \, {\left (2 \, a b^{5} d^{7} + 35 \, a^{2} b^{4} d^{6} e + 140 \, a^{3} b^{3} d^{5} e^{2} + 175 \, a^{4} b^{2} d^{4} e^{3} + 70 \, a^{5} b d^{3} e^{4} + 7 \, a^{6} d^{2} e^{5}\right )} x^{6} + {\left (3 \, a^{2} b^{4} d^{7} + 28 \, a^{3} b^{3} d^{6} e + 63 \, a^{4} b^{2} d^{5} e^{2} + 42 \, a^{5} b d^{4} e^{3} + 7 \, a^{6} d^{3} e^{4}\right )} x^{5} + \frac {1}{4} \, {\left (20 \, a^{3} b^{3} d^{7} + 105 \, a^{4} b^{2} d^{6} e + 126 \, a^{5} b d^{5} e^{2} + 35 \, a^{6} d^{4} e^{3}\right )} x^{4} + {\left (5 \, a^{4} b^{2} d^{7} + 14 \, a^{5} b d^{6} e + 7 \, a^{6} d^{5} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (6 \, a^{5} b d^{7} + 7 \, a^{6} d^{6} e\right )} x^{2} \]

[In]

integrate((b*x+a)*(e*x+d)^7*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")

[Out]

1/14*b^6*e^7*x^14 + a^6*d^7*x + 1/13*(7*b^6*d*e^6 + 6*a*b^5*e^7)*x^13 + 1/4*(7*b^6*d^2*e^5 + 14*a*b^5*d*e^6 +

5*a^2*b^4*e^7)*x^12 + 1/11*(35*b^6*d^3*e^4 + 126*a*b^5*d^2*e^5 + 105*a^2*b^4*d*e^6 + 20*a^3*b^3*e^7)*x^11 + 1/

2*(7*b^6*d^4*e^3 + 42*a*b^5*d^3*e^4 + 63*a^2*b^4*d^2*e^5 + 28*a^3*b^3*d*e^6 + 3*a^4*b^2*e^7)*x^10 + 1/3*(7*b^6

*d^5*e^2 + 70*a*b^5*d^4*e^3 + 175*a^2*b^4*d^3*e^4 + 140*a^3*b^3*d^2*e^5 + 35*a^4*b^2*d*e^6 + 2*a^5*b*e^7)*x^9

+ 1/8*(7*b^6*d^6*e + 126*a*b^5*d^5*e^2 + 525*a^2*b^4*d^4*e^3 + 700*a^3*b^3*d^3*e^4 + 315*a^4*b^2*d^2*e^5 + 42*

a^5*b*d*e^6 + a^6*e^7)*x^8 + 1/7*(b^6*d^7 + 42*a*b^5*d^6*e + 315*a^2*b^4*d^5*e^2 + 700*a^3*b^3*d^4*e^3 + 525*a

^4*b^2*d^3*e^4 + 126*a^5*b*d^2*e^5 + 7*a^6*d*e^6)*x^7 + 1/2*(2*a*b^5*d^7 + 35*a^2*b^4*d^6*e + 140*a^3*b^3*d^5*

e^2 + 175*a^4*b^2*d^4*e^3 + 70*a^5*b*d^3*e^4 + 7*a^6*d^2*e^5)*x^6 + (3*a^2*b^4*d^7 + 28*a^3*b^3*d^6*e + 63*a^4

*b^2*d^5*e^2 + 42*a^5*b*d^4*e^3 + 7*a^6*d^3*e^4)*x^5 + 1/4*(20*a^3*b^3*d^7 + 105*a^4*b^2*d^6*e + 126*a^5*b*d^5

*e^2 + 35*a^6*d^4*e^3)*x^4 + (5*a^4*b^2*d^7 + 14*a^5*b*d^6*e + 7*a^6*d^5*e^2)*x^3 + 1/2*(6*a^5*b*d^7 + 7*a^6*d

^6*e)*x^2

Sympy [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 132085 vs. \(2 (265) = 530\).

