3.13.67 \(\int \frac {(A+B x) (a+c x^2)^3}{\sqrt {d+e x}} \, dx\)

Optimal. Leaf size=348 \[ -\frac {2 c (d+e x)^{7/2} \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{7 e^8}+\frac {6 c^2 (d+e x)^{11/2} \left (a B e^2-2 A c d e+7 B c d^2\right )}{11 e^8}-\frac {2 c^2 (d+e x)^{9/2} \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{9 e^8}+\frac {2 (d+e x)^{3/2} \left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{3 e^8}-\frac {2 \sqrt {d+e x} \left (a e^2+c d^2\right )^3 (B d-A e)}{e^8}-\frac {6 c (d+e x)^{5/2} \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{5 e^8}-\frac {2 c^3 (d+e x)^{13/2} (7 B d-A e)}{13 e^8}+\frac {2 B c^3 (d+e x)^{15/2}}{15 e^8} \]

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Rubi [A]  time = 0.21, antiderivative size = 348, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {772} \begin {gather*} -\frac {2 c (d+e x)^{7/2} \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{7 e^8}+\frac {6 c^2 (d+e x)^{11/2} \left (a B e^2-2 A c d e+7 B c d^2\right )}{11 e^8}-\frac {2 c^2 (d+e x)^{9/2} \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{9 e^8}-\frac {6 c (d+e x)^{5/2} \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{5 e^8}+\frac {2 (d+e x)^{3/2} \left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{3 e^8}-\frac {2 \sqrt {d+e x} \left (a e^2+c d^2\right )^3 (B d-A e)}{e^8}-\frac {2 c^3 (d+e x)^{13/2} (7 B d-A e)}{13 e^8}+\frac {2 B c^3 (d+e x)^{15/2}}{15 e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + c*x^2)^3)/Sqrt[d + e*x],x]

[Out]

(-2*(B*d - A*e)*(c*d^2 + a*e^2)^3*Sqrt[d + e*x])/e^8 + (2*(c*d^2 + a*e^2)^2*(7*B*c*d^2 - 6*A*c*d*e + a*B*e^2)*
(d + e*x)^(3/2))/(3*e^8) - (6*c*(c*d^2 + a*e^2)*(7*B*c*d^3 - 5*A*c*d^2*e + 3*a*B*d*e^2 - a*A*e^3)*(d + e*x)^(5
/2))/(5*e^8) - (2*c*(4*A*c*d*e*(5*c*d^2 + 3*a*e^2) - B*(35*c^2*d^4 + 30*a*c*d^2*e^2 + 3*a^2*e^4))*(d + e*x)^(7
/2))/(7*e^8) - (2*c^2*(35*B*c*d^3 - 15*A*c*d^2*e + 15*a*B*d*e^2 - 3*a*A*e^3)*(d + e*x)^(9/2))/(9*e^8) + (6*c^2
*(7*B*c*d^2 - 2*A*c*d*e + a*B*e^2)*(d + e*x)^(11/2))/(11*e^8) - (2*c^3*(7*B*d - A*e)*(d + e*x)^(13/2))/(13*e^8
) + (2*B*c^3*(d + e*x)^(15/2))/(15*e^8)

Rule 772

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegr
and[(d + e*x)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IGtQ[p, 0]

