3.22.12 \(\int (a+b x) \sqrt {d+e x} (a^2+2 a b x+b^2 x^2)^{5/2} \, dx\) [2112]

Optimal. Leaf size=376 \[ \frac {2 (b d-a e)^6 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x)}-\frac {12 b (b d-a e)^5 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x)}+\frac {30 b^2 (b d-a e)^4 (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x)}-\frac {40 b^3 (b d-a e)^3 (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^7 (a+b x)}+\frac {30 b^4 (b d-a e)^2 (d+e x)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x)}-\frac {12 b^5 (b d-a e) (d+e x)^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^7 (a+b x)}+\frac {2 b^6 (d+e x)^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}}{15 e^7 (a+b x)} \]

[Out]

2/3*(-a*e+b*d)^6*(e*x+d)^(3/2)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)-12/5*b*(-a*e+b*d)^5*(e*x+d)^(5/2)*((b*x+a)^2)^(1/
2)/e^7/(b*x+a)+30/7*b^2*(-a*e+b*d)^4*(e*x+d)^(7/2)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)-40/9*b^3*(-a*e+b*d)^3*(e*x+d)
^(9/2)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)+30/11*b^4*(-a*e+b*d)^2*(e*x+d)^(11/2)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)-12/13
*b^5*(-a*e+b*d)*(e*x+d)^(13/2)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)+2/15*b^6*(e*x+d)^(15/2)*((b*x+a)^2)^(1/2)/e^7/(b*
x+a)

________________________________________________________________________________________

Rubi [A]
time = 0.09, antiderivative size = 376, normalized size of antiderivative = 1.00, 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 = {784, 21, 45} \begin {gather*} \frac {30 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^4}{7 e^7 (a+b x)}-\frac {12 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^5}{5 e^7 (a+b x)}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^6}{3 e^7 (a+b x)}+\frac {2 b^6 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{15/2}}{15 e^7 (a+b x)}-\frac {12 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{13/2} (b d-a e)}{13 e^7 (a+b x)}+\frac {30 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)^2}{11 e^7 (a+b x)}-\frac {40 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^3}{9 e^7 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(b*d - a*e)^6*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^7*(a + b*x)) - (12*b*(b*d - a*e)^5*(d + e
*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^7*(a + b*x)) + (30*b^2*(b*d - a*e)^4*(d + e*x)^(7/2)*Sqrt[a^2 +
2*a*b*x + b^2*x^2])/(7*e^7*(a + b*x)) - (40*b^3*(b*d - a*e)^3*(d + e*x)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(
9*e^7*(a + b*x)) + (30*b^4*(b*d - a*e)^2*(d + e*x)^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^7*(a + b*x)) -
(12*b^5*(b*d - a*e)*(d + e*x)^(13/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*e^7*(a + b*x)) + (2*b^6*(d + e*x)^(15/
2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(15*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*} \int (a+b x) \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int (a+b x) \left (a b+b^2 x\right )^5 \sqrt {d+e x} \, 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 \sqrt {d+e x} \, 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 \sqrt {d+e x}}{e^6}-\frac {6 b (b d-a e)^5 (d+e x)^{3/2}}{e^6}+\frac {15 b^2 (b d-a e)^4 (d+e x)^{5/2}}{e^6}-\frac {20 b^3 (b d-a e)^3 (d+e x)^{7/2}}{e^6}+\frac {15 b^4 (b d-a e)^2 (d+e x)^{9/2}}{e^6}-\frac {6 b^5 (b d-a e) (d+e x)^{11/2}}{e^6}+\frac {b^6 (d+e x)^{13/2}}{e^6}\right ) \, dx}{a b+b^2 x}\\ &=\frac {2 (b d-a e)^6 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x)}-\frac {12 b (b d-a e)^5 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x)}+\frac {30 b^2 (b d-a e)^4 (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x)}-\frac {40 b^3 (b d-a e)^3 (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^7 (a+b x)}+\frac {30 b^4 (b d-a e)^2 (d+e x)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x)}-\frac {12 b^5 (b d-a e) (d+e x)^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^7 (a+b x)}+\frac {2 b^6 (d+e x)^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}}{15 e^7 (a+b x)}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.06, size = 309, normalized size = 0.82 \begin {gather*} \frac {2 \sqrt {(a+b x)^2} (d+e x)^{3/2} \left (15015 a^6 e^6+18018 a^5 b e^5 (-2 d+3 e x)+6435 a^4 b^2 e^4 \left (8 d^2-12 d e x+15 e^2 x^2\right )+2860 a^3 b^3 e^3 \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )+195 a^2 b^4 e^2 \left (128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right )+30 a b^5 e \left (-256 d^5+384 d^4 e x-480 d^3 e^2 x^2+560 d^2 e^3 x^3-630 d e^4 x^4+693 e^5 x^5\right )+b^6 \left (1024 d^6-1536 d^5 e x+1920 d^4 e^2 x^2-2240 d^3 e^3 x^3+2520 d^2 e^4 x^4-2772 d e^5 x^5+3003 e^6 x^6\right )\right )}{45045 e^7 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[(a + b*x)^2]*(d + e*x)^(3/2)*(15015*a^6*e^6 + 18018*a^5*b*e^5*(-2*d + 3*e*x) + 6435*a^4*b^2*e^4*(8*d^2
- 12*d*e*x + 15*e^2*x^2) + 2860*a^3*b^3*e^3*(-16*d^3 + 24*d^2*e*x - 30*d*e^2*x^2 + 35*e^3*x^3) + 195*a^2*b^4*
e^2*(128*d^4 - 192*d^3*e*x + 240*d^2*e^2*x^2 - 280*d*e^3*x^3 + 315*e^4*x^4) + 30*a*b^5*e*(-256*d^5 + 384*d^4*e
*x - 480*d^3*e^2*x^2 + 560*d^2*e^3*x^3 - 630*d*e^4*x^4 + 693*e^5*x^5) + b^6*(1024*d^6 - 1536*d^5*e*x + 1920*d^
4*e^2*x^2 - 2240*d^3*e^3*x^3 + 2520*d^2*e^4*x^4 - 2772*d*e^5*x^5 + 3003*e^6*x^6)))/(45045*e^7*(a + b*x))