Time = 2.14 (sec) , antiderivative size = 132085, normalized size of antiderivative = 364.88 \[ \int (a+b x) (d+e x)^7 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\text {Too large to display} \]

[In]

integrate((b*x+a)*(e*x+d)**7*(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Piecewise((sqrt(a**2 + 2*a*b*x + b**2*x**2)*(b**5*e**7*x**13/14 + x**12*(71*a*b**6*e**7/14 + 7*b**7*d*e**6)/(1

3*b**2) + x**11*(281*a**2*b**5*e**7/14 + 49*a*b**6*d*e**6 - 25*a*(71*a*b**6*e**7/14 + 7*b**7*d*e**6)/(13*b) +

21*b**7*d**2*e**5)/(12*b**2) + x**10*(35*a**3*b**4*e**7 + 147*a**2*b**5*d*e**6 - 12*a**2*(71*a*b**6*e**7/14 +

7*b**7*d*e**6)/(13*b**2) + 147*a*b**6*d**2*e**5 - 23*a*(281*a**2*b**5*e**7/14 + 49*a*b**6*d*e**6 - 25*a*(71*a*

b**6*e**7/14 + 7*b**7*d*e**6)/(13*b) + 21*b**7*d**2*e**5)/(12*b) + 35*b**7*d**3*e**4)/(11*b**2) + x**9*(35*a**

4*b**3*e**7 + 245*a**3*b**4*d*e**6 + 441*a**2*b**5*d**2*e**5 - 11*a**2*(281*a**2*b**5*e**7/14 + 49*a*b**6*d*e*

*6 - 25*a*(71*a*b**6*e**7/14 + 7*b**7*d*e**6)/(13*b) + 21*b**7*d**2*e**5)/(12*b**2) + 245*a*b**6*d**3*e**4 - 2

1*a*(35*a**3*b**4*e**7 + 147*a**2*b**5*d*e**6 - 12*a**2*(71*a*b**6*e**7/14 + 7*b**7*d*e**6)/(13*b**2) + 147*a*

b**6*d**2*e**5 - 23*a*(281*a**2*b**5*e**7/14 + 49*a*b**6*d*e**6 - 25*a*(71*a*b**6*e**7/14 + 7*b**7*d*e**6)/(13

*b) + 21*b**7*d**2*e**5)/(12*b) + 35*b**7*d**3*e**4)/(11*b) + 35*b**7*d**4*e**3)/(10*b**2) + x**8*(21*a**5*b**

2*e**7 + 245*a**4*b**3*d*e**6 + 735*a**3*b**4*d**2*e**5 + 735*a**2*b**5*d**3*e**4 - 10*a**2*(35*a**3*b**4*e**7

+ 147*a**2*b**5*d*e**6 - 12*a**2*(71*a*b**6*e**7/14 + 7*b**7*d*e**6)/(13*b**2) + 147*a*b**6*d**2*e**5 - 23*a*

(281*a**2*b**5*e**7/14 + 49*a*b**6*d*e**6 - 25*a*(71*a*b**6*e**7/14 + 7*b**7*d*e**6)/(13*b) + 21*b**7*d**2*e**

5)/(12*b) + 35*b**7*d**3*e**4)/(11*b**2) + 245*a*b**6*d**4*e**3 - 19*a*(35*a**4*b**3*e**7 + 245*a**3*b**4*d*e*

*6 + 441*a**2*b**5*d**2*e**5 - 11*a**2*(281*a**2*b**5*e**7/14 + 49*a*b**6*d*e**6 - 25*a*(71*a*b**6*e**7/14 + 7

*b**7*d*e**6)/(13*b) + 21*b**7*d**2*e**5)/(12*b**2) + 245*a*b**6*d**3*e**4 - 21*a*(35*a**3*b**4*e**7 + 147*a**

2*b**5*d*e**6 - 12*a**2*(71*a*b**6*e**7/14 + 7*b**7*d*e**6)/(13*b**2) + 147*a*b**6*d**2*e**5 - 23*a*(281*a**2*

b**5*e**7/14 + 49*a*b**6*d*e**6 - 25*a*(71*a*b**6*e**7/14 + 7*b**7*d*e**6)/(13*b) + 21*b**7*d**2*e**5)/(12*b)

+ 35*b**7*d**3*e**4)/(11*b) + 35*b**7*d**4*e**3)/(10*b) + 21*b**7*d**5*e**2)/(9*b**2) + x**7*(7*a**6*b*e**7 +

147*a**5*b**2*d*e**6 + 735*a**4*b**3*d**2*e**5 + 1225*a**3*b**4*d**3*e**4 + 735*a**2*b**5*d**4*e**3 - 9*a**2*(

35*a**4*b**3*e**7 + 245*a**3*b**4*d*e**6 + 441*a**2*b**5*d**2*e**5 - 11*a**2*(281*a**2*b**5*e**7/14 + 49*a*b**

