3.19.89 \(\int \frac {(a+b x) (a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=368 \[ \frac {10 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^4}{e^7 (a+b x)}-\frac {12 b \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^5}{e^7 (a+b x)}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}{e^7 (a+b x) \sqrt {d+e x}}+\frac {2 b^6 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2}}{11 e^7 (a+b x)}-\frac {4 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)}{3 e^7 (a+b x)}+\frac {30 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^2}{7 e^7 (a+b x)}-\frac {8 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^3}{e^7 (a+b x)} \]

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Rubi [A]  time = 0.14, antiderivative size = 368, 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 = {770, 21, 43} \begin {gather*} \frac {2 b^6 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2}}{11 e^7 (a+b x)}-\frac {4 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)}{3 e^7 (a+b x)}+\frac {30 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^2}{7 e^7 (a+b x)}-\frac {8 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^3}{e^7 (a+b x)}+\frac {10 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^4}{e^7 (a+b x)}-\frac {12 b \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^5}{e^7 (a+b x)}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}{e^7 (a+b x) \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^(3/2),x]

[Out]

(-2*(b*d - a*e)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)*Sqrt[d + e*x]) - (12*b*(b*d - a*e)^5*Sqrt[d +
e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)) + (10*b^2*(b*d - a*e)^4*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x
+ b^2*x^2])/(e^7*(a + b*x)) - (8*b^3*(b*d - a*e)^3*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b
*x)) + (30*b^4*(b*d - a*e)^2*(d + e*x)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^7*(a + b*x)) - (4*b^5*(b*d -
a*e)*(d + e*x)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^7*(a + b*x)) + (2*b^6*(d + e*x)^(11/2)*Sqrt[a^2 + 2*a
*b*x + b^2*x^2])/(11*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 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])

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]

Rubi steps

\begin {align*} \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x) \left (a b+b^2 x\right )^5}{(d+e x)^{3/2}} \, 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 \frac {(a+b x)^6}{(d+e x)^{3/2}} \, 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)^6}{e^6 (d+e x)^{3/2}}-\frac {6 b (b d-a e)^5}{e^6 \sqrt {d+e x}}+\frac {15 b^2 (b d-a e)^4 \sqrt {d+e x}}{e^6}-\frac {20 b^3 (b d-a e)^3 (d+e x)^{3/2}}{e^6}+\frac {15 b^4 (b d-a e)^2 (d+e x)^{5/2}}{e^6}-\frac {6 b^5 (b d-a e) (d+e x)^{7/2}}{e^6}+\frac {b^6 (d+e x)^{9/2}}{e^6}\right ) \, dx}{a b+b^2 x}\\ &=-\frac {2 (b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) \sqrt {d+e x}}-\frac {12 b (b d-a e)^5 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x)}+\frac {10 b^2 (b d-a e)^4 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x)}-\frac {8 b^3 (b d-a e)^3 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x)}+\frac {30 b^4 (b d-a e)^2 (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x)}-\frac {4 b^5 (b d-a e) (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x)}+\frac {2 b^6 (d+e x)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x)}\\ \end {align*}

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Mathematica [A]  time = 0.14, size = 163, normalized size = 0.44 \begin {gather*} \frac {2 \sqrt {(a+b x)^2} \left (-154 b^5 (d+e x)^5 (b d-a e)+495 b^4 (d+e x)^4 (b d-a e)^2-924 b^3 (d+e x)^3 (b d-a e)^3+1155 b^2 (d+e x)^2 (b d-a e)^4-1386 b (d+e x) (b d-a e)^5-231 (b d-a e)^6+21 b^6 (d+e x)^6\right )}{231 e^7 (a+b x) \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^(3/2),x]

[Out]

(2*Sqrt[(a + b*x)^2]*(-231*(b*d - a*e)^6 - 1386*b*(b*d - a*e)^5*(d + e*x) + 1155*b^2*(b*d - a*e)^4*(d + e*x)^2
- 924*b^3*(b*d - a*e)^3*(d + e*x)^3 + 495*b^4*(b*d - a*e)^2*(d + e*x)^4 - 154*b^5*(b*d - a*e)*(d + e*x)^5 + 2
1*b^6*(d + e*x)^6))/(231*e^7*(a + b*x)*Sqrt[d + e*x])