________________________________________________________________________________________

Maple [A]
time = 0.04, size = 393, normalized size = 1.05

method	result	size
gosper	\(\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (3003 b^{6} e^{6} x^{6}+20790 a \,b^{5} e^{6} x^{5}-2772 b^{6} d \,e^{5} x^{5}+61425 a^{2} b^{4} e^{6} x^{4}-18900 a \,b^{5} d \,e^{5} x^{4}+2520 b^{6} d^{2} e^{4} x^{4}+100100 a^{3} b^{3} e^{6} x^{3}-54600 a^{2} b^{4} d \,e^{5} x^{3}+16800 a \,b^{5} d^{2} e^{4} x^{3}-2240 b^{6} d^{3} e^{3} x^{3}+96525 a^{4} b^{2} e^{6} x^{2}-85800 a^{3} b^{3} d \,e^{5} x^{2}+46800 a^{2} b^{4} d^{2} e^{4} x^{2}-14400 a \,b^{5} d^{3} e^{3} x^{2}+1920 b^{6} d^{4} e^{2} x^{2}+54054 a^{5} b \,e^{6} x -77220 a^{4} b^{2} d \,e^{5} x +68640 a^{3} b^{3} d^{2} e^{4} x -37440 a^{2} b^{4} d^{3} e^{3} x +11520 a \,b^{5} d^{4} e^{2} x -1536 b^{6} d^{5} e x +15015 e^{6} a^{6}-36036 d \,e^{5} a^{5} b +51480 d^{2} e^{4} a^{4} b^{2}-45760 d^{3} e^{3} a^{3} b^{3}+24960 d^{4} e^{2} a^{2} b^{4}-7680 d^{5} e a \,b^{5}+1024 d^{6} b^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{45045 e^{7} \left (b x +a \right )^{5}}\)	\(393\)
default	\(\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (3003 b^{6} e^{6} x^{6}+20790 a \,b^{5} e^{6} x^{5}-2772 b^{6} d \,e^{5} x^{5}+61425 a^{2} b^{4} e^{6} x^{4}-18900 a \,b^{5} d \,e^{5} x^{4}+2520 b^{6} d^{2} e^{4} x^{4}+100100 a^{3} b^{3} e^{6} x^{3}-54600 a^{2} b^{4} d \,e^{5} x^{3}+16800 a \,b^{5} d^{2} e^{4} x^{3}-2240 b^{6} d^{3} e^{3} x^{3}+96525 a^{4} b^{2} e^{6} x^{2}-85800 a^{3} b^{3} d \,e^{5} x^{2}+46800 a^{2} b^{4} d^{2} e^{4} x^{2}-14400 a \,b^{5} d^{3} e^{3} x^{2}+1920 b^{6} d^{4} e^{2} x^{2}+54054 a^{5} b \,e^{6} x -77220 a^{4} b^{2} d \,e^{5} x +68640 a^{3} b^{3} d^{2} e^{4} x -37440 a^{2} b^{4} d^{3} e^{3} x +11520 a \,b^{5} d^{4} e^{2} x -1536 b^{6} d^{5} e x +15015 e^{6} a^{6}-36036 d \,e^{5} a^{5} b +51480 d^{2} e^{4} a^{4} b^{2}-45760 d^{3} e^{3} a^{3} b^{3}+24960 d^{4} e^{2} a^{2} b^{4}-7680 d^{5} e a \,b^{5}+1024 d^{6} b^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{45045 e^{7} \left (b x +a \right )^{5}}\)	\(393\)