Rubi steps

\begin {align*} \int \frac {(A+B x) \left (a+c x^2\right )^3}{\sqrt {d+e x}} \, dx &=\int \left (\frac {(-B d+A e) \left (c d^2+a e^2\right )^3}{e^7 \sqrt {d+e x}}+\frac {\left (c d^2+a e^2\right )^2 \left (7 B c d^2-6 A c d e+a B e^2\right ) \sqrt {d+e x}}{e^7}+\frac {3 c \left (c d^2+a e^2\right ) \left (-7 B c d^3+5 A c d^2 e-3 a B d e^2+a A e^3\right ) (d+e x)^{3/2}}{e^7}-\frac {c \left (-35 B c^2 d^4+20 A c^2 d^3 e-30 a B c d^2 e^2+12 a A c d e^3-3 a^2 B e^4\right ) (d+e x)^{5/2}}{e^7}+\frac {c^2 \left (-35 B c d^3+15 A c d^2 e-15 a B d e^2+3 a A e^3\right ) (d+e x)^{7/2}}{e^7}-\frac {3 c^2 \left (-7 B c d^2+2 A c d e-a B e^2\right ) (d+e x)^{9/2}}{e^7}+\frac {c^3 (-7 B d+A e) (d+e x)^{11/2}}{e^7}+\frac {B c^3 (d+e x)^{13/2}}{e^7}\right ) \, dx\\ &=-\frac {2 (B d-A e) \left (c d^2+a e^2\right )^3 \sqrt {d+e x}}{e^8}+\frac {2 \left (c d^2+a e^2\right )^2 \left (7 B c d^2-6 A c d e+a B e^2\right ) (d+e x)^{3/2}}{3 e^8}-\frac {6 c \left (c d^2+a e^2\right ) \left (7 B c d^3-5 A c d^2 e+3 a B d e^2-a A e^3\right ) (d+e x)^{5/2}}{5 e^8}-\frac {2 c \left (4 A c d e \left (5 c d^2+3 a e^2\right )-B \left (35 c^2 d^4+30 a c d^2 e^2+3 a^2 e^4\right )\right ) (d+e x)^{7/2}}{7 e^8}-\frac {2 c^2 \left (35 B c d^3-15 A c d^2 e+15 a B d e^2-3 a A e^3\right ) (d+e x)^{9/2}}{9 e^8}+\frac {6 c^2 \left (7 B c d^2-2 A c d e+a B e^2\right ) (d+e x)^{11/2}}{11 e^8}-\frac {2 c^3 (7 B d-A e) (d+e x)^{13/2}}{13 e^8}+\frac {2 B c^3 (d+e x)^{15/2}}{15 e^8}\\ \end {align*}

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Mathematica [A]  time = 0.35, size = 373, normalized size = 1.07 \begin {gather*} \frac {2 \sqrt {d+e x} \left (3 A e \left (15015 a^3 e^6+3003 a^2 c e^4 \left (8 d^2-4 d e x+3 e^2 x^2\right )+143 a c^2 e^2 \left (128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right )+5 c^3 \left (1024 d^6-512 d^5 e x+384 d^4 e^2 x^2-320 d^3 e^3 x^3+280 d^2 e^4 x^4-252 d e^5 x^5+231 e^6 x^6\right )\right )+B \left (15015 a^3 e^6 (e x-2 d)+3861 a^2 c e^4 \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )+195 a c^2 e^2 \left (-256 d^5+128 d^4 e x-96 d^3 e^2 x^2+80 d^2 e^3 x^3-70 d e^4 x^4+63 e^5 x^5\right )-7 c^3 \left (2048 d^7-1024 d^6 e x+768 d^5 e^2 x^2-640 d^4 e^3 x^3+560 d^3 e^4 x^4-504 d^2 e^5 x^5+462 d e^6 x^6-429 e^7 x^7\right )\right )\right )}{45045 e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + c*x^2)^3)/Sqrt[d + e*x],x]

[Out]

(2*Sqrt[d + e*x]*(3*A*e*(15015*a^3*e^6 + 3003*a^2*c*e^4*(8*d^2 - 4*d*e*x + 3*e^2*x^2) + 143*a*c^2*e^2*(128*d^4
- 64*d^3*e*x + 48*d^2*e^2*x^2 - 40*d*e^3*x^3 + 35*e^4*x^4) + 5*c^3*(1024*d^6 - 512*d^5*e*x + 384*d^4*e^2*x^2
- 320*d^3*e^3*x^3 + 280*d^2*e^4*x^4 - 252*d*e^5*x^5 + 231*e^6*x^6)) + B*(15015*a^3*e^6*(-2*d + e*x) + 3861*a^2
*c*e^4*(-16*d^3 + 8*d^2*e*x - 6*d*e^2*x^2 + 5*e^3*x^3) + 195*a*c^2*e^2*(-256*d^5 + 128*d^4*e*x - 96*d^3*e^2*x^
2 + 80*d^2*e^3*x^3 - 70*d*e^4*x^4 + 63*e^5*x^5) - 7*c^3*(2048*d^7 - 1024*d^6*e*x + 768*d^5*e^2*x^2 - 640*d^4*e
^3*x^3 + 560*d^3*e^4*x^4 - 504*d^2*e^5*x^5 + 462*d*e^6*x^6 - 429*e^7*x^7))))/(45045*e^8)