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IntegrateAlgebraic [A]  time = 22.29, size = 466, normalized size = 1.27 \begin {gather*} \frac {2 \sqrt {\frac {(a e+b e x)^2}{e^2}} \left (-231 a^6 e^6+1386 a^5 b e^5 (d+e x)+1386 a^5 b d e^5-3465 a^4 b^2 d^2 e^4+1155 a^4 b^2 e^4 (d+e x)^2-6930 a^4 b^2 d e^4 (d+e x)+4620 a^3 b^3 d^3 e^3+13860 a^3 b^3 d^2 e^3 (d+e x)+924 a^3 b^3 e^3 (d+e x)^3-4620 a^3 b^3 d e^3 (d+e x)^2-3465 a^2 b^4 d^4 e^2-13860 a^2 b^4 d^3 e^2 (d+e x)+6930 a^2 b^4 d^2 e^2 (d+e x)^2+495 a^2 b^4 e^2 (d+e x)^4-2772 a^2 b^4 d e^2 (d+e x)^3+1386 a b^5 d^5 e+6930 a b^5 d^4 e (d+e x)-4620 a b^5 d^3 e (d+e x)^2+2772 a b^5 d^2 e (d+e x)^3+154 a b^5 e (d+e x)^5-990 a b^5 d e (d+e x)^4-231 b^6 d^6-1386 b^6 d^5 (d+e x)+1155 b^6 d^4 (d+e x)^2-924 b^6 d^3 (d+e x)^3+495 b^6 d^2 (d+e x)^4+21 b^6 (d+e x)^6-154 b^6 d (d+e x)^5\right )}{231 e^6 \sqrt {d+e x} (a e+b e x)} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^(3/2),x]

[Out]

(2*Sqrt[(a*e + b*e*x)^2/e^2]*(-231*b^6*d^6 + 1386*a*b^5*d^5*e - 3465*a^2*b^4*d^4*e^2 + 4620*a^3*b^3*d^3*e^3 -
3465*a^4*b^2*d^2*e^4 + 1386*a^5*b*d*e^5 - 231*a^6*e^6 - 1386*b^6*d^5*(d + e*x) + 6930*a*b^5*d^4*e*(d + e*x) -
13860*a^2*b^4*d^3*e^2*(d + e*x) + 13860*a^3*b^3*d^2*e^3*(d + e*x) - 6930*a^4*b^2*d*e^4*(d + e*x) + 1386*a^5*b*
e^5*(d + e*x) + 1155*b^6*d^4*(d + e*x)^2 - 4620*a*b^5*d^3*e*(d + e*x)^2 + 6930*a^2*b^4*d^2*e^2*(d + e*x)^2 - 4
620*a^3*b^3*d*e^3*(d + e*x)^2 + 1155*a^4*b^2*e^4*(d + e*x)^2 - 924*b^6*d^3*(d + e*x)^3 + 2772*a*b^5*d^2*e*(d +
e*x)^3 - 2772*a^2*b^4*d*e^2*(d + e*x)^3 + 924*a^3*b^3*e^3*(d + e*x)^3 + 495*b^6*d^2*(d + e*x)^4 - 990*a*b^5*d
*e*(d + e*x)^4 + 495*a^2*b^4*e^2*(d + e*x)^4 - 154*b^6*d*(d + e*x)^5 + 154*a*b^5*e*(d + e*x)^5 + 21*b^6*(d + e
*x)^6))/(231*e^6*Sqrt[d + e*x]*(a*e + b*e*x))