risch	\(\frac {2 \sqrt {\left (b x +a \right )^{2}}\, \left (3003 b^{6} x^{7} e^{7}+20790 a \,b^{5} e^{7} x^{6}+231 b^{6} d \,e^{6} x^{6}+61425 a^{2} b^{4} e^{7} x^{5}+1890 a \,b^{5} d \,e^{6} x^{5}-252 b^{6} d^{2} e^{5} x^{5}+100100 a^{3} b^{3} e^{7} x^{4}+6825 a^{2} b^{4} d \,e^{6} x^{4}-2100 a \,b^{5} d^{2} e^{5} x^{4}+280 b^{6} d^{3} e^{4} x^{4}+96525 a^{4} b^{2} e^{7} x^{3}+14300 a^{3} b^{3} d \,e^{6} x^{3}-7800 a^{2} b^{4} d^{2} e^{5} x^{3}+2400 a \,b^{5} d^{3} e^{4} x^{3}-320 b^{6} d^{4} e^{3} x^{3}+54054 a^{5} b \,e^{7} x^{2}+19305 a^{4} b^{2} d \,e^{6} x^{2}-17160 a^{3} b^{3} d^{2} e^{5} x^{2}+9360 a^{2} b^{4} d^{3} e^{4} x^{2}-2880 a \,b^{5} d^{4} e^{3} x^{2}+384 b^{6} d^{5} e^{2} x^{2}+15015 a^{6} e^{7} x +18018 a^{5} b d \,e^{6} x -25740 a^{4} b^{2} d^{2} e^{5} x +22880 a^{3} b^{3} d^{3} e^{4} x -12480 a^{2} b^{4} d^{4} e^{3} x +3840 a \,b^{5} d^{5} e^{2} x -512 b^{6} d^{6} e x +15015 d \,a^{6} e^{6}-36036 a^{5} b \,d^{2} e^{5}+51480 a^{4} b^{2} d^{3} e^{4}-45760 a^{3} b^{3} d^{4} e^{3}+24960 a^{2} b^{4} d^{5} e^{2}-7680 a \,b^{5} d^{6} e +1024 b^{6} d^{7}\right ) \sqrt {e x +d}}{45045 \left (b x +a \right ) e^{7}}\)	\(498\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)*(e*x+d)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/45045*(e*x+d)^(3/2)*(3003*b^6*e^6*x^6+20790*a*b^5*e^6*x^5-2772*b^6*d*e^5*x^5+61425*a^2*b^4*e^6*x^4-18900*a*b
^5*d*e^5*x^4+2520*b^6*d^2*e^4*x^4+100100*a^3*b^3*e^6*x^3-54600*a^2*b^4*d*e^5*x^3+16800*a*b^5*d^2*e^4*x^3-2240*
b^6*d^3*e^3*x^3+96525*a^4*b^2*e^6*x^2-85800*a^3*b^3*d*e^5*x^2+46800*a^2*b^4*d^2*e^4*x^2-14400*a*b^5*d^3*e^3*x^
2+1920*b^6*d^4*e^2*x^2+54054*a^5*b*e^6*x-77220*a^4*b^2*d*e^5*x+68640*a^3*b^3*d^2*e^4*x-37440*a^2*b^4*d^3*e^3*x
+11520*a*b^5*d^4*e^2*x-1536*b^6*d^5*e*x+15015*a^6*e^6-36036*a^5*b*d*e^5+51480*a^4*b^2*d^2*e^4-45760*a^3*b^3*d^
3*e^3+24960*a^2*b^4*d^4*e^2-7680*a*b^5*d^5*e+1024*b^6*d^6)*((b*x+a)^2)^(5/2)/e^7/(b*x+a)^5