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IntegrateAlgebraic [A]  time = 0.25, size = 573, normalized size = 1.65 \begin {gather*} \frac {2 \sqrt {d+e x} \left (45045 a^3 A e^7+15015 a^3 B e^6 (d+e x)-45045 a^3 B d e^6+135135 a^2 A c d^2 e^5-90090 a^2 A c d e^5 (d+e x)+27027 a^2 A c e^5 (d+e x)^2-135135 a^2 B c d^3 e^4+135135 a^2 B c d^2 e^4 (d+e x)-81081 a^2 B c d e^4 (d+e x)^2+19305 a^2 B c e^4 (d+e x)^3+135135 a A c^2 d^4 e^3-180180 a A c^2 d^3 e^3 (d+e x)+162162 a A c^2 d^2 e^3 (d+e x)^2-77220 a A c^2 d e^3 (d+e x)^3+15015 a A c^2 e^3 (d+e x)^4-135135 a B c^2 d^5 e^2+225225 a B c^2 d^4 e^2 (d+e x)-270270 a B c^2 d^3 e^2 (d+e x)^2+193050 a B c^2 d^2 e^2 (d+e x)^3-75075 a B c^2 d e^2 (d+e x)^4+12285 a B c^2 e^2 (d+e x)^5+45045 A c^3 d^6 e-90090 A c^3 d^5 e (d+e x)+135135 A c^3 d^4 e (d+e x)^2-128700 A c^3 d^3 e (d+e x)^3+75075 A c^3 d^2 e (d+e x)^4-24570 A c^3 d e (d+e x)^5+3465 A c^3 e (d+e x)^6-45045 B c^3 d^7+105105 B c^3 d^6 (d+e x)-189189 B c^3 d^5 (d+e x)^2+225225 B c^3 d^4 (d+e x)^3-175175 B c^3 d^3 (d+e x)^4+85995 B c^3 d^2 (d+e x)^5-24255 B c^3 d (d+e x)^6+3003 B c^3 (d+e x)^7\right )}{45045 e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((A + B*x)*(a + c*x^2)^3)/Sqrt[d + e*x],x]

[Out]

(2*Sqrt[d + e*x]*(-45045*B*c^3*d^7 + 45045*A*c^3*d^6*e - 135135*a*B*c^2*d^5*e^2 + 135135*a*A*c^2*d^4*e^3 - 135
135*a^2*B*c*d^3*e^4 + 135135*a^2*A*c*d^2*e^5 - 45045*a^3*B*d*e^6 + 45045*a^3*A*e^7 + 105105*B*c^3*d^6*(d + e*x
) - 90090*A*c^3*d^5*e*(d + e*x) + 225225*a*B*c^2*d^4*e^2*(d + e*x) - 180180*a*A*c^2*d^3*e^3*(d + e*x) + 135135
*a^2*B*c*d^2*e^4*(d + e*x) - 90090*a^2*A*c*d*e^5*(d + e*x) + 15015*a^3*B*e^6*(d + e*x) - 189189*B*c^3*d^5*(d +
e*x)^2 + 135135*A*c^3*d^4*e*(d + e*x)^2 - 270270*a*B*c^2*d^3*e^2*(d + e*x)^2 + 162162*a*A*c^2*d^2*e^3*(d + e*
x)^2 - 81081*a^2*B*c*d*e^4*(d + e*x)^2 + 27027*a^2*A*c*e^5*(d + e*x)^2 + 225225*B*c^3*d^4*(d + e*x)^3 - 128700
*A*c^3*d^3*e*(d + e*x)^3 + 193050*a*B*c^2*d^2*e^2*(d + e*x)^3 - 77220*a*A*c^2*d*e^3*(d + e*x)^3 + 19305*a^2*B*
c*e^4*(d + e*x)^3 - 175175*B*c^3*d^3*(d + e*x)^4 + 75075*A*c^3*d^2*e*(d + e*x)^4 - 75075*a*B*c^2*d*e^2*(d + e*
x)^4 + 15015*a*A*c^2*e^3*(d + e*x)^4 + 85995*B*c^3*d^2*(d + e*x)^5 - 24570*A*c^3*d*e*(d + e*x)^5 + 12285*a*B*c
^2*e^2*(d + e*x)^5 - 24255*B*c^3*d*(d + e*x)^6 + 3465*A*c^3*e*(d + e*x)^6 + 3003*B*c^3*(d + e*x)^7))/(45045*e^
8)