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fricas [A]  time = 0.41, size = 365, normalized size = 0.99 \begin {gather*} \frac {2 \, {\left (21 \, b^{6} e^{6} x^{6} - 1024 \, b^{6} d^{6} + 5632 \, a b^{5} d^{5} e - 12672 \, a^{2} b^{4} d^{4} e^{2} + 14784 \, a^{3} b^{3} d^{3} e^{3} - 9240 \, a^{4} b^{2} d^{2} e^{4} + 2772 \, a^{5} b d e^{5} - 231 \, a^{6} e^{6} - 14 \, {\left (2 \, b^{6} d e^{5} - 11 \, a b^{5} e^{6}\right )} x^{5} + 5 \, {\left (8 \, b^{6} d^{2} e^{4} - 44 \, a b^{5} d e^{5} + 99 \, a^{2} b^{4} e^{6}\right )} x^{4} - 4 \, {\left (16 \, b^{6} d^{3} e^{3} - 88 \, a b^{5} d^{2} e^{4} + 198 \, a^{2} b^{4} d e^{5} - 231 \, a^{3} b^{3} e^{6}\right )} x^{3} + {\left (128 \, b^{6} d^{4} e^{2} - 704 \, a b^{5} d^{3} e^{3} + 1584 \, a^{2} b^{4} d^{2} e^{4} - 1848 \, a^{3} b^{3} d e^{5} + 1155 \, a^{4} b^{2} e^{6}\right )} x^{2} - 2 \, {\left (256 \, b^{6} d^{5} e - 1408 \, a b^{5} d^{4} e^{2} + 3168 \, a^{2} b^{4} d^{3} e^{3} - 3696 \, a^{3} b^{3} d^{2} e^{4} + 2310 \, a^{4} b^{2} d e^{5} - 693 \, a^{5} b e^{6}\right )} x\right )} \sqrt {e x + d}}{231 \, {\left (e^{8} x + d e^{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)^(3/2),x, algorithm="fricas")

[Out]

2/231*(21*b^6*e^6*x^6 - 1024*b^6*d^6 + 5632*a*b^5*d^5*e - 12672*a^2*b^4*d^4*e^2 + 14784*a^3*b^3*d^3*e^3 - 9240
*a^4*b^2*d^2*e^4 + 2772*a^5*b*d*e^5 - 231*a^6*e^6 - 14*(2*b^6*d*e^5 - 11*a*b^5*e^6)*x^5 + 5*(8*b^6*d^2*e^4 - 4
4*a*b^5*d*e^5 + 99*a^2*b^4*e^6)*x^4 - 4*(16*b^6*d^3*e^3 - 88*a*b^5*d^2*e^4 + 198*a^2*b^4*d*e^5 - 231*a^3*b^3*e
^6)*x^3 + (128*b^6*d^4*e^2 - 704*a*b^5*d^3*e^3 + 1584*a^2*b^4*d^2*e^4 - 1848*a^3*b^3*d*e^5 + 1155*a^4*b^2*e^6)
*x^2 - 2*(256*b^6*d^5*e - 1408*a*b^5*d^4*e^2 + 3168*a^2*b^4*d^3*e^3 - 3696*a^3*b^3*d^2*e^4 + 2310*a^4*b^2*d*e^
5 - 693*a^5*b*e^6)*x)*sqrt(e*x + d)/(e^8*x + d*e^7)