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 710 vs. \(2 (277) = 554\).
time = 0.29, size = 710, normalized size = 1.89 \begin {gather*} \frac {2}{9009} \, {\left (693 \, b^{5} x^{6} e^{6} - 256 \, b^{5} d^{6} + 1664 \, a b^{4} d^{5} e - 4576 \, a^{2} b^{3} d^{4} e^{2} + 6864 \, a^{3} b^{2} d^{3} e^{3} - 6006 \, a^{4} b d^{2} e^{4} + 3003 \, a^{5} d e^{5} + 63 \, {\left (b^{5} d e^{5} + 65 \, a b^{4} e^{6}\right )} x^{5} - 35 \, {\left (2 \, b^{5} d^{2} e^{4} - 13 \, a b^{4} d e^{5} - 286 \, a^{2} b^{3} e^{6}\right )} x^{4} + 10 \, {\left (8 \, b^{5} d^{3} e^{3} - 52 \, a b^{4} d^{2} e^{4} + 143 \, a^{2} b^{3} d e^{5} + 1287 \, a^{3} b^{2} e^{6}\right )} x^{3} - 3 \, {\left (32 \, b^{5} d^{4} e^{2} - 208 \, a b^{4} d^{3} e^{3} + 572 \, a^{2} b^{3} d^{2} e^{4} - 858 \, a^{3} b^{2} d e^{5} - 3003 \, a^{4} b e^{6}\right )} x^{2} + {\left (128 \, b^{5} d^{5} e - 832 \, a b^{4} d^{4} e^{2} + 2288 \, a^{2} b^{3} d^{3} e^{3} - 3432 \, a^{3} b^{2} d^{2} e^{4} + 3003 \, a^{4} b d e^{5} + 3003 \, a^{5} e^{6}\right )} x\right )} \sqrt {x e + d} a e^{\left (-6\right )} + \frac {2}{45045} \, {\left (3003 \, b^{5} x^{7} e^{7} + 1024 \, b^{5} d^{7} - 6400 \, a b^{4} d^{6} e + 16640 \, a^{2} b^{3} d^{5} e^{2} - 22880 \, a^{3} b^{2} d^{4} e^{3} + 17160 \, a^{4} b d^{3} e^{4} - 6006 \, a^{5} d^{2} e^{5} + 231 \, {\left (b^{5} d e^{6} + 75 \, a b^{4} e^{7}\right )} x^{6} - 63 \, {\left (4 \, b^{5} d^{2} e^{5} - 25 \, a b^{4} d e^{6} - 650 \, a^{2} b^{3} e^{7}\right )} x^{5} + 70 \, {\left (4 \, b^{5} d^{3} e^{4} - 25 \, a b^{4} d^{2} e^{5} + 65 \, a^{2} b^{3} d e^{6} + 715 \, a^{3} b^{2} e^{7}\right )} x^{4} - 5 \, {\left (64 \, b^{5} d^{4} e^{3} - 400 \, a b^{4} d^{3} e^{4} + 1040 \, a^{2} b^{3} d^{2} e^{5} - 1430 \, a^{3} b^{2} d e^{6} - 6435 \, a^{4} b e^{7}\right )} x^{3} + 3 \, {\left (128 \, b^{5} d^{5} e^{2} - 800 \, a b^{4} d^{4} e^{3} + 2080 \, a^{2} b^{3} d^{3} e^{4} - 2860 \, a^{3} b^{2} d^{2} e^{5} + 2145 \, a^{4} b d e^{6} + 3003 \, a^{5} e^{7}\right )} x^{2} - {\left (512 \, b^{5} d^{6} e - 3200 \, a b^{4} d^{5} e^{2} + 8320 \, a^{2} b^{3} d^{4} e^{3} - 11440 \, a^{3} b^{2} d^{3} e^{4} + 8580 \, a^{4} b d^{2} e^{5} - 3003 \, a^{5} d e^{6}\right )} x\right )} \sqrt {x e + d} b e^{\left (-7\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)*(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