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fricas [A]  time = 0.41, size = 454, normalized size = 1.30 \begin {gather*} \frac {2 \, {\left (3003 \, B c^{3} e^{7} x^{7} - 14336 \, B c^{3} d^{7} + 15360 \, A c^{3} d^{6} e - 49920 \, B a c^{2} d^{5} e^{2} + 54912 \, A a c^{2} d^{4} e^{3} - 61776 \, B a^{2} c d^{3} e^{4} + 72072 \, A a^{2} c d^{2} e^{5} - 30030 \, B a^{3} d e^{6} + 45045 \, A a^{3} e^{7} - 231 \, {\left (14 \, B c^{3} d e^{6} - 15 \, A c^{3} e^{7}\right )} x^{6} + 63 \, {\left (56 \, B c^{3} d^{2} e^{5} - 60 \, A c^{3} d e^{6} + 195 \, B a c^{2} e^{7}\right )} x^{5} - 35 \, {\left (112 \, B c^{3} d^{3} e^{4} - 120 \, A c^{3} d^{2} e^{5} + 390 \, B a c^{2} d e^{6} - 429 \, A a c^{2} e^{7}\right )} x^{4} + 5 \, {\left (896 \, B c^{3} d^{4} e^{3} - 960 \, A c^{3} d^{3} e^{4} + 3120 \, B a c^{2} d^{2} e^{5} - 3432 \, A a c^{2} d e^{6} + 3861 \, B a^{2} c e^{7}\right )} x^{3} - 3 \, {\left (1792 \, B c^{3} d^{5} e^{2} - 1920 \, A c^{3} d^{4} e^{3} + 6240 \, B a c^{2} d^{3} e^{4} - 6864 \, A a c^{2} d^{2} e^{5} + 7722 \, B a^{2} c d e^{6} - 9009 \, A a^{2} c e^{7}\right )} x^{2} + {\left (7168 \, B c^{3} d^{6} e - 7680 \, A c^{3} d^{5} e^{2} + 24960 \, B a c^{2} d^{4} e^{3} - 27456 \, A a c^{2} d^{3} e^{4} + 30888 \, B a^{2} c d^{2} e^{5} - 36036 \, A a^{2} c d e^{6} + 15015 \, B a^{3} e^{7}\right )} x\right )} \sqrt {e x + d}}{45045 \, e^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^3/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

2/45045*(3003*B*c^3*e^7*x^7 - 14336*B*c^3*d^7 + 15360*A*c^3*d^6*e - 49920*B*a*c^2*d^5*e^2 + 54912*A*a*c^2*d^4*
e^3 - 61776*B*a^2*c*d^3*e^4 + 72072*A*a^2*c*d^2*e^5 - 30030*B*a^3*d*e^6 + 45045*A*a^3*e^7 - 231*(14*B*c^3*d*e^
6 - 15*A*c^3*e^7)*x^6 + 63*(56*B*c^3*d^2*e^5 - 60*A*c^3*d*e^6 + 195*B*a*c^2*e^7)*x^5 - 35*(112*B*c^3*d^3*e^4 -
120*A*c^3*d^2*e^5 + 390*B*a*c^2*d*e^6 - 429*A*a*c^2*e^7)*x^4 + 5*(896*B*c^3*d^4*e^3 - 960*A*c^3*d^3*e^4 + 312
0*B*a*c^2*d^2*e^5 - 3432*A*a*c^2*d*e^6 + 3861*B*a^2*c*e^7)*x^3 - 3*(1792*B*c^3*d^5*e^2 - 1920*A*c^3*d^4*e^3 +
6240*B*a*c^2*d^3*e^4 - 6864*A*a*c^2*d^2*e^5 + 7722*B*a^2*c*d*e^6 - 9009*A*a^2*c*e^7)*x^2 + (7168*B*c^3*d^6*e -
7680*A*c^3*d^5*e^2 + 24960*B*a*c^2*d^4*e^3 - 27456*A*a*c^2*d^3*e^4 + 30888*B*a^2*c*d^2*e^5 - 36036*A*a^2*c*d*
e^6 + 15015*B*a^3*e^7)*x)*sqrt(e*x + d)/e^8