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giac [B]  time = 0.28, size = 642, normalized size = 1.74 \begin {gather*} \frac {2}{231} \, {\left (21 \, {\left (x e + d\right )}^{\frac {11}{2}} b^{6} e^{70} \mathrm {sgn}\left (b x + a\right ) - 154 \, {\left (x e + d\right )}^{\frac {9}{2}} b^{6} d e^{70} \mathrm {sgn}\left (b x + a\right ) + 495 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{6} d^{2} e^{70} \mathrm {sgn}\left (b x + a\right ) - 924 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{6} d^{3} e^{70} \mathrm {sgn}\left (b x + a\right ) + 1155 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{6} d^{4} e^{70} \mathrm {sgn}\left (b x + a\right ) - 1386 \, \sqrt {x e + d} b^{6} d^{5} e^{70} \mathrm {sgn}\left (b x + a\right ) + 154 \, {\left (x e + d\right )}^{\frac {9}{2}} a b^{5} e^{71} \mathrm {sgn}\left (b x + a\right ) - 990 \, {\left (x e + d\right )}^{\frac {7}{2}} a b^{5} d e^{71} \mathrm {sgn}\left (b x + a\right ) + 2772 \, {\left (x e + d\right )}^{\frac {5}{2}} a b^{5} d^{2} e^{71} \mathrm {sgn}\left (b x + a\right ) - 4620 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{5} d^{3} e^{71} \mathrm {sgn}\left (b x + a\right ) + 6930 \, \sqrt {x e + d} a b^{5} d^{4} e^{71} \mathrm {sgn}\left (b x + a\right ) + 495 \, {\left (x e + d\right )}^{\frac {7}{2}} a^{2} b^{4} e^{72} \mathrm {sgn}\left (b x + a\right ) - 2772 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{2} b^{4} d e^{72} \mathrm {sgn}\left (b x + a\right ) + 6930 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} b^{4} d^{2} e^{72} \mathrm {sgn}\left (b x + a\right ) - 13860 \, \sqrt {x e + d} a^{2} b^{4} d^{3} e^{72} \mathrm {sgn}\left (b x + a\right ) + 924 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{3} b^{3} e^{73} \mathrm {sgn}\left (b x + a\right ) - 4620 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{3} b^{3} d e^{73} \mathrm {sgn}\left (b x + a\right ) + 13860 \, \sqrt {x e + d} a^{3} b^{3} d^{2} e^{73} \mathrm {sgn}\left (b x + a\right ) + 1155 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{4} b^{2} e^{74} \mathrm {sgn}\left (b x + a\right ) - 6930 \, \sqrt {x e + d} a^{4} b^{2} d e^{74} \mathrm {sgn}\left (b x + a\right ) + 1386 \, \sqrt {x e + d} a^{5} b e^{75} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-77\right )} - \frac {2 \, {\left (b^{6} d^{6} \mathrm {sgn}\left (b x + a\right ) - 6 \, a b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{2} b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) - 20 \, a^{3} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{4} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) - 6 \, a^{5} b d e^{5} \mathrm {sgn}\left (b x + a\right ) + a^{6} e^{6} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-7\right )}}{\sqrt {x e + d}} \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)^(3/2),x, algorithm="giac")

[Out]

2/231*(21*(x*e + d)^(11/2)*b^6*e^70*sgn(b*x + a) - 154*(x*e + d)^(9/2)*b^6*d*e^70*sgn(b*x + a) + 495*(x*e + d)
^(7/2)*b^6*d^2*e^70*sgn(b*x + a) - 924*(x*e + d)^(5/2)*b^6*d^3*e^70*sgn(b*x + a) + 1155*(x*e + d)^(3/2)*b^6*d^
4*e^70*sgn(b*x + a) - 1386*sqrt(x*e + d)*b^6*d^5*e^70*sgn(b*x + a) + 154*(x*e + d)^(9/2)*a*b^5*e^71*sgn(b*x +
a) - 990*(x*e + d)^(7/2)*a*b^5*d*e^71*sgn(b*x + a) + 2772*(x*e + d)^(5/2)*a*b^5*d^2*e^71*sgn(b*x + a) - 4620*(
x*e + d)^(3/2)*a*b^5*d^3*e^71*sgn(b*x + a) + 6930*sqrt(x*e + d)*a*b^5*d^4*e^71*sgn(b*x + a) + 495*(x*e + d)^(7
/2)*a^2*b^4*e^72*sgn(b*x + a) - 2772*(x*e + d)^(5/2)*a^2*b^4*d*e^72*sgn(b*x + a) + 6930*(x*e + d)^(3/2)*a^2*b^
4*d^2*e^72*sgn(b*x + a) - 13860*sqrt(x*e + d)*a^2*b^4*d^3*e^72*sgn(b*x + a) + 924*(x*e + d)^(5/2)*a^3*b^3*e^73
*sgn(b*x + a) - 4620*(x*e + d)^(3/2)*a^3*b^3*d*e^73*sgn(b*x + a) + 13860*sqrt(x*e + d)*a^3*b^3*d^2*e^73*sgn(b*
x + a) + 1155*(x*e + d)^(3/2)*a^4*b^2*e^74*sgn(b*x + a) - 6930*sqrt(x*e + d)*a^4*b^2*d*e^74*sgn(b*x + a) + 138
6*sqrt(x*e + d)*a^5*b*e^75*sgn(b*x + a))*e^(-77) - 2*(b^6*d^6*sgn(b*x + a) - 6*a*b^5*d^5*e*sgn(b*x + a) + 15*a
^2*b^4*d^4*e^2*sgn(b*x + a) - 20*a^3*b^3*d^3*e^3*sgn(b*x + a) + 15*a^4*b^2*d^2*e^4*sgn(b*x + a) - 6*a^5*b*d*e^
5*sgn(b*x + a) + a^6*e^6*sgn(b*x + a))*e^(-7)/sqrt(x*e + d)