2/9009*(693*b^5*x^6*e^6 - 256*b^5*d^6 + 1664*a*b^4*d^5*e - 4576*a^2*b^3*d^4*e^2 + 6864*a^3*b^2*d^3*e^3 - 6006*
a^4*b*d^2*e^4 + 3003*a^5*d*e^5 + 63*(b^5*d*e^5 + 65*a*b^4*e^6)*x^5 - 35*(2*b^5*d^2*e^4 - 13*a*b^4*d*e^5 - 286*
a^2*b^3*e^6)*x^4 + 10*(8*b^5*d^3*e^3 - 52*a*b^4*d^2*e^4 + 143*a^2*b^3*d*e^5 + 1287*a^3*b^2*e^6)*x^3 - 3*(32*b^
5*d^4*e^2 - 208*a*b^4*d^3*e^3 + 572*a^2*b^3*d^2*e^4 - 858*a^3*b^2*d*e^5 - 3003*a^4*b*e^6)*x^2 + (128*b^5*d^5*e
- 832*a*b^4*d^4*e^2 + 2288*a^2*b^3*d^3*e^3 - 3432*a^3*b^2*d^2*e^4 + 3003*a^4*b*d*e^5 + 3003*a^5*e^6)*x)*sqrt(
x*e + d)*a*e^(-6) + 2/45045*(3003*b^5*x^7*e^7 + 1024*b^5*d^7 - 6400*a*b^4*d^6*e + 16640*a^2*b^3*d^5*e^2 - 2288
0*a^3*b^2*d^4*e^3 + 17160*a^4*b*d^3*e^4 - 6006*a^5*d^2*e^5 + 231*(b^5*d*e^6 + 75*a*b^4*e^7)*x^6 - 63*(4*b^5*d^
2*e^5 - 25*a*b^4*d*e^6 - 650*a^2*b^3*e^7)*x^5 + 70*(4*b^5*d^3*e^4 - 25*a*b^4*d^2*e^5 + 65*a^2*b^3*d*e^6 + 715*
a^3*b^2*e^7)*x^4 - 5*(64*b^5*d^4*e^3 - 400*a*b^4*d^3*e^4 + 1040*a^2*b^3*d^2*e^5 - 1430*a^3*b^2*d*e^6 - 6435*a^
4*b*e^7)*x^3 + 3*(128*b^5*d^5*e^2 - 800*a*b^4*d^4*e^3 + 2080*a^2*b^3*d^3*e^4 - 2860*a^3*b^2*d^2*e^5 + 2145*a^4
*b*d*e^6 + 3003*a^5*e^7)*x^2 - (512*b^5*d^6*e - 3200*a*b^4*d^5*e^2 + 8320*a^2*b^3*d^4*e^3 - 11440*a^3*b^2*d^3*
e^4 + 8580*a^4*b*d^2*e^5 - 3003*a^5*d*e^6)*x)*sqrt(x*e + d)*b*e^(-7)

________________________________________________________________________________________

Fricas [A]
time = 0.47, size = 413, normalized size = 1.10 \begin {gather*} \frac {2}{45045} \, {\left (1024 \, b^{6} d^{7} + {\left (3003 \, b^{6} x^{7} + 20790 \, a b^{5} x^{6} + 61425 \, a^{2} b^{4} x^{5} + 100100 \, a^{3} b^{3} x^{4} + 96525 \, a^{4} b^{2} x^{3} + 54054 \, a^{5} b x^{2} + 15015 \, a^{6} x\right )} e^{7} + {\left (231 \, b^{6} d x^{6} + 1890 \, a b^{5} d x^{5} + 6825 \, a^{2} b^{4} d x^{4} + 14300 \, a^{3} b^{3} d x^{3} + 19305 \, a^{4} b^{2} d x^{2} + 18018 \, a^{5} b d x + 15015 \, a^{6} d\right )} e^{6} - 12 \, {\left (21 \, b^{6} d^{2} x^{5} + 175 \, a b^{5} d^{2} x^{4} + 650 \, a^{2} b^{4} d^{2} x^{3} + 1430 \, a^{3} b^{3} d^{2} x^{2} + 2145 \, a^{4} b^{2} d^{2} x + 3003 \, a^{5} b d^{2}\right )} e^{5} + 40 \, {\left (7 \, b^{6} d^{3} x^{4} + 60 \, a b^{5} d^{3} x^{3} + 234 \, a^{2} b^{4} d^{3} x^{2} + 572 \, a^{3} b^{3} d^{3} x + 1287 \, a^{4} b^{2} d^{3}\right )} e^{4} - 320 \, {\left (b^{6} d^{4} x^{3} + 9 \, a b^{5} d^{4} x^{2} + 39 \, a^{2} b^{4} d^{4} x + 143 \, a^{3} b^{3} d^{4}\right )} e^{3} + 384 \, {\left (b^{6} d^{5} x^{2} + 10 \, a b^{5} d^{5} x + 65 \, a^{2} b^{4} d^{5}\right )} e^{2} - 512 \, {\left (b^{6} d^{6} x + 15 \, a b^{5} d^{6}\right )} e\right )} \sqrt {x e + d} e^{\left (-7\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)*(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