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giac [A]  time = 0.22, size = 504, normalized size = 1.45 \begin {gather*} \frac {2}{45045} \, {\left (15015 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} B a^{3} e^{\left (-1\right )} + 9009 \, {\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 a^{2} c e^{\left (-2\right )} + 3861 \, {\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 )} B a^{2} c e^{\left (-3\right )} + 429 \, {\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 a c^{2} e^{\left (-4\right )} + 195 \, {\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 )} B a c^{2} e^{\left (-5\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 )} A c^{3} e^{\left (-6\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 c^{3} e^{\left (-7\right )} + 45045 \, \sqrt {x e + d} A a^{3}\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^3/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

2/45045*(15015*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*B*a^3*e^(-1) + 9009*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/
2)*d + 15*sqrt(x*e + d)*d^2)*A*a^2*c*e^(-2) + 3861*(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)*B*a^2*c*e^(-3) + 429*(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*a*c^2*e^(-4) + 195*(63*(x*e + d)^(11/2) - 38
5*(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)*B*a*c^2*e^(-5) + 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)*A*c
^3*e^(-6) + 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*c^3*e^(-7) + 45045*sqrt(x*e + d)*A*a^3)*e^(-1)

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maple [A]  time = 0.05, size = 489, normalized size = 1.41 \begin {gather*} \frac {2 \sqrt {e x +d}\, \left (3003 B \,c^{3} x^{7} e^{7}+3465 A \,c^{3} e^{7} x^{6}-3234 B \,c^{3} d \,e^{6} x^{6}-3780 A \,c^{3} d \,e^{6} x^{5}+12285 B a \,c^{2} e^{7} x^{5}+3528 B \,c^{3} d^{2} e^{5} x^{5}+15015 A a \,c^{2} e^{7} x^{4}+4200 A \,c^{3} d^{2} e^{5} x^{4}-13650 B a \,c^{2} d \,e^{6} x^{4}-3920 B \,c^{3} d^{3} e^{4} x^{4}-17160 A a \,c^{2} d \,e^{6} x^{3}-4800 A \,c^{3} d^{3} e^{4} x^{3}+19305 B \,a^{2} c \,e^{7} x^{3}+15600 B a \,c^{2} d^{2} e^{5} x^{3}+4480 B \,c^{3} d^{4} e^{3} x^{3}+27027 A \,a^{2} c \,e^{7} x^{2}+20592 A a \,c^{2} d^{2} e^{5} x^{2}+5760 A \,c^{3} d^{4} e^{3} x^{2}-23166 B \,a^{2} c d \,e^{6} x^{2}-18720 B a \,c^{2} d^{3} e^{4} x^{2}-5376 B \,c^{3} d^{5} e^{2} x^{2}-36036 A \,a^{2} c d \,e^{6} x -27456 A a \,c^{2} d^{3} e^{4} x -7680 A \,c^{3} d^{5} e^{2} x +15015 B \,a^{3} e^{7} x +30888 B \,a^{2} c \,d^{2} e^{5} x +24960 B a \,c^{2} d^{4} e^{3} x +7168 B \,c^{3} d^{6} e x +45045 A \,a^{3} e^{7}+72072 A \,d^{2} a^{2} c \,e^{5}+54912 A a \,c^{2} d^{4} e^{3}+15360 A \,c^{3} d^{6} e -30030 B \,a^{3} d \,e^{6}-61776 B \,d^{3} a^{2} c \,e^{4}-49920 B a \,c^{2} d^{5} e^{2}-14336 B \,c^{3} d^{7}\right )}{45045 e^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+a)^3/(e*x+d)^(1/2),x)

[Out]

2/45045*(e*x+d)^(1/2)*(3003*B*c^3*e^7*x^7+3465*A*c^3*e^7*x^6-3234*B*c^3*d*e^6*x^6-3780*A*c^3*d*e^6*x^5+12285*B
*a*c^2*e^7*x^5+3528*B*c^3*d^2*e^5*x^5+15015*A*a*c^2*e^7*x^4+4200*A*c^3*d^2*e^5*x^4-13650*B*a*c^2*d*e^6*x^4-392
0*B*c^3*d^3*e^4*x^4-17160*A*a*c^2*d*e^6*x^3-4800*A*c^3*d^3*e^4*x^3+19305*B*a^2*c*e^7*x^3+15600*B*a*c^2*d^2*e^5
*x^3+4480*B*c^3*d^4*e^3*x^3+27027*A*a^2*c*e^7*x^2+20592*A*a*c^2*d^2*e^5*x^2+5760*A*c^3*d^4*e^3*x^2-23166*B*a^2
*c*d*e^6*x^2-18720*B*a*c^2*d^3*e^4*x^2-5376*B*c^3*d^5*e^2*x^2-36036*A*a^2*c*d*e^6*x-27456*A*a*c^2*d^3*e^4*x-76
80*A*c^3*d^5*e^2*x+15015*B*a^3*e^7*x+30888*B*a^2*c*d^2*e^5*x+24960*B*a*c^2*d^4*e^3*x+7168*B*c^3*d^6*e*x+45045*
A*a^3*e^7+72072*A*a^2*c*d^2*e^5+54912*A*a*c^2*d^4*e^3+15360*A*c^3*d^6*e-30030*B*a^3*d*e^6-61776*B*a^2*c*d^3*e^
4-49920*B*a*c^2*d^5*e^2-14336*B*c^3*d^7)/e^8