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maple [A]  time = 0.06, size = 393, normalized size = 1.07 \begin {gather*} -\frac {2 \left (-21 b^{6} e^{6} x^{6}-154 a \,b^{5} e^{6} x^{5}+28 b^{6} d \,e^{5} x^{5}-495 a^{2} b^{4} e^{6} x^{4}+220 a \,b^{5} d \,e^{5} x^{4}-40 b^{6} d^{2} e^{4} x^{4}-924 a^{3} b^{3} e^{6} x^{3}+792 a^{2} b^{4} d \,e^{5} x^{3}-352 a \,b^{5} d^{2} e^{4} x^{3}+64 b^{6} d^{3} e^{3} x^{3}-1155 a^{4} b^{2} e^{6} x^{2}+1848 a^{3} b^{3} d \,e^{5} x^{2}-1584 a^{2} b^{4} d^{2} e^{4} x^{2}+704 a \,b^{5} d^{3} e^{3} x^{2}-128 b^{6} d^{4} e^{2} x^{2}-1386 a^{5} b \,e^{6} x +4620 a^{4} b^{2} d \,e^{5} x -7392 a^{3} b^{3} d^{2} e^{4} x +6336 a^{2} b^{4} d^{3} e^{3} x -2816 a \,b^{5} d^{4} e^{2} x +512 b^{6} d^{5} e x +231 a^{6} e^{6}-2772 a^{5} b d \,e^{5}+9240 a^{4} b^{2} d^{2} e^{4}-14784 a^{3} b^{3} d^{3} e^{3}+12672 a^{2} b^{4} d^{4} e^{2}-5632 a \,b^{5} d^{5} e +1024 b^{6} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{231 \sqrt {e x +d}\, \left (b x +a \right )^{5} e^{7}} \end {gather*}

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)^(3/2),x)

[Out]

-2/231/(e*x+d)^(1/2)*(-21*b^6*e^6*x^6-154*a*b^5*e^6*x^5+28*b^6*d*e^5*x^5-495*a^2*b^4*e^6*x^4+220*a*b^5*d*e^5*x
^4-40*b^6*d^2*e^4*x^4-924*a^3*b^3*e^6*x^3+792*a^2*b^4*d*e^5*x^3-352*a*b^5*d^2*e^4*x^3+64*b^6*d^3*e^3*x^3-1155*
a^4*b^2*e^6*x^2+1848*a^3*b^3*d*e^5*x^2-1584*a^2*b^4*d^2*e^4*x^2+704*a*b^5*d^3*e^3*x^2-128*b^6*d^4*e^2*x^2-1386
*a^5*b*e^6*x+4620*a^4*b^2*d*e^5*x-7392*a^3*b^3*d^2*e^4*x+6336*a^2*b^4*d^3*e^3*x-2816*a*b^5*d^4*e^2*x+512*b^6*d
^5*e*x+231*a^6*e^6-2772*a^5*b*d*e^5+9240*a^4*b^2*d^2*e^4-14784*a^3*b^3*d^3*e^3+12672*a^2*b^4*d^4*e^2-5632*a*b^
5*d^5*e+1024*b^6*d^6)*((b*x+a)^2)^(5/2)/e^7/(b*x+a)^5