2/45045*(1024*b^6*d^7 + (3003*b^6*x^7 + 20790*a*b^5*x^6 + 61425*a^2*b^4*x^5 + 100100*a^3*b^3*x^4 + 96525*a^4*b
^2*x^3 + 54054*a^5*b*x^2 + 15015*a^6*x)*e^7 + (231*b^6*d*x^6 + 1890*a*b^5*d*x^5 + 6825*a^2*b^4*d*x^4 + 14300*a
^3*b^3*d*x^3 + 19305*a^4*b^2*d*x^2 + 18018*a^5*b*d*x + 15015*a^6*d)*e^6 - 12*(21*b^6*d^2*x^5 + 175*a*b^5*d^2*x
^4 + 650*a^2*b^4*d^2*x^3 + 1430*a^3*b^3*d^2*x^2 + 2145*a^4*b^2*d^2*x + 3003*a^5*b*d^2)*e^5 + 40*(7*b^6*d^3*x^4
+ 60*a*b^5*d^3*x^3 + 234*a^2*b^4*d^3*x^2 + 572*a^3*b^3*d^3*x + 1287*a^4*b^2*d^3)*e^4 - 320*(b^6*d^4*x^3 + 9*a
*b^5*d^4*x^2 + 39*a^2*b^4*d^4*x + 143*a^3*b^3*d^4)*e^3 + 384*(b^6*d^5*x^2 + 10*a*b^5*d^5*x + 65*a^2*b^4*d^5)*e
^2 - 512*(b^6*d^6*x + 15*a*b^5*d^6)*e)*sqrt(x*e + d)*e^(-7)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b x\right ) \sqrt {d + e x} \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(b**2*x**2+2*a*b*x+a**2)**(5/2)*(e*x+d)**(1/2),x)

[Out]

Integral((a + b*x)*sqrt(d + e*x)*((a + b*x)**2)**(5/2), x)