6*d*e**6 - 25*a*(71*a*b**6*e**7/14 + 7*b**7*d*e**6)/(13*b) + 21*b**7*d**2*e**5)/(12*b**2) + 245*a*b**6*d**3*e*

*4 - 21*a*(35*a**3*b**4*e**7 + 147*a**2*b**5*d*e**6 - 12*a**2*(71*a*b**6*e**7/14 + 7*b**7*d*e**6)/(13*b**2) +

147*a*b**6*d**2*e**5 - 23*a*(281*a**2*b**5*e**7/14 + 49*a*b**6*d*e**6 - 25*a*(71*a*b**6*e**7/14 + 7*b**7*d*e**

6)/(13*b) + 21*b**7*d**2*e**5)/(12*b) + 35*b**7*d**3*e**4)/(11*b) + 35*b**7*d**4*e**3)/(10*b**2) + 147*a*b**6*

d**5*e**2 - 17*a*(21*a**5*b**2*e**7 + 245*a**4*b**3*d*e**6 + 735*a**3*b**4*d**2*e**5 + 735*a**2*b**5*d**3*e**4

- 10*a**2*(35*a**3*b**4*e**7 + 147*a**2*b**5*d*e**6 - 12*a**2*(71*a*b**6*e**7/14 + 7*b**7*d*e**6)/(13*b**2) +

147*a*b**6*d**2*e**5 - 23*a*(281*a**2*b**5*e**7/14 + 49*a*b**6*d*e**6 - 25*a*(71*a*b**6*e**7/14 + 7*b**7*d*e*

*6)/(13*b) + 21*b**7*d**2*e**5)/(12*b) + 35*b**7*d**3*e**4)/(11*b**2) + 245*a*b**6*d**4*e**3 - 19*a*(35*a**4*b

**3*e**7 + 245*a**3*b**4*d*e**6 + 441*a**2*b**5*d**2*e**5 - 11*a**2*(281*a**2*b**5*e**7/14 + 49*a*b**6*d*e**6

- 25*a*(71*a*b**6*e**7/14 + 7*b**7*d*e**6)/(13*b) + 21*b**7*d**2*e**5)/(12*b**2) + 245*a*b**6*d**3*e**4 - 21*a

*(35*a**3*b**4*e**7 + 147*a**2*b**5*d*e**6 - 12*a**2*(71*a*b**6*e**7/14 + 7*b**7*d*e**6)/(13*b**2) + 147*a*b**

6*d**2*e**5 - 23*a*(281*a**2*b**5*e**7/14 + 49*a*b**6*d*e**6 - 25*a*(71*a*b**6*e**7/14 + 7*b**7*d*e**6)/(13*b)

+ 21*b**7*d**2*e**5)/(12*b) + 35*b**7*d**3*e**4)/(11*b) + 35*b**7*d**4*e**3)/(10*b) + 21*b**7*d**5*e**2)/(9*b

) + 7*b**7*d**6*e)/(8*b**2) + x**6*(a**7*e**7 + 49*a**6*b*d*e**6 + 441*a**5*b**2*d**2*e**5 + 1225*a**4*b**3*d*

*3*e**4 + 1225*a**3*b**4*d**4*e**3 + 441*a**2*b**5*d**5*e**2 - 8*a**2*(21*a**5*b**2*e**7 + 245*a**4*b**3*d*e**

6 + 735*a**3*b**4*d**2*e**5 + 735*a**2*b**5*d**3*e**4 - 10*a**2*(35*a**3*b**4*e**7 + 147*a**2*b**5*d*e**6 - 12

*a**2*(71*a*b**6*e**7/14 + 7*b**7*d*e**6)/(13*b**2) + 147*a*b**6*d**2*e**5 - 23*a*(281*a**2*b**5*e**7/14 + 49*

a*b**6*d*e**6 - 25*a*(71*a*b**6*e**7/14 + 7*b**7*d*e**6)/(13*b) + 21*b**7*d**2*e**5)/(12*b) + 35*b**7*d**3*e**

4)/(11*b**2) + 245*a*b**6*d**4*e**3 - 19*a*(35*a**4*b**3*e**7 + 245*a**3*b**4*d*e**6 + 441*a**2*b**5*d**2*e**5

- 11*a**2*(281*a**2*b**5*e**7/14 + 49*a*b**6*d*e**6 - 25*a*(71*a*b**6*e**7/14 + 7*b**7*d*e**6)/(13*b) + 21*b*