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maxima [B]  time = 0.63, size = 603, normalized size = 1.64 \begin {gather*} \frac {2 \, {\left (7 \, b^{5} e^{5} x^{5} + 256 \, b^{5} d^{5} - 1152 \, a b^{4} d^{4} e + 2016 \, a^{2} b^{3} d^{3} e^{2} - 1680 \, a^{3} b^{2} d^{2} e^{3} + 630 \, a^{4} b d e^{4} - 63 \, a^{5} e^{5} - 5 \, {\left (2 \, b^{5} d e^{4} - 9 \, a b^{4} e^{5}\right )} x^{4} + 2 \, {\left (8 \, b^{5} d^{2} e^{3} - 36 \, a b^{4} d e^{4} + 63 \, a^{2} b^{3} e^{5}\right )} x^{3} - 2 \, {\left (16 \, b^{5} d^{3} e^{2} - 72 \, a b^{4} d^{2} e^{3} + 126 \, a^{2} b^{3} d e^{4} - 105 \, a^{3} b^{2} e^{5}\right )} x^{2} + {\left (128 \, b^{5} d^{4} e - 576 \, a b^{4} d^{3} e^{2} + 1008 \, a^{2} b^{3} d^{2} e^{3} - 840 \, a^{3} b^{2} d e^{4} + 315 \, a^{4} b e^{5}\right )} x\right )} a}{63 \, \sqrt {e x + d} e^{6}} + \frac {2 \, {\left (63 \, b^{5} e^{6} x^{6} - 3072 \, b^{5} d^{6} + 14080 \, a b^{4} d^{5} e - 25344 \, a^{2} b^{3} d^{4} e^{2} + 22176 \, a^{3} b^{2} d^{3} e^{3} - 9240 \, a^{4} b d^{2} e^{4} + 1386 \, a^{5} d e^{5} - 7 \, {\left (12 \, b^{5} d e^{5} - 55 \, a b^{4} e^{6}\right )} x^{5} + 10 \, {\left (12 \, b^{5} d^{2} e^{4} - 55 \, a b^{4} d e^{5} + 99 \, a^{2} b^{3} e^{6}\right )} x^{4} - 2 \, {\left (96 \, b^{5} d^{3} e^{3} - 440 \, a b^{4} d^{2} e^{4} + 792 \, a^{2} b^{3} d e^{5} - 693 \, a^{3} b^{2} e^{6}\right )} x^{3} + {\left (384 \, b^{5} d^{4} e^{2} - 1760 \, a b^{4} d^{3} e^{3} + 3168 \, a^{2} b^{3} d^{2} e^{4} - 2772 \, a^{3} b^{2} d e^{5} + 1155 \, a^{4} b e^{6}\right )} x^{2} - {\left (1536 \, b^{5} d^{5} e - 7040 \, a b^{4} d^{4} e^{2} + 12672 \, a^{2} b^{3} d^{3} e^{3} - 11088 \, a^{3} b^{2} d^{2} e^{4} + 4620 \, a^{4} b d e^{5} - 693 \, a^{5} e^{6}\right )} x\right )} b}{693 \, \sqrt {e x + d} e^{7}} \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)^(3/2),x, algorithm="maxima")

[Out]

2/63*(7*b^5*e^5*x^5 + 256*b^5*d^5 - 1152*a*b^4*d^4*e + 2016*a^2*b^3*d^3*e^2 - 1680*a^3*b^2*d^2*e^3 + 630*a^4*b
*d*e^4 - 63*a^5*e^5 - 5*(2*b^5*d*e^4 - 9*a*b^4*e^5)*x^4 + 2*(8*b^5*d^2*e^3 - 36*a*b^4*d*e^4 + 63*a^2*b^3*e^5)*
x^3 - 2*(16*b^5*d^3*e^2 - 72*a*b^4*d^2*e^3 + 126*a^2*b^3*d*e^4 - 105*a^3*b^2*e^5)*x^2 + (128*b^5*d^4*e - 576*a
*b^4*d^3*e^2 + 1008*a^2*b^3*d^2*e^3 - 840*a^3*b^2*d*e^4 + 315*a^4*b*e^5)*x)*a/(sqrt(e*x + d)*e^6) + 2/693*(63*
b^5*e^6*x^6 - 3072*b^5*d^6 + 14080*a*b^4*d^5*e - 25344*a^2*b^3*d^4*e^2 + 22176*a^3*b^2*d^3*e^3 - 9240*a^4*b*d^
2*e^4 + 1386*a^5*d*e^5 - 7*(12*b^5*d*e^5 - 55*a*b^4*e^6)*x^5 + 10*(12*b^5*d^2*e^4 - 55*a*b^4*d*e^5 + 99*a^2*b^
3*e^6)*x^4 - 2*(96*b^5*d^3*e^3 - 440*a*b^4*d^2*e^4 + 792*a^2*b^3*d*e^5 - 693*a^3*b^2*e^6)*x^3 + (384*b^5*d^4*e
^2 - 1760*a*b^4*d^3*e^3 + 3168*a^2*b^3*d^2*e^4 - 2772*a^3*b^2*d*e^5 + 1155*a^4*b*e^6)*x^2 - (1536*b^5*d^5*e -
7040*a*b^4*d^4*e^2 + 12672*a^2*b^3*d^3*e^3 - 11088*a^3*b^2*d^2*e^4 + 4620*a^4*b*d*e^5 - 693*a^5*e^6)*x)*b/(sqr
t(e*x + d)*e^7)