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 970 vs. \(2 (277) = 554\).
time = 2.72, size = 970, normalized size = 2.58 \begin {gather*} \frac {2}{45045} \, {\left (90090 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a^{5} b d e^{\left (-1\right )} \mathrm {sgn}\left (b x + a\right ) + 45045 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} a^{4} b^{2} d e^{\left (-2\right )} \mathrm {sgn}\left (b x + a\right ) + 25740 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} a^{3} b^{3} d e^{\left (-3\right )} \mathrm {sgn}\left (b x + a\right ) + 2145 \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} a^{2} b^{4} d e^{\left (-4\right )} \mathrm {sgn}\left (b x + a\right ) + 390 \, {\left (63 \, {\left (x e + d\right )}^{\frac {11}{2}} - 385 \, {\left (x e + d\right )}^{\frac {9}{2}} d + 990 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {x e + d} d^{5}\right )} a b^{5} d e^{\left (-5\right )} \mathrm {sgn}\left (b x + a\right ) + 15 \, {\left (231 \, {\left (x e + d\right )}^{\frac {13}{2}} - 1638 \, {\left (x e + d\right )}^{\frac {11}{2}} d + 5005 \, {\left (x e + d\right )}^{\frac {9}{2}} d^{2} - 8580 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{3} + 9009 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{4} - 6006 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{5} + 3003 \, \sqrt {x e + d} d^{6}\right )} b^{6} d e^{\left (-6\right )} \mathrm {sgn}\left (b x + a\right ) + 18018 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} a^{5} b e^{\left (-1\right )} \mathrm {sgn}\left (b x + a\right ) + 19305 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} a^{4} b^{2} e^{\left (-2\right )} \mathrm {sgn}\left (b x + a\right ) + 2860 \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} a^{3} b^{3} e^{\left (-3\right )} \mathrm {sgn}\left (b x + a\right ) + 975 \, {\left (63 \, {\left (x e + d\right )}^{\frac {11}{2}} - 385 \, {\left (x e + d\right )}^{\frac {9}{2}} d + 990 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {x e + d} d^{5}\right )} a^{2} b^{4} e^{\left (-4\right )} \mathrm {sgn}\left (b x + a\right ) + 90 \, {\left (231 \, {\left (x e + d\right )}^{\frac {13}{2}} - 1638 \, {\left (x e + d\right )}^{\frac {11}{2}} d + 5005 \, {\left (x e + d\right )}^{\frac {9}{2}} d^{2} - 8580 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{3} + 9009 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{4} - 6006 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{5} + 3003 \, \sqrt {x e + d} d^{6}\right )} a b^{5} e^{\left (-5\right )} \mathrm {sgn}\left (b x + a\right ) + 7 \, {\left (429 \, {\left (x e + d\right )}^{\frac {15}{2}} - 3465 \, {\left (x e + d\right )}^{\frac {13}{2}} d + 12285 \, {\left (x e + d\right )}^{\frac {11}{2}} d^{2} - 25025 \, {\left (x e + d\right )}^{\frac {9}{2}} d^{3} + 32175 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{4} - 27027 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{5} + 15015 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{6} - 6435 \, \sqrt {x e + d} d^{7}\right )} b^{6} e^{\left (-6\right )} \mathrm {sgn}\left (b x + a\right ) + 45045 \, \sqrt {x e + d} a^{6} d \mathrm {sgn}\left (b x + a\right ) + 15015 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a^{6} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)*(e*x+d)^(1/2),x, algorithm="giac")

[Out]

2/45045*(90090*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^5*b*d*e^(-1)*sgn(b*x + a) + 45045*(3*(x*e + d)^(5/2) -
10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^4*b^2*d*e^(-2)*sgn(b*x + a) + 25740*(5*(x*e + d)^(7/2) - 21*(x*
e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*a^3*b^3*d*e^(-3)*sgn(b*x + a) + 2145*(35*(x*e
+ d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4
)*a^2*b^4*d*e^(-4)*sgn(b*x + a) + 390*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 -
1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*a*b^5*d*e^(-5)*sgn(b*x + a) + 15
*(231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 8580*(x*e + d)^(7/2)*d^3 + 9009*
(x*e + d)^(5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*b^6*d*e^(-6)*sgn(b*x + a) + 18018*(3*
(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^5*b*e^(-1)*sgn(b*x + a) + 19305*(5*(x*e + d)^
(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*a^4*b^2*e^(-2)*sgn(b*x + a) + 28
60*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(
x*e + d)*d^4)*a^3*b^3*e^(-3)*sgn(b*x + a) + 975*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(
7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*a^2*b^4*e^(-4)*sgn(b*x
+ a) + 90*(231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 8580*(x*e + d)^(7/2)*d
^3 + 9009*(x*e + d)^(5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*a*b^5*e^(-5)*sgn(b*x + a) +
7*(429*(x*e + d)^(15/2) - 3465*(x*e + d)^(13/2)*d + 12285*(x*e + d)^(11/2)*d^2 - 25025*(x*e + d)^(9/2)*d^3 +
32175*(x*e + d)^(7/2)*d^4 - 27027*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6 - 6435*sqrt(x*e + d)*d^7)*b^
6*e^(-6)*sgn(b*x + a) + 45045*sqrt(x*e + d)*a^6*d*sgn(b*x + a) + 15015*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a
^6*sgn(b*x + a))*e^(-1)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \left (a+b\,x\right )\,\sqrt {d+e\,x}\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)*(d + e*x)^(1/2)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2),x)

[Out]

int((a + b*x)*(d + e*x)^(1/2)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2), x)

________________________________________________________________________________________