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maxima [A]  time = 0.56, size = 453, normalized size = 1.30 \begin {gather*} \frac {2 \, {\left (3003 \, {\left (e x + d\right )}^{\frac {15}{2}} B c^{3} - 3465 \, {\left (7 \, B c^{3} d - A c^{3} e\right )} {\left (e x + d\right )}^{\frac {13}{2}} + 12285 \, {\left (7 \, B c^{3} d^{2} - 2 \, A c^{3} d e + B a c^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {11}{2}} - 5005 \, {\left (35 \, B c^{3} d^{3} - 15 \, A c^{3} d^{2} e + 15 \, B a c^{2} d e^{2} - 3 \, A a c^{2} e^{3}\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 6435 \, {\left (35 \, B c^{3} d^{4} - 20 \, A c^{3} d^{3} e + 30 \, B a c^{2} d^{2} e^{2} - 12 \, A a c^{2} d e^{3} + 3 \, B a^{2} c e^{4}\right )} {\left (e x + d\right )}^{\frac {7}{2}} - 27027 \, {\left (7 \, B c^{3} d^{5} - 5 \, A c^{3} d^{4} e + 10 \, B a c^{2} d^{3} e^{2} - 6 \, A a c^{2} d^{2} e^{3} + 3 \, B a^{2} c d e^{4} - A a^{2} c e^{5}\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 15015 \, {\left (7 \, B c^{3} d^{6} - 6 \, A c^{3} d^{5} e + 15 \, B a c^{2} d^{4} e^{2} - 12 \, A a c^{2} d^{3} e^{3} + 9 \, B a^{2} c d^{2} e^{4} - 6 \, A a^{2} c d e^{5} + B a^{3} e^{6}\right )} {\left (e x + d\right )}^{\frac {3}{2}} - 45045 \, {\left (B c^{3} d^{7} - A c^{3} d^{6} e + 3 \, B a c^{2} d^{5} e^{2} - 3 \, A a c^{2} d^{4} e^{3} + 3 \, B a^{2} c d^{3} e^{4} - 3 \, A a^{2} c d^{2} e^{5} + B a^{3} d e^{6} - A a^{3} e^{7}\right )} \sqrt {e x + d}\right )}}{45045 \, e^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^3/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

2/45045*(3003*(e*x + d)^(15/2)*B*c^3 - 3465*(7*B*c^3*d - A*c^3*e)*(e*x + d)^(13/2) + 12285*(7*B*c^3*d^2 - 2*A*
c^3*d*e + B*a*c^2*e^2)*(e*x + d)^(11/2) - 5005*(35*B*c^3*d^3 - 15*A*c^3*d^2*e + 15*B*a*c^2*d*e^2 - 3*A*a*c^2*e
^3)*(e*x + d)^(9/2) + 6435*(35*B*c^3*d^4 - 20*A*c^3*d^3*e + 30*B*a*c^2*d^2*e^2 - 12*A*a*c^2*d*e^3 + 3*B*a^2*c*
e^4)*(e*x + d)^(7/2) - 27027*(7*B*c^3*d^5 - 5*A*c^3*d^4*e + 10*B*a*c^2*d^3*e^2 - 6*A*a*c^2*d^2*e^3 + 3*B*a^2*c
*d*e^4 - A*a^2*c*e^5)*(e*x + d)^(5/2) + 15015*(7*B*c^3*d^6 - 6*A*c^3*d^5*e + 15*B*a*c^2*d^4*e^2 - 12*A*a*c^2*d
^3*e^3 + 9*B*a^2*c*d^2*e^4 - 6*A*a^2*c*d*e^5 + B*a^3*e^6)*(e*x + d)^(3/2) - 45045*(B*c^3*d^7 - A*c^3*d^6*e + 3
*B*a*c^2*d^5*e^2 - 3*A*a*c^2*d^4*e^3 + 3*B*a^2*c*d^3*e^4 - 3*A*a^2*c*d^2*e^5 + B*a^3*d*e^6 - A*a^3*e^7)*sqrt(e
*x + d))/e^8