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mupad [B]  time = 3.09, size = 396, normalized size = 1.08 \begin {gather*} \frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (\frac {2\,b^5\,x^6}{11\,e}-\frac {462\,a^6\,e^6-5544\,a^5\,b\,d\,e^5+18480\,a^4\,b^2\,d^2\,e^4-29568\,a^3\,b^3\,d^3\,e^3+25344\,a^2\,b^4\,d^4\,e^2-11264\,a\,b^5\,d^5\,e+2048\,b^6\,d^6}{231\,b\,e^7}+\frac {x\,\left (2772\,a^5\,b\,e^6-9240\,a^4\,b^2\,d\,e^5+14784\,a^3\,b^3\,d^2\,e^4-12672\,a^2\,b^4\,d^3\,e^3+5632\,a\,b^5\,d^4\,e^2-1024\,b^6\,d^5\,e\right )}{231\,b\,e^7}+\frac {8\,b^2\,x^3\,\left (231\,a^3\,e^3-198\,a^2\,b\,d\,e^2+88\,a\,b^2\,d^2\,e-16\,b^3\,d^3\right )}{231\,e^4}+\frac {4\,b^4\,x^5\,\left (11\,a\,e-2\,b\,d\right )}{33\,e^2}+\frac {10\,b^3\,x^4\,\left (99\,a^2\,e^2-44\,a\,b\,d\,e+8\,b^2\,d^2\right )}{231\,e^3}+\frac {x^2\,\left (2310\,a^4\,b^2\,e^6-3696\,a^3\,b^3\,d\,e^5+3168\,a^2\,b^4\,d^2\,e^4-1408\,a\,b^5\,d^3\,e^3+256\,b^6\,d^4\,e^2\right )}{231\,b\,e^7}\right )}{x\,\sqrt {d+e\,x}+\frac {a\,\sqrt {d+e\,x}}{b}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*x)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2))/(d + e*x)^(3/2),x)

[Out]

((a^2 + b^2*x^2 + 2*a*b*x)^(1/2)*((2*b^5*x^6)/(11*e) - (462*a^6*e^6 + 2048*b^6*d^6 + 25344*a^2*b^4*d^4*e^2 - 2
9568*a^3*b^3*d^3*e^3 + 18480*a^4*b^2*d^2*e^4 - 11264*a*b^5*d^5*e - 5544*a^5*b*d*e^5)/(231*b*e^7) + (x*(2772*a^
5*b*e^6 - 1024*b^6*d^5*e + 5632*a*b^5*d^4*e^2 - 9240*a^4*b^2*d*e^5 - 12672*a^2*b^4*d^3*e^3 + 14784*a^3*b^3*d^2
*e^4))/(231*b*e^7) + (8*b^2*x^3*(231*a^3*e^3 - 16*b^3*d^3 + 88*a*b^2*d^2*e - 198*a^2*b*d*e^2))/(231*e^4) + (4*
b^4*x^5*(11*a*e - 2*b*d))/(33*e^2) + (10*b^3*x^4*(99*a^2*e^2 + 8*b^2*d^2 - 44*a*b*d*e))/(231*e^3) + (x^2*(2310
*a^4*b^2*e^6 + 256*b^6*d^4*e^2 - 1408*a*b^5*d^3*e^3 - 3696*a^3*b^3*d*e^5 + 3168*a^2*b^4*d^2*e^4))/(231*b*e^7))
)/(x*(d + e*x)^(1/2) + (a*(d + e*x)^(1/2))/b)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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)**(3/2),x)

[Out]

Timed out

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