*7*d**2*e**5)/(12*b**2) + 245*a*b**6*d**3*e**4 - 21*a*(35*a**3*b**4*e**7 + 147*a**2*b**5*d*e**6 - 12*a**2*(71*

a*b**6*e**7/14 + 7*b**7*d*e**6)/(13*b**2) + 147*a*b**6*d**2*e**5 - 23*a*(281*a**2*b**5*e**7/14 + 49*a*b**6*d*e

**6 - 25*a*(71*a*b**6*e**7/14 + 7*b**7*d*e**6)/(13*b) + 21*b**7*d**2*e**5)/(12*b) + 35*b**7*d**3*e**4)/(11*b)

+ 35*b**7*d**4*e**3)/(10*b) + 21*b**7*d**5*e**2)/(9*b**2) + 49*a*b**6*d**6*e - 15*a*(7*a**6*b*e**7 + 147*a**5*

b**2*d*e**6 + 735*a**4*b**3*d**2*e**5 + 1225*a**3*b**4*d**3*e**4 + 735*a**2*b**5*d**4*e**3 - 9*a**2*(35*a**4*b

**3*e**7 + 245*a**3*b**4*d*e**6 + 441*a**2*b**5*d**2*e**5 - 11*a**2*(281*a**2*b**5*e**7/14 + 49*a*b**6*d*e**6

- 25*a*(71*a*b**6*e**7/14 + 7*b**7*d*e**6)/(13*b) + 21*b**7*d**2*e**5)/(12*b**2) + 245*a*b**6*d**3*e**4 - 21*a

*(35*a**3*b**4*e**7 + 147*a**2*b**5*d*e**6 - 12*a**2*(71*a*b**6*e**7/14 + 7*b**7*d*e**6)/(13*b**2) + 147*a*b**

6*d**2*e**5 - 23*a*(281*a**2*b**5*e**7/14 + 49*a*b**6*d*e**6 - 25*a*(71*a*b**6*e**7/14 + 7*b**7*d*e**6)/(13*b)

+ 21*b**7*d**2*e**5)/(12*b) + 35*b**7*d**3*e**4)/(11*b) + 35*b**7*d**4*e**3)/(10*b**2) + 147*a*b**6*d**5*e**2

- 17*a*(21*a**5*b**2*e**7 + 245*a**4*b**3*d*e**6 + 735*a**3*b**4*d**2*e**5 + 735*a**2*b**5*d**3*e**4 - 10*a**

2*(35*a**3*b**4*e**7 + 147*a**2*b**5*d*e**6 - 12*a**2*(71*a*b**6*e**7/14 + 7*b**7*d*e**6)/(13*b**2) + 147*a*b*

*6*d**2*e**5 - 23*a*(281*a**2*b**5*e**7/14 + 49*a*b**6*d*e**6 - 25*a*(71*a*b**6*e**7/14 + 7*b**7*d*e**6)/(13*b

) + 21*b**7*d**2*e**5)/(12*b) + 35*b**7*d**3*e**4)/(11*b**2) + 245*a*b**6*d**4*e**3 - 19*a*(35*a**4*b**3*e**7

+ 245*a**3*b**4*d*e**6 + 441*a**2*b**5*d**2*e**5 - 11*a**2*(281*a**2*b**5*e**7/14 + 49*a*b**6*d*e**6 - 25*a*(7

1*a*b**6*e**7/14 + 7*b**7*d*e**6)/(13*b) + 21*b**7*d**2*e**5)/(12*b**2) + 245*a*b**6*d**3*e**4 - 21*a*(35*a**3

*b**4*e**7 + 147*a**2*b**5*d*e**6 - 12*a**2*(71*a*b**6*e**7/14 + 7*b**7*d*e**6)/(13*b**2) + 147*a*b**6*d**2*e*

*5 - 23*a*(281*a**2*b**5*e**7/14 + 49*a*b**6*d*e**6 - 25*a*(71*a*b**6*e**7/14 + 7*b**7*d*e**6)/(13*b) + 21*b**

7*d**2*e**5)/(12*b) + 35*b**7*d**3*e**4)/(11*b) + 35*b**7*d**4*e**3)/(10*b) + 21*b**7*d**5*e**2)/(9*b) + 7*b**

7*d**6*e)/(8*b) + b**7*d**7)/(7*b**2) + x**5*(7*a**7*d*e**6 + 147*a**6*b*d**2*e**5 + 735*a**5*b**2*d**3*e**4 +

1225*a**4*b**3*d**4*e**3 + 735*a**3*b**4*d**5*e**2 + 147*a**2*b**5*d**6*e - 7*a**2*(7*a**6*b*e**7 + 147*a**5*