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mupad [B]  time = 0.14, size = 324, normalized size = 0.93 \begin {gather*} \frac {{\left (d+e\,x\right )}^{7/2}\,\left (6\,B\,a^2\,c\,e^4+60\,B\,a\,c^2\,d^2\,e^2-24\,A\,a\,c^2\,d\,e^3+70\,B\,c^3\,d^4-40\,A\,c^3\,d^3\,e\right )}{7\,e^8}+\frac {{\left (d+e\,x\right )}^{11/2}\,\left (42\,B\,c^3\,d^2-12\,A\,c^3\,d\,e+6\,B\,a\,c^2\,e^2\right )}{11\,e^8}+\frac {2\,{\left (c\,d^2+a\,e^2\right )}^2\,{\left (d+e\,x\right )}^{3/2}\,\left (7\,B\,c\,d^2-6\,A\,c\,d\,e+B\,a\,e^2\right )}{3\,e^8}+\frac {2\,B\,c^3\,{\left (d+e\,x\right )}^{15/2}}{15\,e^8}+\frac {2\,c^2\,{\left (d+e\,x\right )}^{9/2}\,\left (-35\,B\,c\,d^3+15\,A\,c\,d^2\,e-15\,B\,a\,d\,e^2+3\,A\,a\,e^3\right )}{9\,e^8}+\frac {2\,c^3\,\left (A\,e-7\,B\,d\right )\,{\left (d+e\,x\right )}^{13/2}}{13\,e^8}+\frac {2\,{\left (c\,d^2+a\,e^2\right )}^3\,\left (A\,e-B\,d\right )\,\sqrt {d+e\,x}}{e^8}+\frac {6\,c\,\left (c\,d^2+a\,e^2\right )\,{\left (d+e\,x\right )}^{5/2}\,\left (-7\,B\,c\,d^3+5\,A\,c\,d^2\,e-3\,B\,a\,d\,e^2+A\,a\,e^3\right )}{5\,e^8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + c*x^2)^3*(A + B*x))/(d + e*x)^(1/2),x)

[Out]

((d + e*x)^(7/2)*(70*B*c^3*d^4 + 6*B*a^2*c*e^4 - 40*A*c^3*d^3*e + 60*B*a*c^2*d^2*e^2 - 24*A*a*c^2*d*e^3))/(7*e
^8) + ((d + e*x)^(11/2)*(42*B*c^3*d^2 - 12*A*c^3*d*e + 6*B*a*c^2*e^2))/(11*e^8) + (2*(a*e^2 + c*d^2)^2*(d + e*
x)^(3/2)*(B*a*e^2 + 7*B*c*d^2 - 6*A*c*d*e))/(3*e^8) + (2*B*c^3*(d + e*x)^(15/2))/(15*e^8) + (2*c^2*(d + e*x)^(
9/2)*(3*A*a*e^3 - 35*B*c*d^3 - 15*B*a*d*e^2 + 15*A*c*d^2*e))/(9*e^8) + (2*c^3*(A*e - 7*B*d)*(d + e*x)^(13/2))/
(13*e^8) + (2*(a*e^2 + c*d^2)^3*(A*e - B*d)*(d + e*x)^(1/2))/e^8 + (6*c*(a*e^2 + c*d^2)*(d + e*x)^(5/2)*(A*a*e
^3 - 7*B*c*d^3 - 3*B*a*d*e^2 + 5*A*c*d^2*e))/(5*e^8)

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sympy [A]  time = 139.19, size = 1284, normalized size = 3.69

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+a)**3/(e*x+d)**(1/2),x)

[Out]

Piecewise(((-2*A*a**3*d/sqrt(d + e*x) - 2*A*a**3*(-d/sqrt(d + e*x) - sqrt(d + e*x)) - 6*A*a**2*c*d*(d**2/sqrt(
d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e**2 - 6*A*a**2*c*(-d**3/sqrt(d + e*x) - 3*d**2*sqrt(d + e*
x) + d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5)/e**2 - 6*A*a*c**2*d*(d**4/sqrt(d + e*x) + 4*d**3*sqrt(d + e*x) -
2*d**2*(d + e*x)**(3/2) + 4*d*(d + e*x)**(5/2)/5 - (d + e*x)**(7/2)/7)/e**4 - 6*A*a*c**2*(-d**5/sqrt(d + e*x)
- 5*d**4*sqrt(d + e*x) + 10*d**3*(d + e*x)**(3/2)/3 - 2*d**2*(d + e*x)**(5/2) + 5*d*(d + e*x)**(7/2)/7 - (d +
e*x)**(9/2)/9)/e**4 - 2*A*c**3*d*(d**6/sqrt(d + e*x) + 6*d**5*sqrt(d + e*x) - 5*d**4*(d + e*x)**(3/2) + 4*d**
3*(d + e*x)**(5/2) - 15*d**2*(d + e*x)**(7/2)/7 + 2*d*(d + e*x)**(9/2)/3 - (d + e*x)**(11/2)/11)/e**6 - 2*A*c*
*3*(-d**7/sqrt(d + e*x) - 7*d**6*sqrt(d + e*x) + 7*d**5*(d + e*x)**(3/2) - 7*d**4*(d + e*x)**(5/2) + 5*d**3*(d
+ e*x)**(7/2) - 7*d**2*(d + e*x)**(9/2)/3 + 7*d*(d + e*x)**(11/2)/11 - (d + e*x)**(13/2)/13)/e**6 - 2*B*a**3*
d*(-d/sqrt(d + e*x) - sqrt(d + e*x))/e - 2*B*a**3*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3
)/e - 6*B*a**2*c*d*(-d**3/sqrt(d + e*x) - 3*d**2*sqrt(d + e*x) + d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5)/e**3
- 6*B*a**2*c*(d**4/sqrt(d + e*x) + 4*d**3*sqrt(d + e*x) - 2*d**2*(d + e*x)**(3/2) + 4*d*(d + e*x)**(5/2)/5 -
(d + e*x)**(7/2)/7)/e**3 - 6*B*a*c**2*d*(-d**5/sqrt(d + e*x) - 5*d**4*sqrt(d + e*x) + 10*d**3*(d + e*x)**(3/2)
/3 - 2*d**2*(d + e*x)**(5/2) + 5*d*(d + e*x)**(7/2)/7 - (d + e*x)**(9/2)/9)/e**5 - 6*B*a*c**2*(d**6/sqrt(d + e
*x) + 6*d**5*sqrt(d + e*x) - 5*d**4*(d + e*x)**(3/2) + 4*d**3*(d + e*x)**(5/2) - 15*d**2*(d + e*x)**(7/2)/7 +
2*d*(d + e*x)**(9/2)/3 - (d + e*x)**(11/2)/11)/e**5 - 2*B*c**3*d*(-d**7/sqrt(d + e*x) - 7*d**6*sqrt(d + e*x) +
7*d**5*(d + e*x)**(3/2) - 7*d**4*(d + e*x)**(5/2) + 5*d**3*(d + e*x)**(7/2) - 7*d**2*(d + e*x)**(9/2)/3 + 7*d
*(d + e*x)**(11/2)/11 - (d + e*x)**(13/2)/13)/e**7 - 2*B*c**3*(d**8/sqrt(d + e*x) + 8*d**7*sqrt(d + e*x) - 28*
d**6*(d + e*x)**(3/2)/3 + 56*d**5*(d + e*x)**(5/2)/5 - 10*d**4*(d + e*x)**(7/2) + 56*d**3*(d + e*x)**(9/2)/9 -
28*d**2*(d + e*x)**(11/2)/11 + 8*d*(d + e*x)**(13/2)/13 - (d + e*x)**(15/2)/15)/e**7)/e, Ne(e, 0)), ((A*a**3*
x + A*a**2*c*x**3 + 3*A*a*c**2*x**5/5 + A*c**3*x**7/7 + B*a**3*x**2/2 + 3*B*a**2*c*x**4/4 + B*a*c**2*x**6/2 +
B*c**3*x**8/8)/sqrt(d), True))

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